X µŒ ˘˘ µ Œ 6 µ œ ﬁµÙ Fˇµ2013 c 4maths.nju.edu.cn/~yangfei/papers/ph.d thesis.pdf · The study of the topological and geometric properties of the Julia sets of holomorphic

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1.1 Vã . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Ì½n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Julia 8 Cantor ±kn¼ê . . . . . . . . . . . . . . . . . . 2

1.2.2 McMullen N Julia 8[é¡x . . . . . . . . . . . . . . . . . 5

1.2.3 x¼êÄåXÚ . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2.4 ng tableaux ¢y½n . . . . . . . . . . . . . . . . . . . . . . . 9

1.2.5 McMullen NÃ¡áÚ>. Hausdorff ê . . . . . . 9

1.3 ÎÒ(² . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1Ù E©ÛÚEÄåXÚ£ý 11

2.1 EA, [/N, ÿ Riemann N½n . . . . . . . . . . . . . . . . 11

2.2 Riemann N½n, Koebe ½n, . . . . . . . . . . . . . . . 13

2.3 ±Ï:, Fatou 8, Julia 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.4 Ak^Ún . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1nÙ Julia 8 Cantor ±kn¼ê 18

3.1 Úó . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2 .: ±9V/ . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.3 Cantor ±. Julia 8mÿÀÝ . . . . . . . . . . . . . . . . . . . . 30

3.4 Julia 8þÿÀd Cantor ±kn¼êÝa . . . . . . . . . . . . 34

3.5 Cantor ±kn¼ê Julia 8þÿÀÝaêO . . . . . . . . . 36

3.6 Julia 8 Cantor ±Vkn¼ê . . . . . . . . . . . . . . . . . . 40

3.7 õV Cantor ±. Julia 8~f . . . . . . . . . . . . . . . . . 46

3.8 Ún 3.7.2 y² . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

1oÙ McMullen N Julia 8[é¡x 56

4.1 cóÚÄ½Â . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.2 gd.:<º/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

i

4.2.1 <ºn©½nÚëêm©) . . . . . . . . . . . . . . . . . . . . 59

4.2.2 Cantor 8[é¡üz . . . . . . . . . . . . . . . . . . . . . . . 60

4.2.3 Cantor ±[é¡üz . . . . . . . . . . . . . . . . . . . . . 60

4.2.4 Sierpinski /#[é¡üz . . . . . . . . . . . . . . . . . . . . 63

4.3 gd.:<º/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.3.1 nØ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.3.2 ÓN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.3.3 <º¹e Sierpinski /#[é¡üz . . . . . . . . . . . . 66

4.4 Cantor ±[é¡AÛ . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

1ÊÙ x¼êÄåXÚ 71

5.1 cóÚy²g´Vã . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.2 Ä(J . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.3 5z« . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.4 äk û[ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.5 ÃâNEÚ§ëY5 . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.6 ½n 5.1.1 y² . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

18Ù ng tableaux ¢y½n#y² 96

6.1 Ä tableaux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

6.2 äþÄåXÚ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

6.3 ng children Nòÿ . . . . . . . . . . . . . . . . . . . . . . . . . . 100

6.4 lIP:Ðng children N . . . . . . . . . . . . . . . . . . . . 101

6.5 IP:ÚÐng children Nê8 . . . . . . . . . . . . . . . . . 104

1ÔÙ McMullen NÃ¡áÚ>. Hausdorff ê 105

7.1 ½nã . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

7.2 úªí . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

7.3 ON¹ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

ë©z 114

®uLÚ®¤Ø© 119

120

Julia 8ÿÀÚAÛ5ïÄ

¥©Á

X¼ê Julia8ÿÀÚAÛ5ïÄÉÄåXÚ¥¯K.ùÙ

¥Ì) Julia8 (½öÙf8)ëÏ5,ÛÜëÏ5±9K5SN.éuõª

Julia 8ÿÀ5, éõöó. Äk´gõª fc(z) = z2 + c, Ù¥

c áu Mandelbrot 8, Yoccoz y²XJ fc Ø´Ã¡, K fc Julia 8 J(fc)

´ÛÜëÏ [38]. Lyubichy² c¢ê, J(fc)ÛÜëÏ [46]. ëê c¦ fc

äk Siegel, Peterseny²XJ fc 3 Siegelþ^=êk..,K J(fc)

´ÛÜëÏ [57]. XJ^=ê David a., Petersen Ú Zakeri y² Siegel

>.´ Jordan éA Julia 8´ÛÜëÏ [58]. d, Avila, Buff Ú Cheritat

y²3>.1w Siegel ~f [4].

éungõª, Branner Ú Hubbard y²ØêõëÏ©|, Ù Julia 8

ëÏ©|Ñ´ü: [11]. C, ©z [63] òù(Øí2?¿õª. ékn¼ê

5`, XJ§ÁÛk, @oØêõëÏ©|±, Ù Julia 8ëÏ©|o

´ü:, o´^ Jordan [59].

1¦ Julia 8ØëÏ¿Ù¤këÏ©|Ñ´ Jordan kn¼ê~

f´ McMullen é [49]. ¦y²XJ f(z) = z2 + λ/z3 ¥ λ v, K f

Julia8ÓuIOn© Cantor8ü ±¦È.ù«a. Julia8y3¡

Cantor±. , õöÄ¼êx fλ(z) = zm +λ/zl, Ù¥ l,m ≥ 2

λ ∈ C \ 0, y3Ñ¡ McMullen N. N´`² 1/l + 1/m < 1 λ é,

fλ Julia 8´ Cantor ± ([49, §7], [24, §3]).

ùØ©1Ü©ÌïÄ Julia 8 Cantor ±kn¼ê. 8c=3

·ëê¦ McMullen N (½3Ù Fatou 8þ6Äkn¼ê) Julia 8´

Cantor ±. ùr¦·g´Ä3Ù§kn¼ê¦Ù Julia 8 Cantor

±. , HaıssinskyÚ PilgrimÏL[/ÃâEÑa “þ” McMullenN

ØÓ Julia 8 Cantor ±kn¼ê, ¦¿vkÑùkn¼êLª

[35]. Ï g,¯KÒÚÄÑùkn¼êLª.

¯¢þ, 3ùØ©¥, ·ØÑùa.kn¼êäNLª,

3 “þ” é¤k Julia 8 Cantor ±kn¼ê. ùp¤¢ “þ” ´

3Ä Julia þÿÀÝa¿Âe. äN/`, ·éxkn¼êL

ª (McMullen N´ùx¼êAÏ/), ¦·ëê, § Julia 8Ñ´

Cantor ±. ,¡, éu?¿½ Julia 8 Cantor ±kn¼ê, ·o

±3éùxkn¼ê¥é¼ê¯k½¼ê¦§3éA Julia

8þ´ÿÀÝ (½n 3.1.1 Ú 3.1.2).

3dóÄ:þ, ·é Julia 8 Cantor ±kn¼ê3 Julia 8þÿ

ÀÝaê8ÑOúªÚþe.O, ¿é 5 ≤ d ≤ 36 /ÑäNê

L, Ù¥ d kn¼êNÝ.

Vkn¼êäküÄåXÚ5, ·éXVkn¼ê¦

§ Julia 8´ Cantor ±. â·¤, ù´1 Julia 8 Cantor ±V

kn¼ê~f. ÙÌEg´ò5Vkn¼êáÚ^üëÏÔ

O.

Ø©1Ü©ÌïÄ McMullen N Julia 8AÛ5. â<ºn©½

n, XJ McMullen Ngd.:ÑÃ¡¤áÚ, @o§ Julia 8U´

Cantor 8, Cantor ±½ö´ Sierpinski /# [24]. ·y²3ù«¹

e, McMullen N Julia 8o[é¡duIOn© Cantor 8, IO

Cantor ±½ö´ Sierpinski /# (3,«¿ÂeÍO).

éu McMullen N fλ ¥ëê λ áu¢ê/, ©z [62, 76] ¥Ñ fλ

Julia8 Jλ ´ Sierpinski/#¿^.3dÄ:þ,ö3©¥Ñ Jλ [é

¡du Sierpinski /#¿^. AO/, 3Vkn¼ê, ¦Ù Julia

8[é¡du Sierpinski/#. d, éuëê λáuEê/, ·Ñ Jλ

[é¡du Sierpinski /#¿©^ (d¦ fλ Lª¥üê

÷v l = m ≥ 3).

lÿÀÝ5w, ¤k Cantor ±Ñ´, ù´Ï§ÑÿÀd (Ó

) u “IO” Cantor ± C × S1, Ù¥ C n© Cantor 8 S1 ü ±. Ï

d, ¤k Cantor ±¤8Ü´L(, ·±^[é¡AÛÝ

5w. ¯¢þ, [é¡AÛp¡Ä¯KÒ´äü½ÓÝþm´

Ä´*d[é¡d. Ýþm/ê½Â¤k[é¡dÝþ

m Hausdorff êe(.. âéØ©1Ü©éÑ Julia 8 Cantor ±

kn¼êx|Ü©Û, ·`²3ùx¼ê¥3~f, ¦§ Julia 8

McMullen Julia 8´ÓØ´[é¡d. AO/, ù~f±^5äN/

y3 Julia 8 Cantor ±Vkn¼ê¦Ù/ê?¿Cu 2.

Ø©1nÜ©ïÄx¼ê Siegel >.K5. X¼ê Siegel

>. (´ Julia 8f8) ÿÀÚAÛ5´~¯K. é^=ê

\þ½â^, Douady ß Siegel >.½´ Jordan . 3ù

¯Kþ, Douady, Zakeri, Shishikura Ú Zhang éukn¼ê/Ñz (

[26, 77, 66, 80]). éu¼ê/ù¯Kvk)û ( [32, 41, 78, 79]).

öÄ¼êx fα(z) = e2πiθ sin(z) + α sin3(z) : α ∈ C \ 0, ¿y² θ k

.., ¼ê fα ±:% Siegel >.o´[±²L 2 , 4 , ½ 6

.:.

Ø©1oÜ©Ä. tableau ¢y¯K. .IP tableau ´ Branner Ú

Hubbard 3ïÄngõªÚ\ [11]. ù´~k^óä, §3ïÄ Julia 8

ÿÀ5¥åX©^. Branner Ú Hubbard y²: XJÄ.I

P tableau ÷vK, @où tableau ½±dngõª¢y. ·^d

õªpûäþÄåXÚÚäÄåXÚ¢y½n ( [30, 18]) Ñ Branner Ú

Hubbard tableau ¢y½n#y².

d, 3Ø©Ü©öïÄ McMullen NÃ¡áÚ>

.5, ¿ÑÙ Hausdorff êìCLª.

'c: Julia 8, Fatou 8, Cantor ±, Sierpinski /#, McMullen N, [é¡d,

Hausdorff ê, Siegel , aõªä, tableau

2000 MR ÌK©a: 37F45, 37F20, 37F10, 37F25, 32G05

¥ã©aÒ: O174.5

The Study of the Topological and GeometricProperties of Julia Sets

ABSTRACT

The study of the topological and geometric properties of the Julia sets of holomorphic

maps is an important problem in complex dynamics. It mainly includes the connected-

ness, locally connectivity and the regularity of Julia sets (or their subsets), etc. For the

topological properties of the Julia sets of polynomials, many authors contributed on this.

For the quadratic polynomials fc(z) = z2 + c, where c belongs to the Mandelbrot set,

Yoccoz proved that if fc is not infinitely renormalizable, then the Julia set J(fc) of fc is

locally connected [38]. Lyubich proved that if c is real, then J(fc) is locally connected

[46]. For the case when fc has a Siegel disk, Petersen proved that if the rotation number

on the Siegel disk is of bounded type, then J(fc) is also locally connected [57]. If the

rotation number is of David type, then Petersen and Zakeri proved that the boundary of

the Siegel disk is a Jordan curve and the Julia set is locally connected [58]. Moreover,

Avila, Buff and Cheritat proved that there exists Siegel disk whose boundary is a smooth

curve [4].

For cubic polynomials, Branner and Hubbard proved that all but countably many

components of the Julia set are single points. Recently, this result has been extended to

all polynomials in [63]. For the rational maps, if it is geometrically finite, then, with the

possible exception of countable components, every Julia component is either a point or a

Jordan curve [59].

The first example of rational map with disconnected Julia set whose components are

all Jordan curves was discovered by McMullen [49]. He showed that if f(z) = z2 + λ/z3

and λ is small enough, then the Julia set of f is homeomorphic to the product of the

middle third Cantor set and the unit circle. These types of Julia sets are called Cantor

circles. Later, many authors focus on the family, which is commonly referred as the

McMullen maps: fλ(z) = zm + λ/zl, where l,m ≥ 2 and λ ∈ C \ 0. It can be shown

that when 1/l + 1/m < 1 and λ is small enough, the Julia set of fλ is a Cantor set of

circles (see [49, §7] and [24, §3]).

The first part of this thesis mainly focuses on the rational maps whose Julia sets

are Cantor circles. Up to now, it is known that McMullen maps (or the rational maps

after perturbing on their Fatou sets) are the only examples such the Julia sets of rational

maps are Cantor circles. This motivates us to think about the question whether there

exist other rational maps whose Julia sets are Cantor circles. Although Haıssinsky and

Pilgrim have constructed a class of rational maps whose Julia sets are Cantor circle which

is different from McMullen maps “essentially” by quasiconformal surgery. However, they

did not give the specific expressions of these maps [35]. Therefore, a natural question is

whether one can give the specific expressions of these maps.

In fact, in this thesis, we not only give the specific expressions of these types of

rational maps, but also find out all the rational maps whose Julia sets are Cantor circles

“essentially”. The word “essentially” means we consider this problem in the sense of the

topological conjugacy classes on the Julia sets. Specifically, we find the specific expressions

of a family of rational maps (McMullen maps are the special cases) such that their Julia

sets are Cantor circles after choosing appropriate parameters. On the other hand, for

each given rational map whose Julia set is a Cantor set of circles, we can always find a

map in the family which we have found such they are topologically conjugate on their

corresponding Julia sets (Theorems 3.1.1 and 3.1.2).

Based on this, we calculate and give a lower and upper bound of the number of

different topological conjugacy classes on the Julia sets of rational maps whose Julia sets

are Cantor circles. Moreover, we give the specific numbers of the different topological

conjugacy classes for 5 ≤ d ≤ 36, where d is the degree of the rational maps.

Hyperbolic rational maps possess simple dynamical properties. We find out a series

of non-hyperbolic rational maps whose Julia sets are Cantor circles. As far as we known,

this is the first example of non-hyperbolic rational maps whose Julia sets are Cantor

circles. The idea behind the construction is the attracting basin of the original rational

maps has been replaced by a simply connected parabolic basin.

The second part of this thesis mainly focuses on the study of the geometric properties

of the Julia sets of McMullen maps. According to the Escape Trichotomy Theorem, if all

the critical points of a McMullen map are attracted by ∞, then the corresponding Julia

set is either a Cantor set, a Cantor set of circles or a Sierpinski carpet [24]. We prove

that in this case, the Julia set of a McMullen is quasisymmetrically equivalent either to a

standard Cantor set, a standard Cantor set of circles or a round Sierpinski carpet (which

is also standard in some sense).

For the case when the parameter of the McMulle family is real, a sufficient and

necessary condition such the Julia set Jλ of McMullen map fλ is a Sierpinski carpet was

gave in [62, 76]. Based on this, we give a sufficient and necessary condition to guarantee

that Jλ is quasisymmetrically equivalent to a round Sierpinski carpet. In particular, there

exists non-hyperbolic rational map whose Julia set is quasisymmetrically equivalent to

a round Sierpinski carpet. Moreover, for case when λ is complex, we give a sufficient

condition to guarantee that Jλ is quasisymmetrically equivalent to a round Sierpinski

carpet (but we need to set l = m ≥ 3, where l,m is the integers in fλ).

From the topological point of view, all Cantor sets of circles are the same since they

are all topologically equivalent (homeomorphic) to the “stand” Cantor set of circles C×S1,

where C is the middle third Cantor set and S1 is the unit circle. Therefore, to obtain much

richer structure of all Cantor sets of circles, we can look at the Cantor circles equipped

with metric from the point of view of quasisymmetric geometry. In fact, a basic problem

in the quasisymmetric geometry is to determine whether two given homeomorphic metric

spaces are quasisymmetrically equivalent to each other. The conformal dimension of a

metric space is the infimum of the Hausdorff dimensions of all metric spaces which are

quasisymmetrically equivalent to the original metric space. According to the combina-

torial analysis on the rational maps which we have found in the first part of this thesis,

we show that this family gives a series of specific examples, such the Julia sets of them

are homeomorphic, but not quasisymmetrically equivalent to the Julia sets of McMullen

maps. In particular, these examples can be used to verify that there exist hyperbolic

rational maps whose Julia sets are Cantor circles and whose conformal dimensions are

arbitrarily close to 2.

The third part of this thesis studies the regularity of the boundaries of a family of

entire functions. The study of the topological and geometric properties of the boundaries

of Siegel disks (subsets of Julia sets) is an important problem. After assuming the rotation

number satisfies some arithmetical condition, Douady conjectured that the boundary of

the Siegel disk must be a Jordan curve. For this problem, Douady, Zaker, Shishikura and

Zhang contributed to the rational case (see [26, 77, 66, 80]). But this problem is far from

being solved for transcendental case (see [32, 41, 78, 79]). We consider the one-dimensional

family fα(z) = e2πiθ sin(z) + α sin3(z) : α ∈ C \ 0, and prove that the boundary of the

Siegel disk centered at the origin of fα is a quasicircle which passes through 2, 4 or 6

critical points of fα counted with multiplicity if θ is of bounded type.

The fourth part of this thesis considers the realize problem of the critical tableau. As

a powerful tool, the critical tableau was introduce by Branner and Hubbard when they

studied the cubic polynomials [11]. It plays a central role in the study of locally connec-

tivity of the Julia sets. Branner and Hubbard proved that: If an abstract critical marked

tableau satisfies some rules, then this tableau must be realized by a cubic polynomial. We

give a new proof of the realization theorem of tableaux by using tree dynamics induced

by polynomials and the realization theorem of tree dynamics (see [30, 18]).

In the last part of this thesis, we study the boundary behavior of the immediate basin

of infinity of McMullen maps, and give an asymptotic formula of its Hausdorff dimension.

Keywords: Julia set, Fatou set, Cantor circles, Sierpinski carpet, McMullen map, qua-

sisymmetrically equivalent, Hausdorff dimension, Siegel disk, polynomial-like tree, tableau

2000 MR Subject Classification: 37F45, 37F20, 37F10, 37F25, 32G05

Chinese Library Classification: O174.5

1Ù XØ

§1.1 Vã

ùÆ¬Ø©Ìde¡ASN|¤:

1. Julia 8 Cantor ±kn¼êÄåXÚ.

éuùÜ©,·éÑxkn¼ê fp,d1,··· ,dn ,¦Ù¥z3À·

ëê Julia 8Ñ´ Cantor ±, Ù¥ p ∈ 0, 1, n ≥ 2 d1, · · · , dn n ¦∑ni=1

1di< 1 ê. ·òy²XJ n ≥ 3, Kz fp,d1,··· ,dn 3 Julia 8þØ´ÿ

ÀÝu McMullen N. ùL²·éakn¼êäNLª¦§

Julia 8´ Cantor ±, § McMullen N3þ´ØÓ.

éu?¿½ Julia 8 Cantor ±kn¼ê, ·o±3éùxk

n¼ê¥é fp,d1,··· ,dn ¯k½¼ê3éA Julia 8þ´ÿÀÝ. ùL

²·3þé “¤k” Julia 8 Cantor ±kn¼ê. dd, ·é Julia

8 Cantor ±kn¼ê3 Julia 8þÿÀÝaê8ÑOúªÚ

þe.O, ¿é 5 ≤ d ≤ 36 /ÑäNêL, Ù¥ d kn¼êNÝ.

Vkn¼êÄåXÚ5'ü, ·éXVkn¼ê¦

§ Julia 8´ Cantor ±. â·¤, ù´1 Julia 8 Cantor ±

Vkn¼ê~f. ÙÌEg´ò5Vkn¼êáÚ^üëÏ

Ô5O.

2. McMullen N Julia 8[é¡x.

ùÜ©Ì) McMullenN fλ(z) = zm +λ/zl Julia8[é¡AÛx,

Ù¥ l,m ≥ 2 λ ∈ C \ 0 . ·y²XJ fλ gd.:ÑÃ¡¤áÚ, @o

§ Julia 8o[é¡duIOn© Cantor 8, IO Cantor ±½

öÍO Sierpinski /#.

éu fλ ¥ëê λ áu¢ê/, ·3©¥Ñ Jλ [é¡duIO

Sierpinski /#¿^. éuëê λ Eê l = m ≥ 3 /, ·Ñ Jλ [

é¡du Sierpinski /#¿©^.

du?¿ Cantor ±ÑÓuIOn© Cantor 8ü ±¦È. ·

Ä Cantor ±[é¡[é¡d¯K. âØ©1Ü©é¼êx, ·Ñ

þVkn¼ê Julia ÓØ[é¡d~f. d, âé·éÑ Julia 8

Cantor±kn¼êx|Ü©Û,·`²ù¼êxTÐÑäN~

f5y3 Julia 8 Cantor ±Vkn¼ê¦Ù/ê?¿Cu 2.

3. x¼êÄåXÚ.

1

2 1Ù XØ

3ùÜ©¥, ·ïÄx¼ê Siegel >.K5. ·Ä¼ê

x fα(z) = e2πiθ sin(z) + α sin3(z) : α ∈ C \ 0, ¿y² θ k.., ¼ê fα ±

:% Siegel >.o´[±²L 2 , 4 , ½ 6 .:.

4. ng tableaux ¢y#y².

3ùÜ©¥, ·Ä. tableau ¢y¯K. Branner Ú Hubbard y²: XJ

Ä.IP tableau÷v,K,@où tableau½±dngõ

ª¢y. ·^dõªpûäþÄåXÚÚäÄåXÚ¢y½nÑ Branner

Ú Hubbard tableau ¢y½n#y².

5. McMullen NÃ¡áÚ>. Hausdorff êìCúª.

Ø©Ü©,·Ñëê λªu 0, McMullenN fλ(z) = zn+λ/zn

Ã¡áÚ>. Hausdorff êìCúª, Ù¥ n ≥ 3.

§1.2 Ì½n

§1.2.1 Julia 8 Cantor ±kn¼ê

XJ Riemann ¥¡ C f8Óu C × S1, Ù¥ C Cantor n©8 S1

ü ±, KTf8¡´ Cantor ±. éu McMullen N:

fλ(z) = zm + λ/zl, (1.2.1)

®² 1/l + 1/m < 1 λ é, ¼ê fλ Julia 8´ Cantor ± (l =

3,m = 2 ¹ [49, §7] ¹ [24, §3]).

y3kXeng,¯K:

(1) Ø McMullen N, ´Ä3Ù§kn¼êÙ Julia 8 Cantor ±?

(2) XJ1¯KY´½, @§ó? ó, ·UÄéÑ§

Lª?

(3) ·UÄ3½§ÝþéÑ¤k Julia 8 Cantor ±kn¼ê?

e¡½n 1.2.1 Ú 1.2.2 Ñþ¡ù¯K½£. ·ò31nÙÑù

¯K[?Ø. - p ∈ 0, 1, n ≥ 2, d1, · · · , dn n ¦∑n

i=11di< 1

ê. ½Â

fp,d1,··· ,dn(z) = z(−1)n−pd1

n−1∏i=1

(zdi+di+1 − adi+di+1

i )(−1)n−i−p , (1.2.2)

Ù¥ a1, · · · , an−1 n− 1 ÷v 0 < |a1| < · · · < |an−1| < 1 Eê.

§1.2. Ì½n 3

½n 1.2.1. é? p ∈ 0, 1, n ≥ 2, ±9¦∑n

i=11di< 1 n ê

d1, · · · , dn, Ñ3·ëê ai, ¦ fp,d1,··· ,dn Julia 8´ Cantor ±, Ù¥ 1 ≤i ≤ n− 1.

3 §3.2 ¥, ·òÑëê ai äN. d, ·òy²XJ n ≥ 3, K?

Û fp,d1,··· ,dn ÚMcMullenN3§éA Julia8þÑØ´ÿÀÝ (½n 3.2.6).

ùL²·éakn¼êäNLª¦§ Julia 8´ Cantor ±, §

McMullen N3þ´ØÓ.

~X, - p = 1, n = 4, d1 = d2 = d3 = d4 = 5 ½Â

f1,5,5,5,5(z) =(z10 − a10

1 )(z10 − a103 )

z5(z10 − a102 )

, (1.2.3)

Ù¥ a1 = 0.00025, a2 = 0.005 a3 = 0.1. ÏLO½ö|^½n 3.1.1,

±`² f1,5,5,5,5 Julia 8´ Cantor ±. Ï f1,5,5,5,5 3 Julia 8þÄåXÚ

Ýu 4ÎÒm Σn := 0, 1, 2, 3N ü>²£, fλ 3 Julia8þÄåXÚÝu

2 ÎÒm Σ2 := 0, 1N ü>²£. ùL² f1,5,5,5,5 Ú fλ 3§éA Julia þØ

´ÿÀÝ.

ã 1.1 ¼ê f1,5,5,5,5 Julia 8 (ã), fλ(z) = z3 + 0.001/z3 Julia 8

(mã) Ø´ÿÀÝ. ü Julia 8Ñ´ Cantor ±.

½n 1.2.2. f kn¼ê, Ù Julia 8 Cantor ±. K3 p ∈ 0, 1, ê n ≥ 2 ±9÷v

∑ni=1

1di< 1 d1, · · · , dn Ú·ëê ai ¦ f Ú fp,d1,··· ,dn 3

§éA Julia 8þ´ÿÀÝ, Ù¥ 1 ≤ i ≤ n− 1.

4 1Ù XØ

d½n 1.2.2, ·±é Julia 8 Cantor ±kn¼ê3 Julia 8þÿÀ

Ýaê8ÑOúª.

½n 1.2.3. NÝ d ≥ 5 Cantor ±kn¼ê Julia 8þØÓÿÀ

Ýaê

N(d) =∑n≥2

](d1, · · · , dn) :n∑i=1

di = d n∑i=1

1

di< 1

+∑n≥3

](d1, · · · , dn) :n∑i=1

di = d,

n∑i=1

1

di< 1, (d1, · · · , dn) = (dn, · · · , d1) n Ûê .

·3L 3.1 ¥Ñ 5 ≤ d ≤ 36 N(d) äNê. d, ·kXeO.


Ýaê N(d) ÷v

n0∑n=1

(d− n2 − 3n− 1)n

n!≤ N(d) < 2

n1∑n=1

C nd−n−2, (1.2.4)

Ù¥ n0 = [(√

4d+ 5− 3)/2] n1 = [√d− 1]− 1.

XJkn¼ê.:Ñá5±Ï;¤áÚ, KTkn¼ê¡´V.

éu fp,d1,··· ,dn Julia 8©|K5, ±y²XJ fp,d1,··· ,dn ´V, K fp,d1,··· ,dn

z Julia 8ëÏ©|Ñ´[ (íØ 3.3.3).

XJ λ v, K fλ ´V ( [24]). ·±EVkn¼ê¦

§ Julia 8´ Cantor ±. - m,n ≥ 2 ü÷v 1/m + 1/n < 1 ê

λ ∈ C \ 0, ·½Â

Pλ(z) =1n((1 + z)n − 1) + λm+nzm+n

1− λm+nzm+n. (1.2.5)

y 0 ´ Pλ ¦f 1 ÔØÄ:. ·k

½n 1.2.5. XJ 0 < |λ| ≤ 1/(210mn3), K Pλ ´VÙ Julia 8 Cantor

±.

?Ú,·±EõVkn¼ê¦§ Julia8´ Cantor±.

éz n ≥ 2 ±9 1 ≤ i ≤ n, - di = n+ 1. ·½Â

Pn(z) = An(n+ 1)z(−1)n+1(n+1)

nzn+1 + 1

n−1∏i=1

(z2n+2 − b2n+2i )(−1)i−1

+Bn, (1.2.6)

§1.2. Ì½n 5

Ù¥ b1, · · · , bn−1 ÷v 1 > |b1| > · · · > |bn−1| > 0 n− 1 Eê

An =1

1 + (2n+ 2)Cn

n−1∏i=1

(1−b2n+2i )(−1)i , Bn =

(2n+ 2)Cn1 + (2n+ 2)Cn

Cn =n−1∑i=1

(−1)i−1b2n+2i

1− b2n+2i

.

ùp An Ú Bn y Pn(1) = 1 P ′n(1) = 1. =, 1 Pn ¦f 1 Ô

ØÄ: (Ún 3.7.1).

½n 1.2.6. éz n ≥ 2 Ú 1 ≤ i ≤ n− 1, XJ |bi| = si, Ù¥ 0 < s ≤ 1/(25n2),

K Pn ´VÙ Julia 8 Cantor ±.

§1.2.2 McMullen N Julia 8[é¡x

duMcMullenN Julia8ÿÀ5®²þ/ïÄ,·éMcMullen

N Julia 8?1AÛþx.

XJ®²üÝþm´Ó, ·'%ùüm´Ä´*d[é¡d

. Ýþm (X, dX) Ú (Y, dY ) ¡´*d[é¡d´3üÓ f : X → Y

Ú η : [0,∞)→ [0,∞) ¦éuzØÓ: x, y, z ∈ X k dY (f(x),f(y))dY (f(x),f(z))

≤ η(dX(x,y)dX(x,z)

).

- l,m ≥ 2 ü÷v 1/l + 1/m < 1 ê, IO Cantor 8 Cl,m ½ÂS

¼êXÚ g0, g1 áÚf, Ù¥ g0(x) = (1− x)/l g1(x) = 1 + (x− 1)/m þ½Â

3ü «m [0, 1]þ. AO/, C3,3 IOn© Cantor8. IO Cantor± Al,m ½

Â Cl,m × S1, Ù¥ S1 := z ∈ C : |z| = 1 ´ü ±.

éu McMullen N fλ Julia 8 Jλ ÿÀ5, Devaney, Look Ú Uminsky y

²<ºn©½n. ù½n`XJ fλ ¤k.:Ñ ∞ áÚ, K Jλ o´

Cantor 8, Cantor ±½ö´ Sierpinski /# ( [24, ½n 0.1] ½ö½n

4.2.1). éu Jλ [é¡AÛ5, ·ke¡½n.

½n 1.2.7. b fλ ¤k.:Ñ ∞ áÚ. Ke¡nö¤á:

(1) Jλ ´ Cantor 8, K Jλ [é¡duIOn© Cantor 8 C3,3;

(2) Jλ ´ Cantor ±, K Jλ [é¡duIO Cantor ± Al,m, Ù¥ l

Ú m (1.2.1) ¥ê;

(3) Jλ ´ëÏ, K Jλ [é¡du Sierpinski /#.

ùp Sierpinski/#´ Sierpinski/#z©|>.Ñ´î¼ (½ö

¥¡) ±.

Ø 0Ú∞, N fλ e l+m.:¡´gd.:. ùgd.:

þkÓ; ( §4.2). §oÓáÚ ∞ o;Ók.. XJgd

.:vk ∞ áÚ, K Jλ ´ëÏ ( [25]). 3ù«¹e, üg,¯KÑy:

6 1Ù XØ

(1) Jλ oÿ´ Sierpinski /#?

(2) §´Ä[é¡du/#?

éþ¡¯K, ©z [62] ¥é λ > 0 /Ä. P Kλ = C \ A(∞) fλ

“W¿” Julia 8, Ù¥ A(∞) ∞ áÚ. 31oÙ¥, ·òy²

½n 1.2.8. 3ü¢ê λ0 < λ1 ¦XJ λ ∈ [λ0, λ1], K Jλ [é¡du

/#=e¡÷v:

(I) Kλ SÜ Int(Kλ) = ∅ λ 6= λ0;

(II) λ ∈ H, Ù¥ H ´ M 3 Λ ¥Ó primitive V©|¿ HØ´ H♥ 3ÓNe.

AO/, XJ λ ∈ (λ0, λ1) ¦ fλ Ù¥.:´îý±Ï, K Jλ [é

¡du/#.

ÓNÚÓ½Â §4.3.2. á=íØ, ·k

íØ 1.2.9. 3Vkn¼ê, Ù Julia 8[é¡du/#.

¯¢þ, â½n 1.2.8 y², ·$3Ã¡kn¼ê, Ù Julia

8[é¡du Sierpinski /#.

·òÄ/¦ Jλ ´ Sierpinski /#. AO/, ëê λ ±3E

²¡þCz Ø==´3¢¶þ. , , ù¦·b l = m ≥ 3.

½n 1.2.10. b l = m ≥ 3 fλ gd.;´k.. XJ λ ÷ve¡

^, K Jλ ´ Sierpinski /#.

(a) SÜ Int(Kλ) = ∅ gd.:;4 ∞ áÚ>.Ø;

(b) λ ∈ H ∪ rH, Ù¥ H ´ M 3 Λ ¥Ó primitive V©|¿

H Ø´ H♥ 3ÓNe.

d, XJ λ ÷v (a) ½ö (b) λ 6= rH, K Jλ [é¡du/#.

éz n ≥ 2, - Jp,d1,··· ,dn §1.2.1 ¥½Â fp,d1,··· ,dn Julia 8. 3e5ã

½n¥, ·o´b½ ai ®²ÀÐ¦ Jp,d1,··· ,dn ´ Cantor ±, Ï·=

éù«¹a,. Ó, ·b½ λ v¦ (1.2.1) ¥½Â McMullen N fλ

Julia 8 Jl,m Cantor ±, Ù¥ 1/l + 1/m < 1. XJéz 1 ≤ i ≤ n k

di = n+ 1, ·^ Jn 5L« fp,n+1,··· ,n+1 Julia 8. Ýþm X /êu

¤k X [é¡dÝþm Hausdorff êe(..

§1.2. Ì½n 7

½n 1.2.11. Jp,d1,··· ,dn /êu 1 + αd1,··· ,dn, Ù¥ αd1,··· ,dn e¡§

:n∑i=1

d−αd1,··· ,dni = 1.

AO/, XJéz 1 ≤ i ≤ nk di = n+1, K αn := αd1,··· ,dn = log(n)/ log(n+1). XJ

k 6= n, K αk 6= αn. XJ n ≥ 3, Kéz¦ 1/l + 1/m < 1 l,m ≥ 2 k αn 6= αl,m.

â½n 1.2.1 y², ·¤k fp,d1,··· ,dn Ñ´V. 5¿/ê´

Ýþm[é¡ØCþ ( [47]) ?¿Vkn¼ê Julia 8 Hausdorff

êîu 2 ( [70, ½n 4 ÚíØ]). Ïd, ½n 1.2.11 á=ke¡üíØ.

íØ 1.2.12. é?¿ k, n ≥ 2, Julia 8 Jk Ú Jn *d[é¡d= k = n.

d, XJ n ≥ 3, K Jn Ø[é¡du?Û Jl,m, Ù¥ 1/l + 1/m < 1.

íØ 1.2.13. Jn Hausdorff ê Hdim(Jn) ÷v

1 + log(n)/ log(n+ 1) ≤ Hdim(Jn) < 2.

§1.2.3 x¼êÄåXÚ

- f 3:äkÃn¥5ØÄ:5X¼ê. =, f(0) = 0

f ′(0) = e2πiθ, Ù¥ 0 < θ < 1 ´Ãnê. XJ3X¼êò f Ýf5^

= Rθ : z 7→ e2πiθz, K¡ f 3ØÄ: 0 ?´ÛÜ5z. ¼ê f Ýu Rθ

«´üëÏ« ∆f , ¡´ f ±:% Siegel . Ï f 3 Siegel

∆f p¡ÄåXÚ´ü. éõÿ, ·'%>. ∂∆f AÛÚÿÀ5. ~X,

§´Ä´^ Jordan ? ½ö?Ú, §´Ä´[?

XJ^=ê θ ´k.., Douady, Zakeri Ú Shishikura, ©O/y²gõª,

ngõª±9¤kNÝØu 2 õªk.. Siegel >.´[ [26,

77, 66]. éõöé¼ê/Äù¯K. Geyer, Keen Ú Zhang ©O

Ä z 7→ e2πiθzez Ú z 7→ (e2πiθz + αz2)ez [32, 41]. Ó, Cheritat Ä“

ü”¼ê [14].

C, Zakeri Ú Zhang ©O(J. éu5¼ê f(z) =

P (z) exp(Q(z)), Ù¥ f(0) = 0 f ′(0) = e2πiθ, P Ú Q Ñ´õª, Zakeri y²

XJ 0 < θ < 1 ´k..Ãnê, @o f ±:% Siegel >.´

[¹ f .: [78]. 3 [80] ¥, Zhang y²ù(Øé¤kNÝ

Ø$u 2 kn¼ê¤á.

8 1Ù XØ

31ÊÙ¥, ·ò?Ø±Ï¼êx. - 0 < θ < 1 k..Ãn

ê. P

F = fα(z) = e2πiθ sin(z) + α sin3(z) : α ∈ C \ 0. (1.2.7)

éz fα ∈ F , ·y²e¡

½n 1.2.14. ¼ê fα ±:% Siegel >.´[, 3Pê

¿Âe²L fα 2 , 4 ½ö 6 .:.

½ÂBf

Λ = c : |Re(c)| ≤ π/2 \ 0. (1.2.8)

e¡½n 1.2.15 Ñaqu Zakeri ng Siegel õª [77] ±9 Keen-Zhang

¼êx [41] ëêmaqy© (ã 1.2).

ã 1.2 α–²¡ÛÜã, Ù¥ θ = (√

5− 1)/2 ^=ê. ¥mçÚÜ©L«

Γint, ÚÜ©K¹3 Γext ¥. >. Γ ´^ßü4. ã¡

: [−0.20, 0.35]× [−0.25, 0.30].

½n 1.2.15. - F = fα : α ∈ C \ 0 (1.2.7) ¥½Â¼êx. K3ü

H : C \ 0 → Λ Ú^²L −λ/3 ©l 0 Ú ∞ ü4 Γ ⊂ C \ 0 ¦(1) XJ α ∈ Γint, K ∂∆fα TÐ²L 2 .: π/2 Ú −π/2.

(2) XJ α ∈ Γext, K ∂∆fα TÐ²L 2 .: H(α) Ú −H(α).

(3) XJ α ∈ Γ \ −λ/3, K ∂∆fα TÐ²L 4 .: ±π/2 Ú ±H(α).

(4) XJ α = −λ/3, K ∂∆fα TÐ²L 2 ê 3 .: π/2 Ú −π/2.

§1.3. ÎÒ(² 9

§1.2.4 ng tableaux ¢y½n

zäkØëÏ Julia 8ngõª f Ñ±½Â N ng tableau

½IP: (marked grid), §´÷v,^ M(j, k) ∈ 0, 1 | j, k ≥0 j + k ≤ N. Branner Ú Hubbard [11] y²ngõgIP:7L÷v

K. Ó, ±½Â N ÄIP: M = M(j, k) ∈ 0, 1 | j, k ≥0 j + k ≤ N [11, 19].

XJngõª f tableau ÄIP: M Ó, K¡ M ù

ngõª f ¤¢y. Branner Ú Hubbard y²e¡½n.

½n 1.2.16 (Tableaux ¢y, [11]). ?ÛIP:Ñ±dngõª¤¢

y.

ÄäkØëÏ Julia 8õª f 3Ã¡áÚ Ω(f) þ. XJPò f

3 Ω(f) þ³ëÏ©|l¤:ûm T , K f : Ω(f)→ Ω(f) p

üXä T þÄåXÚ F : T → T ( [30, 18]). d F : T → T ÄåXÚ

±Ä/½Âaõªä (½Â 6.2.1).

318Ù¥, ·ò^äþÄåXÚÑ½n 1.2.16 #y². ÙÌg

´: k`²?ÛIP:±dÓ “Ð” 3 g children N¢y,

ù 3 g children N±òÿNÝu 3 aõªä. duz 3 g

aõªäÑ±d 3 gõª¤¢y (½n 6.2.2), l ½n 1.2.16 y.

§1.2.5 McMullen NÃ¡áÚ>. Hausdorff ê

·3Ø©ÙÑëêªu 0 , McMullen NÃ¡áÚ

>. Hausdorff êìCúª.

½n 1.2.17. - n ≥ 3, bëê λ ¿©¦ McMullen N fλ(z) = zn + λ/zn

Julia 8´ Cantor ±. K fλ Ã¡áÚ>. ∂Bλ Hausdorff

ê´

dimH(∂Bλ) = 1 +|λ|2

log n+O(|λ|3). (1.2.9)

AO/, XJ n 6= 4, Kp O(|λ|3) ±d O(|λ|4) O.

§1.3 ÎÒ(²

·Ñ©ÎÒ(². P

• E²¡ C;

10 1Ù XØ

• Riemann ¥¡ (¡*¿E²¡) C = C ∪ ∞;• þE²¡ H := z ∈ C : Im(z) > 0;• % u a ∈ C » r > 0 î¼ D(a, r) := z ∈ C : |z − a| < r;• % u 0 » r Dr := D(0, r);

• % u 0 » r ± Tr;• ü D := D1;

• ü ± S1 := ∂D;

• A(r1, r2) z ∈ C : r1 < |z| < r2;• R ¢, Z ê8, N∗ ê8.

éu C ¥?Ûüm8 A Ú B, ·^ A ⊂⊂ B L« A ⊂ B. X, Y C (½

C) þ;f8 x ∈ X, ½Â x Y ¥¡ (½î¼) ål d(x, Y ) = infy∈Y |x− y|.8Ü X Ú Y m¥¡ (½î¼) ål d(X, Y ) = infx∈X,y∈Y |x− y|, Ù¥ | · | L«¥¡ (½î¼) Ýþ. 8Ü X ¥¡ (½î¼) »½Â diam(X) = supx1,x2∈X |x1 − x2|.©¥ÑyëêmÚÜ© Julia 8ã´æ^ A. Cheritat JøÑ

. 'u Julia 8ãëw©z [12, 55] Ú [53, N¹ H].

1Ù E©ÛÚEÄåXÚ£ý

ùÙ·OE©ÛÚEÄåXÚ¥ÄVgÚ½n. Ì)[/

Nüd½Â, ÿ Riemann N½n, Koebe ½n, Fatou 8, Julia 8½

Â, kn¼êÄåXÚ©a.

§2.1 EA, [/N, ÿ Riemann N½n

E¼ê f(z) ´« Ω ⊂ C þëY¼ê, = f ¢Ü u ÚJÜ v

¢Cþ x, y ¼êþäkëY ê. d du = uxdx + uydy, dv = vxdx + vydy,

f äkE©/ª: df = du + idv = (ux + ivx)dx + (uy + ivy)dy. P dz =

dx+ idy, dz = dx− idy, d dx = (dz + dz)/2, dy = (dz − dz)/(2i), ·±ò df U

df = fzdz + fzdz, Ù¥

fz = (ux + vy)/2 + i(vx − uy)/2,

fz = (ux − vy)/2 + i(uy + vx)/2.(2.1.1)

½Â 2.1.1. ·¡ (2.1.1) ª¥ fz 9 fz f ü/ª ê.

½Â 2.1.2. f ´« Ω ⊂ C SëY~¼ê z0 ∈ Ω, e

fz(z0) 6= 0, ½Â f 3 z0 EA (complex dilatation ) µf (z0) = fz(z0)/fz(z0).

âEA½Â, e f /N, K µf = 0.

·Äk0[/NAÛ½Â, dk7kÚ?/£. \

[SNëw©z [44] 1nÙ.

P Q C ¥ Jordan « (>.ü4), 3Ù>.þUgÀ½

ØÓo: z1, z2, z3, z4, K Q ëÓùo:¤ÿÀo>/, P Q(z1, z2, z3, z4). ù

po:¡o>/º:.

½Â 2.1.3. éu*¿E²¡ C ¥?ÛÿÀo>/ Q(z1, z2, z3, z4), Ñ3¢

ê a > 0, b > 0,¦ Q(z1, z2, z3, z4)/duÝ/ R(0, a, a+bi, bi). ùÝ/>

' a/bdÿÀo>/ Q(z1, z2, z3, z4)û½. Ù¥ a/b¡ÿÀo>/ Q(z1, z2, z3, z4)

/ (conformal modulus ), ¡, P Mod(Q(z1, z2, z3, z4)). 3²(ÿÀo>

/oº:ÚgS, Q(z1, z2, z3, z4)P Q, Mod(Q(z1, z2, z3, z4))PMod(Q).

½Â 2.1.4 (AÛ½Â). f : D → G ´ C ¥« D G Ó, e

3~ê K ≥ 1, ¦éu D ¥zÿÀo>/ Q, Q ⊂ D, þk Mod(f(Q)) ≤K Mod(Q),K¡ f ´ K [/N (quasiconformal mapping ),P K–qcN.

11

12 1Ù E©ÛÚEÄåXÚ£ý

N´Ñ, f : D → G ´ K–qc N¿^´µ

Mod(Q)/K ≤ Mod(f(Q)) ≤ K Mod(Q), Ù¥ Q ⊂ D.

Ñ[/N©Û½Â. d·kÚ\¼êäk ACL 5Vg.

½Â 2.1.5. f ´½Â3 D ⊂ C SEëY¼ê. XJéu?¿Ý/ R =

x + iy : a < x < b, c < y < d, R ⊂ D\∞, f−1(∞), N f(x, y) éA¤k½

x ∈ (a, b) Ñ´ y ýéëY¼ê, éA¤k½ y ∈ (c, d) Ñ´ x ýéëY¼

ê. K¡¼ê f 3 D SãþýéëY, ¡ f äk ACL 5.

Ún 2.1.6 ([44, p. 77-79]). f : D → G « D G [/N, K f 3

D Säk ACL 5, Ï f A??k ê, A?? .

½Â 2.1.7 (©Û½Â, [2, p. 16]). D,G þ C ¥«, f : D → G Ó

, XJ f ÷ve^µ

( 1 ) f 3 D þ´ ACL ¶

( 2 ) |µf (z)| ≤ (K − 1)/(K + 1) 3 D SA??¤á.

K¡ f : D → G ´ K–qc N.

[/N©Û½ÂÚAÛ½Â´d ([2, p. 20]).

íØ 2.1.8. XJ[/N f ÷v µf = 0 a.e., K f /N.

y². dÚn 2.1.6 f ÷v ACL 5, 2â µf = 0 a.e., éì[/N

©Û½Â K = 1. Ï f 1–qc N, = f /N.

½Â 2.1.9. éu« ΩþE¼ê µ,XJ µ´ Lebesgueÿ÷v ‖µ‖∞ <

1, K¡ µ Ω þ Beltrami Xê. § fz = µ(z)fz K¡ Beltrami §.

éu½ Ω þ Beltrami Xê µ Ú ν, µ Ú ν 3 Ω þA??, ·ò

µ Ú ν wÓ Beltrami Xê, P µ = ν. XJ f ´[/N, KÙEA

µf = fz/fz ´ Beltrami Xê.

éu½ Beltrami Xê, ´Ä½3[/N±ÙEAQ? e

¡ÿRiemannN½n (Measurable Riemann Mapping Theorem) Ñ£.

½n 2.1.10 (Ahlfors and Bers, [3]). - µ C þ Beltrami Xê, z1, z2, z3

C þØÓn:, w1, w2, w3 C þØÓn:. K3[/N

f : C→ C, ¦ fz = µ(z)fz 3 C þA??¤á, k f(zn) = wn, n = 1, 2, 3.

§2.2. RIEMANN N½n, KOEBE ½n, 13

e ϕ ψ äkÓ Beltrami Xêü[/N, K ϕ Ú ψ =

EÜ/N, = ϕ = φ ψ, Ù¥ φ /N.

éõ©z¥ò Beltrami § L2 2ÂÓ)½Â[/N. ½n 2.1.10 ¥

ÿ¼ê µ(z) ¡[/N f EA, K(f) = (1 + ‖µf‖)/(1−‖µf‖) ¡ f

() û, Ù¥ ‖µf‖ L« f Beltrami Xê3 L∞ m¥ê.

½n 2.1.11 (Ahlfors and Bers, [3]). µ = µ(z, t) ´½Â3 C×D þ (½ C× [0, 1]

þ) k.ÿ¼ê, |µ(z, t)| ≤ k < 1 éA¤k z ∈ C Úé?¿ t ∈ D (½

[0, 1])Ñ¤á, Ù¥ k ~ê . d, b½éA¤k½ z ∈ C, µ(z, t) ´ t ∈ D)Û¼ê (½ [0, 1] þëY¼ê). qéu?¿½ t, ω = f(z, t) ´ C g[/N, ÙEA µ(z, t), ± 0, 1 ∞ ØÄ. Kéu?¿½ z ∈ C,

ω = f(z, t) ´ t ∈ D )Û¼ê (½ [0, 1] þëY¼ê) .

§2.2 Riemann N½n, Koebe ½n,

Riemann N½n´ü¼êØ¥~½n.

½n 2.2.1 (Riemann N½n). Ω ⊂ C ´üëÏ«, >.õu:.

q z0 ∈ Ω ´?¿½:, K3/N f : Ω→ D, ò Ω Nü . X

J¦ f ′(z0) = 0 f ′(z0) > 0, Kù/N f ´.

½Â 2.2.2. X ´ C þk.8Ü, XJé?¿ ε > 0, 3 δ > 0 ¦é X

¥?¿ü: a, b, |a − b| < δ, Ò½3 X ëÏf8 E ÷v: a, b ∈ E

diam(E) < ε, K·¡ X ´ÛÜëÏ.

½n 2.2.3 (Caratheodory òÿ½n). f : D→ Ω ´/N, K f ±ëYò

ÿ>.= Ω ´ÛÜëÏ.

e¡Ñ3ü¼êØ¥~k^ ½n.

½n 2.2.4 (Koebe ½n, [60, p. 9]). f : D → C ´ü)Û¼ê, Ké

?¿ z ∈ D, k

|f ′(0)| |z|(1 + |z|)2

≤ |f(z)− f(0)| ≤ |f ′(0)| |z|(1− |z|)2

;

|f ′(0)| 1− |z|(1 + |z|)3

≤ |f ′(z)| ≤ |f ′(0)| 1 + |z|(1− |z|)3

.


½Â 2.2.5. A C þëÏ«, XJ A /du A1,R = z ∈ C : 1 <

|z| < R, K¡ A ÿÀ¿½Â A ½/

mod(A) =1

2πlogR.

e A /du C \ 0 ½ D \ 0, K½Â§ ∞.

½n 2.2.6 (Grotzsch Øª). A,A1, A2 þëÏ, A1 ∩ A2 = ∅, Ai ⊂ A

C \ A ü©|©Oá\ C \ Ai ü©|¥, i = 1, 2. Kk

mod(A1) + mod(A2) ≤ mod(A).

§2.3 ±Ï:, Fatou 8, Julia 8

·ÄX¼ê f : X → X ÄåXÚ, Ù¥ X o Riemann ¥¡ (k

n¼ê/) oE²¡ (¼ê/). e f : C → C knN, K f

L« P (z)/Q(z), Ù¥ P Ú Q põª, f NÝ (degree)½Â

deg(f) = maxdeg(P ), deg(Q).Ä: z0 3 f : X → X eS: z0 7→ z1 = f(z0) 7→ z2 = f 2(z0) 7→ · · · , ¡S

f n(z)n≥0 z0 3 f ec; (forward orbit), P O+f (z0). e3

ê p ¦ f p(z0) = z0, K¡ z0 f ±Ï: (periodic point), ¤k p ¥ö¡

z0 ±Ï (period). d z0 ;P z0, z1, · · · , zp−1, ¡d; f ±Ï;

(periodic orbit ½ cycle). w,±Ï;Sz:Ñ´±Ï:¿äkÓ±Ï. A

O/, e p = 1, K¡ z0 f ØÄ: (fixed point).

½Â 2.3.1. éu±Ï p ±Ï; f : z0 7→ z1 7→ · · · 7→ zp−1 7→ zp = z0,

½ÂEê λ = (f p)′(zi) =∏p−1

i=0 f′(zi) T;¦f (multiplier). ±Ï;¥:

¹ ∞ , d·½ÂT;¦f λ = 1/(f p)′(∞), Ù¥ (f p)′(∞) ½Â4

limz→∞

(f p)′(z)([6], p.41).

±Ï;¦fXJ©O÷v |λ| < 1!|λ| > 1 Ú |λ| = 1, K¡ù;´á

5 (attracting)!½5 (repelling)Ú¥5 (indifferent). AO/, λ = 0 ¡T;

´á5 (superattracting), 0 < |λ| < 1 K¡T;ÁÛá5 (geometrically

attracting). λ = e2πiθ θ knê, ¡T±Ï;´kn¥5 (rational indiffer-

ent), XJ ∀n, f n ÑØðN, Kkn¥5±Ï;¡´Ô (parabolic).

A θ Ãnê, K¡T;¡Ãn¥5 (irrational indifferent)±Ï;.

§2.3. ±Ï:, FATOU 8, JULIA 8 15

- f Ú g ´ü/ÝX¼ê,=3MobiusC h,¦ hf h−1 =

g. 1e f äk±Ï; C, K g äkA±Ï; h(C), â½Â§äk

Ó¦f. ÏdX¼ê3/Ýê±Ï;5ØC.

½Â 2.3.2. F ´lÝþm X Ýþm Y ¼êx, XJéu X ¥

z: x ±9 F ¥?ÛÃ¡S, Ñ3ÙfS±9 x U ¦TfS

3 U þ´Âñ, K¡¼êx F X þ5x (normal family).

½Â 2.3.3. f : X → X Riemann ¥¡½E²¡gXN, ½Â¤k

¦ f S¼êx f nn≥0 5«¿ f Fatou 8, P F (f), Ù8

X \ F (f) ½Â f Julia 8, P J(f).

d½Â Fatou 8m8, Julia 8K48.

½Â 2.3.4. F Ýþm (X1, d1)Ýþm (X2, d2)ëY¼êx, ∀ε > 0,X

Jéu X1 ¥: x0, 3ê δ, ¦ d1(x, x0) < δ , k d2(f(x), f(x0)) < ε

é F ¥?¿¼ê f Ñ¤á, Kd·¡¼êx F 3 x0 ù:´ÝëY

(equicontinuous). XJéu X1 ¥z:Ñ±éA δ,@o·Ò¡¼

êx F 3 X1 þ´ÝëY.

â Arzela-Ascoli ½n ( [1]), kn¼êx3«þ55duT¼êx

ÝëY5, ¤±: z áu f Fatou 8du3¹ z m U , ¦ f S

¼êS f nn≥0 3 U þ´ÝëY.

âÝëY5½Â, éu?¿ ε > 0, e z ∈ F (f), K3 δ > 0, ¦

é?¿g,ê n, ± z %, δ »m3 f n êá3»Ø

L ε ¥. ùL²éu Fatou 8¥:5`, T:¿©3 f ?¿gS

ê´¿©. Julia 8¥:3 f SeKwéÐ©'¯a, =e

z ∈ J(f), Kéu z ?¿m U , 73 U ¥ü:, ¦ùü:3 f Se¥

yØÓÄåXÚ5.

â Montel ½n, X¼êx F = fα : D → C, α ∈ A ´« D þ5

x= F 3 D þ´ÛÜk..

½Â 2.3.5. - U F (f) ëÏ©|, K U ¡ f Fatou©| (Fatou

component). e3ê p ≥ 1, ¦ f p(U) = U , K¡ U f ±Ï Fatou ©

| (periodic Fatou component); XJ3 n ≥ 0, ¦ f n(U) ´±Ï Fatou ©|, K¡

U ´ª±Ï (eventually periodic). XJ U Ø´ª±Ï, ·¡ U f

i (wandering domain).


½n 2.3.6 ([71]). kn¼êvki.

ù½n`²?ÛNÝu 1kn¼ê Fatou©|ª7N±Ï

Fatou ©|¥, Ï éukn¼ê Fatou 8©aIÄ±Ï Fatou ©|Ò1.

éukn¼ê±Ï Fatou ©|, ·kXe©a½n.

½n 2.3.7 ([51]). U kn¼ê f ±Ï p Fatou©|,Ù¥ deg(f) ≥2, K U =XeÊ«/ (ã2.1) :

() áÚ Ô Siegel Herman

ã 2.1 Ê« Fatou ©|.

(AB) áÚ, U ¹ f ±Ï p á5±Ï: z0, 3 U ?Û;f8

þk f np(z) Âñ: z0 (n→∞)¶

(SAB) áÚ, (AB)aq, =òá5±Ï:Uá5±Ï:¶

(PB) Ô, 3 U >.þ3Ô±Ï: z0, ¦ f p(z0) = z0, f p 3 z0

¦fu 1, 3 U ?Û;f8þk f np(z) Âñ: z0 (n→∞)¶

(SD) Siegel,3/N ψ : U → DÚÃnê θ,¦ ψf pψ−1(z) =

e2πiθz, = f p 3 U þ^/Ýuü þÃn^=¶

(HR) Herman , (SD)aq, =Iò D ¤ A(r, 1) = z : r < |z| < 1,Ù¥ 0 < r < 1.

5¿¼ê±ki [5], ±Ï Fatou ©|©aØÓukn¼ê. 'X

¼êvk Herman ±k Baker .

k' Julia 8Ú Fatou 8Ù§5IOë©z [6, 13, 53] ½ö [8,

28, 40, 45, 68].

§2.4 Ak^Ún

éu?Ûf8 A ⊂ C, A î¼»½Â Diam(A) = supx,y∈A |x − y|. ?^Jordan C, TþüØÓ:ò§©¤üã. ^ I L«äkáî¼»

@ã, éu½~ê k, XJ Diam(I)/|x − y| ≤ k é C þ¤kØÓ x Ú y ¤á,

§2.4. Ak^Ún 17

K·¡ C ´ k–k.ò=. XJ^ Jordan ´ k–k.ò=, K§´ K–[,

Ù¥~ê K =6u k, ½, ( [43, p.100]).

·^ |I| L«©¡1w½öl I î¼Ý. 5¿ Tr = z : |z| = r, Ù¥r > 0.

Ún 2.4.1. b f : D → C ´ü. éu? 0 < r < 1, K f(Tr) ´[ f(Tr) ûk=6u r þ..

y². é?ÛØÓ x, y ∈ f(Tr), - I, J f(Tr) \ x, y ü¦ |I| ≤ |J | ëÏ©|. ·äó3Ø6u f, xÚ y ~ê C(r) > 0,¦ |I| ≤ C(r)|x−y|.

- L = [x, y] ë x Ú y ã, K |L| = |x − y|. e¡?Øò©¤ü«¹:

L ⊂ D Ú L 6⊂ D, Ù¥ D = f(D). - x1, y1 ∈ D Ú I1 ⊂ D ©O x, y, I c. X

J L ⊂ D, P L1 = f−1(L), §´ D ¥^1w. ½ r0 ∈ (r, 1), ·^ Ci(r0)

L«=6u r0 ~ê (Ï =6u r). XJ L1 ⊂ Dr0 , Kâ Koebe ½nk

|L| ≥ C1(r0)|x1 − y1| |I| ≤ C2(r0)|I1| ≤ 2C2(r0)|x1 − y1|. ùL² |I| ≤ C3(r0)|L|. XJ L1 6⊂ Dr0 , - L2 L1 ∩ Dr0 ¹ x1 ©|. â Koebe ½n, |I| ≤ C2(r0)|I1|,|I1| ≤ C4(r0)|L2| ±9 |L| ≥ C5(r0)|L2|. Ïd |I| ≤ C6(r0)|L|. XJ L 6⊂ D, ?Øa

qu L1 6⊂ Dr0 ¹. Úny..

íØ 2.4.2. - D Jordan «. b γ ⊂ D ´[ f : D → C ´ü. K f(γ) ´ K–[, Ù¥ K = K(D, γ) ´=6u D Ú γ Ø6u

f ~ê.

Ún 2.4.3. - 0 < x < 1 Ãnê, K8Ü M = e2πkxi | k ∈ N 3ü ±þÈ.

y². b3 S1 þml M Ø. À¥Ù¥^, P I, ¦

I ∩M = ∅. ùL²é¤k k ≥ 1, Ik = e−2πkxiz | z ∈ I M Ø. Ï I ´

, K73 k ∈ N ¦ I = Ik. l x ´knê. ùgñL²Ún¤á.

1nÙ Julia 8 Cantor ±kn¼ê

ékn¼ê Julia8ÿÀ5ïÄÉÄåXÚ¥¯K.éõö,

'X Avila, Buff, Cheritat, Branner, Douady, Hubbard, Lyubich, Pertersen, Shishikura,

Yoccoz, Zakeri ÑédÑïÄ [4, 11, 38, 46, 57, 58, 66]. ézNÝØu 2

õª, ©z [63] ¥®²y²ØêõëÏ©|, Ù§ Julia 8ëÏ©|Ñ

´ü:. éukn¼ê, Julia 8±ÑyE,ÿÀ. Pilgrim Ú Tan y²: XJ

V (/, AÛk) kn¼ê Julia 8Ø´ëÏ, KØkõ©|Ú§

êc±, Julia ©|o´ü:o´ Jordan [59, ½n 1.2]. ù

Ù¥, ·òÄakn¼ê, Ù Julia 8äküÿÀ: z Julia 8©|Ñ´

Jordan .

§3.1 Úó

XJ Riemann ¥¡ C f8Óu C × S1, Ù¥ C Cantor n©8 S1

ü ±,KTf8¡´ Cantor ± (Cantor circles). 1 Julia8 Cantor

±~f´d McMullen uy ( [49, §7]). ¦y²XJ f(z) = z2 + λ/z3 ¥ λ v

, K f Julia 8´ Cantor ±. , õÖör5¿å8¥3e¡¼

êx, y3Ñ¡ McMullen N:

fλ(z) = zm + λ/zl, (3.1.1)

Ù¥ l,m ≥ 2 λ ∈ C \ 0 ( [24, 69, 61] Úp¡ë©z). λ éÿ, ùx

AÏkn¼ê±w¤´üõª f0(z) = zm 6Ä. ®² 1/l + 1/m < 1

, 3:B M, ¡´ McMullen , ¦ λ ∈ M , ¼ê fλ

Julia 8´ Cantor ± ( [49, §7] Ú [24, §3]).

y3kXeng,¯K: (1) Ø McMullen N, ´Ä3Ù§kn¼

êÙ Julia 8 Cantor ±? (2) XJ1¯KY´½, @§ó?

ó, ·UÄéÑ§Lª? (3) ·UÄ3½§ÝþéÑ¤k Julia 8

Cantor ±kn¼ê? ùÙòÑù¯K½£.

d[/Ãâ,ò fλ ¹∞áÚ6Ä¡AÛáÚ,·±

éõ#kn¼ê. ½Ù¥,@oùN3 Riemann¥¡þ fλ Ø´ÿ

ÀÝ. §3éA Julia8þ´ÿÀÝ. AO/, gε,λ(z) = 1z(zm+ε) 1

z+λ/zl

´~f, Ù¥ 1/l+ 1/m < 1 ε, λ ∈ C \ 0 Ñ¿©. , , Ïùa.kn

¼ê±w¤´3ý McMullen N Fatou 8þÃâ, ¤±3

18

§3.1. Úó 19

þ§´±w¤´ McMullen N, ùØ´·é. Ïd~k¯

K´, éÙ§¦ Julia 8´ Cantor ±kn¼ê, §3 Julia 8þØÿÀ

Ýu?Û McMullen N.

®² “þ” McMullen NØÓkn¼ê´3 ( [35, §§1,2]). ù

Ù¥, ·òÑùa.kn¼êäNLª, Ø=¹ [35] ¥?Ø¹,

3 “þ” ¹¤k Julia 8 Cantor ±kn¼ê (½n 3.1.2). -

p ∈ 0, 1, n ≥ 2 ê d1, · · · , dn n ¦∑n

i=11di< 1 ê. ·½Â

fp,d1,··· ,dn(z) = z(−1)n−pd1

n−1∏i=1

(zdi+di+1 − adi+di+1

i )(−1)n−i−p , (3.1.2)

Ù¥ a1, · · · , an−1 n − 1 ÷v 0 < |a1| < · · · < |an−1| < 1 Eê. AO/, XJ

n = 2, K f1,d1,d2(z) = zd2 − ad1+d21 /zd1 ´éõö®²ïÄ McMullen N. d

, f0,d1,d2(z) = zd1/(zd1+d2 − ad1+d21 ) /Ýu McMullen N z 7→ zd1 + λ/zd2 , Ù¥

λ 6= 0. ¼ê fp,d1,··· ,dn 3 0Ú∞?NÝ©O´ d1 Ú dn deg(fp,d1,··· ,dn) =∑n

i=1 di.

éx (3.1.2) ¥z, Ný 0 Ú ∞ áu fp,d1,··· ,dn Fatou 8. - D0 Ú D∞

©O¹ 0 Ú ∞ Fatou ©|. y3ke¡o«¹:

(1) XJ p = 1 n Ûê, K f(D0) = D0 f(D∞) = D∞;

(2) XJ p = 1 n óê, K f(D0) = D∞ f(D∞) = D∞;

(3) XJ p = 0 n Ûê, K f(D0) = D∞ f(D∞) = D0;

(4) XJ p = 0 n óê, K f(D0) = D0 f(D∞) = D0.

Äk·òéÑ (3.1.2) ¥·ëê ai, Ù¥ 1 ≤ i ≤ n− 1, ¦þ¡o«¹

¥ fp,d1,··· ,dn Julia 8Ñ´ Cantor ±. - ξ =∑n

i=11di K ≥ 3 d1, · · · , dn ¥

ê.

½n 3.1.1. XJ |an−1| = (s1K−2)1/dn, é 1 ≤ i ≤ n − 2 k |ai| = (s1K

−5)1/di+1

|ai+1|, Ù¥ s1 > 0 ¿©, K f1,d1,··· ,dn Julia 8´ Cantor ±. XJ |an−1| =

(s1/dn+(1−ξ)/30 )1/dn éu 1 ≤ i ≤ n−2 k |ai| = (s

1+1/dn+2(1−ξ)/30 )1/di+1|ai+1|, Ù¥ s0 > 0

¿©, K f1,d1,··· ,dn Julia 8´ Cantor ±.

3,«§Ýþ, ½n 3.1.1 L«·®²éxäkëê s1 Ú s0 kn¼ê,

§ Julia 8 Cantor ±. Ï s1 Ú s0 ±?¿, ùkn¼êw¤´ zdn

Ú z−dn 6Ä (éAu p = 1 Ú 0). §3.2 ¥Ñëê s1 Ú s0 äN

( (3.2.1) Ú (3.2.2)). d, ·òy²XJ n ≥ 3, Kz fp,d1,··· ,dn 3 Julia 8þØ´

ÿÀÝu McMullen N (½n 3.2.6). ùL²·éakn¼êäNL

ª¦§ Julia 8´ Cantor ±, § McMullen N3þ´ØÓ.

20 1nÙ Julia 8 Cantor ±kn¼ê

5¿XJkn¼ê f Julia 8 J(f) ´ Cantor ±, Ï z Julia ©

|Ñ´ Jordan , @o3 J(f) þBØ3.: (Ún 3.3.1). ùL² f z±

Ï Fatou ©|o´á5o´Ô. ¯¢þ, ·kXe½n

½n 3.1.2. f kn¼ê, Ù Julia 8 Cantor ±. K3 p ∈ 0, 1, ê n ≥ 2 ±9÷v

∑ni=1

1di< 1 d1, · · · , dn ¦ f Ú fp,d1,··· ,dn 3§éA Julia

8þ´ÿÀÝ, fp,d1,··· ,dn ëê ai d½n 3.1.1 (½, Ù¥ 1 ≤ i ≤ n− 1.

â½n 3.1.2· “A”é¤k Julia8 Cantor±kn¼ê. X

Jkn¼ê.:Ñá5±Ï;¤áÚ, KTkn¼ê¡´V. éu

fp,d1,··· ,dn Julia 8©|K5, ±y²XJ fp,d1,··· ,dn ´V, K fp,d1,··· ,dn z

Julia 8©|Ñ´[ (íØ 3.3.3).

XJ λ v, K fλ ´V ( [24]). y3·EVkn¼ê¦

§ Julia 8´ Cantor ±. - m,n ≥ 2 ü÷v 1/m + 1/n < 1 ê

λ ∈ C \ 0, ·½Â

Pλ(z) =1n((1 + z)n − 1) + λm+nzm+n

1− λm+nzm+n. (3.1.3)

y 0 ´ Pλ ¦f 1 ÔØÄ:. ·k

½n 3.1.3. XJ 0 < |λ| ≤ 1/(210mn3), K Pλ ´VÙ Julia 8 Cantor

±.

É½n 3.1.1 éu, ·±EõVkn¼ê¦§ Julia 8´

Cantor ±. üå, éz n ≥ 2 ±9 1 ≤ i ≤ n, ·=Ä¹ di = n+ 1. é

z n ≥ 2, ·½Â

Pn(z) = An(n+ 1)z(−1)n+1(n+1)

nzn+1 + 1

n−1∏i=1

(z2n+2 − b2n+2i )(−1)i−1

+Bn, (3.1.4)

Ù¥ b1, · · · , bn−1 n− 1 ÷v 1 > |b1| > · · · > |bn−1| > 0 Eê

An =1

1 + (2n+ 2)Cn

n−1∏i=1

(1−b2n+2i )(−1)i , Bn =

(2n+ 2)Cn1 + (2n+ 2)Cn

Cn =n−1∑i=1

(−1)i−1b2n+2i

1− b2n+2i

.

(3.1.5)

ùp An Ú Bn y Pn(1) = 1 P ′n(1) = 1. =, 1 Pn ¦f 1 Ô

ØÄ: (Ún 3.7.1).

½n 3.1.4. éz n ≥ 2 Ú 1 ≤ i ≤ n− 1, XJ |bi| = si, Ù¥ 0 < s ≤ 1/(25n2),

K Pn ´VÙ Julia 8 Cantor ±.

§3.2. .: ±9V/ 21

±wÑéz n ≥ 2, Pn 3Ù Julia 8þÄåXÚÝu fp,n+1,··· ,n+1 3

Julia 8þÄåXÚ (p = 1). §3 Fatou 8þÄåXÚ«O´ fp,n+1,··· ,n+1

3 ∞ áÚ¤ Pn Ô.

ùÙ´ùSü: 31 §3.2¥,·O,y²½n 3.1.1. 3 §3.3¥,

·y²½n 3.1.2. 3 §3.4 ¥, ·òOÿÀd Cantor ±kn¼êÝa

ê8¿3 §3.5¥Ñùê8þe.O.3 §3.6¥,·y² λvÿ,

Pλ Julia 8´ Cantor ±, ùBy²½n 3.1.3. ·ò3 §3.7 ¥y²½n

3.1.4 ¿r'Ún3 §3.8.

§3.2 .: ±9V/

·ÄkÑÄÚkÔ.

Ún 3.2.1. - n ≥ 2 ê, a ∈ C \ 0 0 < ε < 1/2.

(1) XJ |z − a| ≤ ε|a|, K |zn − an| ≤ ((1 + ε)n − 1) |a|n;

(2) XJ |zn − an| ≤ ε|a|n, K |a/z|n < 1 + 2ε 3, 1 ≤ j ≤ n ¦ |z −ae2πij/n| < ε|a|;

(3) XJ 0 < ε < 1/n, K nε < (1 + ε)n − 1 < 3nε nε/3 < 1− (1− ε)n < nε.

y². (1) - z = a(1 + reiθ), Ù¥ 0 ≤ r ≤ ε 0 ≤ θ < 2π, K

|zn − an| = |(1 + reiθ)n − 1| · |a|n ≤ ((1 + ε)n − 1) |a|n.

(2) XJ 0 < ε < 1/2, K |a/z|n ≤ 1/(1− ε) < 1 + 2ε, u´ (2) ¥1Øª¤

á. éu1(Ø, - zn = an(1 + reiθ), Ù¥ 0 ≤ r ≤ ε 0 ≤ θ < 2π, K3,

1 ≤ j ≤ n ¦ z = ae2πij/n(1 + reiθ)1/n. d, XJ n ≥ 2, K

|z − ae2πij/n| = |(1 + reiθ)1/n − 1| · |a| ≤ ((1 + ε)1/n − 1) · |a| < ε|a|.

(3) é x 7→ xn3«m [1, 1 + ε] Ú [1− ε, 1] þ©O|^.KF¥½n.

½ n ≥ 2 - d1, · · · , dn ≥ 2 n ¦ ξ =∑n

i=11di< 1 ê. - K ≥ 3

d1, · · · , dn ¥ê. - u1 = s1K−5 v1 = s1K

−2, Ù¥

0 < s1 ≤ minK−5ξ/(1−ξ), K5−2K < 1. (3.2.1)

- u0 = s1+1/dn+2(1−ξ)/30 , v0 = s

1/dn+(1−ξ)/30 , Ù¥

0 < s0 ≤ min2−(1−ξ)−1(1+1/dn−2ξ/3)−1

, (4K)−3/(1−ξ), K−2K(1+1/dn+2(1−ξ)/3)−1 < 1. (3.2.2)


éu p ∈ 0, 1,- |an−1,p| = v1/dnp |ai,p| = u

1/di+1p |ai+1,p|¼êx fp,d1,··· ,dn ¥ n− 1

ëê, Ù¥ 1 ≤ i ≤ n− 2. Ï3¹e, éu p = 0 Ú p = 1 ùü«¹±Ó

?1?Ø, Bå, ØuÚå· ¹e, ·^ s, u, v Ú ai ©O5L«

sp, up, vp Ú ai,p, Ù¥ 1 ≤ i ≤ n− 1.

Ún 3.2.2. (1) u2/K ≤ K−4.

(2) XJ 1 ≤ j ≤ i ≤ n− 1, K |aj/ai| ≤ ui−jK .

(3) XJ p = 1, K (3a) (s/|a1|)d1 < su/(2v) = sK−3/2 (3b) (|a1|/s)d1v/2 > K.

(4) XJ p = 0, K (4a) 2Ku/v < s 1/(2Kv) > (2/s)1/dn; (4b) (s/|a1|)d1 <sv/2 < u1/2/2 (4c) (|a1|/s)d1u/(2v) > (2/s)1/dn.

y². (1)â (3.2.1)Ú (3.2.2),·k s1 ≤ K5−2K Ú s0 ≤ K−2K(1+1/dn+2(1−ξ)/3)−1.

ùL² u2/K1 = (s1K

−5)2/K ≤ K−4 u2/K0 ≤ K−4.

(2) XJ j = i, K(Ø´w,. b 1 ≤ j < i ≤ n − 1, Ïéu 1 ≤ i ≤ n k

K ≥ di, K

|aj/ai| = u1

dj+1+···+ 1

di ≤ ui−jK .

ùÒy² (2).

(3)XJ p = 1,K u = sK−5 v = sK−2. Ï s ≤ K−5ξ/(1−ξ),·k s1−ξK5ξ ≤ 1,

u´

s1− 1

d1 s−( 1

d2+···+ 1

dn)K

5( 1d2

+···+ 1dn−1

)+ 2dn 2

1d1K

3d1 < 1.

Ï

|a1| = u1d2

+···+ 1dn−1 v

1dn = s

1d2

+···+ 1dn /K

5( 1d2

+···+ 1dn−1

)+ 2dn ,

þªdu s1− 1

d1 21d1K

3d1 /|a1| < 1. Ïd·k (s/|a1|)d1 < su/(2v) = sK−3/2 (3a)

y². d, Ï (|a1|/s)d1 > 2K3/s = 2K/v, (3b) ±d (3a) íÑ.

(4) XJ p = 0, K u = s1+1/dn+2(1−ξ)/3, v = s1/dn+(1−ξ)/3. â (3.2.2), ·

4Ks(1−ξ)/3 ≤ 1, ù¿X 2Ku/v = 2Ks1+(1−ξ)/3 < s. 5¿ 21+1/dnKs(1−ξ)/3 < 1, ù

du 1/(2Kv) > (2/s)1/dn . ù(å (4a) y².

d (3.2.2), ·

1 ≥ 2s(1−ξ)(1+1/dn−2ξ/3) > 21d1 s(1−ξ)(1+1/dn−2ξ/3)

= 21d1 s

1− 1d1 /s

( 1d2

+···+ 1dn−1

)+ 1dn

( 1d1

+···+ 1dn

)+2ξ(1−ξ)

3

> 21d1 s

1− 1d1 /s

( 1d2

+···+ 1dn−1

)+ 1dn

( 1d1

+···+ 1dn

)+ 1−ξ3

( 1d1

+2( 1d2

+···+ 1dn−1

)+ 1dn

)

= s1− 1

d1 (2/v)1d1 /|a1|.

ù¿X (s/|a1|)d1 < sv/2 = u1/2s(1+1/dn)/2/2 < u1/2/2. Ïd (4b) ¤á.

§3.2. .: ±9V/ 23

(4c) y²aqu(4b). ·I5¿

1 ≥ 2s(1−ξ)(1+1/dn−2ξ/3) > 21d1

(1+ 1dn

)s(1−ξ)(1+1/dn−2ξ/3) > (s/|a1|)(2v/u)

1d1 (2/s)

1d1dn .

ùL² (|a1|/s)d1u/(2v) > (2/s)1/dn .

e5, üå, ·^ f L« fp,d1,··· ,dn . 5¿ 0 Ú ∞ ´ f ê d1 − 1

Ú dn − 1 .:, f NÝ´∑n

i=1 di. P Di = di + di+1, ·k 5 ≤ Di ≤ 2K,

Ù¥ 1 ≤ i ≤ n− 1. Ø 0 Ú ∞, f ∑n−1

i=1 Di .:é¡§)

(−1)p zf ′(z)

f(z)=

n−1∑i=1

(−1)n−iDizDi

zDi − aDii+ (−1)nd1 = 0. (3.2.3)

éu 1 ≤ i ≤ n− 1, - CP i := wi,j = riai exp(πi2j−1Di

) : 1 ≤ j ≤ Di þ!á3± Tri|ai| þ Di :8Ü, Ù¥ ri = Di

√di/di+1. e¡ÚnL« f

∑n−1i=1 Di

gd .: “~C”⋃n−1i=1 CP i.

Ún 3.2.3. éz wi,j ∈ CP i, Ù¥ 1 ≤ i ≤ n− 1 1 ≤ j ≤ Di, 3 (3.2.3)

) wi,j, ¦ |wi,j − wi,j| < u2K |ai|. d, wi1,j1 = wi2,j2 = (i1, j1) = (i2, j2).

y². 5¿ (3.2.3) m>ªdu

(−1)n−i(

DizDi

zDi − aDii− di

)+Gi(z) = 0, (3.2.4)

Ù¥

Gi(z) =∑

1≤j≤n−1, j 6=i

(−1)n−jDjzDj

zDj − aDjj+ (−1)nd1 + (−1)n−idi. (3.2.5)

3 (3.2.4) ü>Ó¦± (zDi − aDii )/di+1, Ù¥ 1 ≤ i ≤ n− 1, ·k

(−1)n−i(zDi + diaDii /di+1) + (zDi − aDii )Gi(z)/di+1 = 0. (3.2.6)

- Ωi = z : |zDi + diaDii /di+1| ≤ ε |ai|Di, Ù¥ ε = u

2K 1 ≤ i ≤ n − 1. éz

z ∈ Ωi, ÏâÚn 3.2.2(1) ε ≤ K−4, ·k

K−1 < di/di+1 − ε ≤ |z/ai|Di ≤ di/di+1 + ε < K − 1 < K. (3.2.7)

ùL²

K−1 < |ai/z|Di < K l K−1 < |ai/z|5 < K. (3.2.8)

XJ 1 ≤ j < i Ú z ∈ Ωi, ·k

|aj/z|Di ≤ |ai/z|Di |ai−1/ai|Di < Ku1+di+1/di < 1. (3.2.9)


Ïd, |aj/z| < 1. daq?Ø, XJ i < j ≤ n − 1 z ∈ Ωi, K± |z/aj| < 1.

XJ 1 ≤ j < i, âÚn 3.2.2(1)(2) Ú (3.2.8), ·k

|aj/z|Dj ≤ |ai/z|5|aj/ai|5 < K ε5(i−j)/2 ≤ K−9. (3.2.10)

aq/, XJ i < j ≤ n− 1, ·k

|z/aj|Dj ≤ |z/ai|5|ai/aj|5 < K ε5(j−i)/2 ≤ K−9. (3.2.11)

â½Â, ·k ∑1≤j<i

(−1)n−jDj + (−1)nd1 + (−1)n−idi = 0. (3.2.12)

â (3.2.5), (3.2.10), (3.2.11), (3.2.12) ±9 ε5/2 ≤ K−10, ·k

|Gi(z)| =

∣∣∣∣∣ ∑1≤j<i

(−1)n−jDj

1− (aj/z)Dj+

∑i<j≤n−1

(−1)n−j−1Dj(z/aj)Dj

1− (z/aj)Dj+ (−1)nd1 + (−1)n−idi

∣∣∣∣∣≤ 2K

∣∣∣∣∣ ∑1≤j<i

(−1)n−j(aj/z)Dj

1− (aj/z)Dj+

∑i<j≤n−1

(−1)n−j−1(z/aj)Dj

1− (z/aj)Dj

∣∣∣∣∣<

4K2

1−K−9

n−1∑k=1

ε5k/2 <4K2

1−K−9

ε5/2

1− ε5/2< 5K2 ε5/2,

ùL²XJ z ∈ Ωi, â (3.2.7) ÚÚn 3.2.2(1), ·k

|zDi − aDii | · |Gi(z)|/di+1 < 3K3 ε5/2|ai|Di < ε|ai|Di . (3.2.13)

â (3.2.6) ±9 Rouche ½n, 3§ (3.2.3) ) wi,j ¦éz 1 ≤ j ≤ Di

k wi,j ∈ Ωi. AO/, âÚn 3.2.1(2) 1(Ø |wi,j − wi,j| < ε|ai|. 5¿é1 ≤ i ≤ n− 2, ·k

|ai+1| − |ai| − 2ε|ai| − 2ε|ai+1| > |ai+1|(1− 2ε− (1 + 2ε)K−2) > 0. (3.2.14)

âÚn 3.2.2(1) Ú ri = Di

√di/di+1 ≤ (K/2)1/5, ·k,

ri|ai| sin(π/Di)

ε|ai|≥ K4(

2

K)1/5 · 2

π· π

2K> K2 > 1. (3.2.15)

ùL² wi1,j1 = wi2,j2 = (i1, j1) = (i2, j2). Úny²..

éu 1 ≤ i ≤ n− 1, - CPi := wi,j : 1 ≤ j ≤ Di Ún 3.2.3 ¥ f Cu±

Tri|ai| Di gd.:8Ü¿P CVi = f(CPi).

§3.2. .: ±9V/ 25

Ún 3.2.4. éz 1 ≤ i ≤ n− 1, 3¹ CPi ∪ Tri|ai| ∪ T|ai| Ai, ¦

(1)XJ p = 1, KéuÛê n− ik f(Ai) ⊂ Ds éuóê n− ikf(Ai) ⊂ C\DK.

AO/, f .¤8Ü÷v⋃n−1i=1 CVi ⊂ Ds ∪C \DK. d, Ds C \DK

á3 f Fatou 8¥¿ f−1(As,K) ⊂ As,K.

(2)XJ p = 0, Kéuóê n−ik f(Ai) ⊂ Ds éuÛê n−ikf(Ai) ⊂ C\DM ,

Ù¥ M = (2/s)1/dn. AO/, f .¤8Ü÷v⋃n−1i=1 CVi ⊂ Ds ∪ C \ DM . d

, Ds C \ DM á3 f Fatou 8¥¿ f−1(As,M) ⊂ As,M .

y². - ε = u2K ≤ K−4 Ñy3Ún 3.2.3 ¥ê. éz 1 ≤ i ≤ n− 1, ½Â

Ai = z : (minri, 1 − 2ε)|ai| < |z| < (maxri, 1+ 2ε)|ai| (3.2.16)

Ù¥ ri = Di

√di/di+1. w,, Ai ⊃ CPi ∪ Tri|ai| ∪ T|ai|. â½Â, ·k

(2/K)1Di ≤ minri, 1 ≤ maxri, 1 ≤ (K/2)

1Di . (3.2.17)

XJ z ∈ Ai, ·k

|ai/z| ≤1

(2/K)1Di − 2ε

≤ (K/2)1Di

1− 2K−4(K/2)1/5< (K/2)

1Di (1 + 4/K19/5). (3.2.18)

|z/ai| ≤ (K/2)1Di + 2ε ≤ (K/2)

1Di + 2/K4 < (K/2)

1Di (1 + 1/K3). (3.2.19)

ù¿X

|ai/z|5 < (K/2)5Di (1 + 4/K19/5)5 < (K/2) e20/K19/5

< (K/2) e20/319/5 < 7K/10. (3.2.20)

¿Ó,

|z/ai|5 < (K/2)5Di (1 + 1/K3)5 < (K/2) e5/K3

< (K/2) e5/27 < 7K/10. (3.2.21)

d, aqu (3.2.20) Ú (3.2.21) ?Ø, ·k

|ai/z|di + |z/ai|di+1 < 7K/5. (3.2.22)

5¿éz 1 ≤ i ≤ n−2k |ai/ai+1|di+1 = u |an−1|dn = v. - 1 ≤ i1 ≤ i2 ≤ n−1


Ú p ∈ 0, 1, ·k

i2∏j=i1

|aj|(−1)n−j−pDj = |ai1|(−1)n−i1−pdi1 |ai2|(−1)n−i2−pdi2+1

i2−1∏j=i1

∣∣∣∣ ajaj+1

∣∣∣∣(−1)n−j−pdj+1

= |ai1|(−1)n−i1−pdi1 |ai2|(−1)n−i2−pdi2+1 u(−1)n−i1−p−(−1)n−i2−p

2

=

(|a1|d1u/v)(−1)p if i1 = 1 and i2 = n− 1 is even

(|a1|−d1/v)(−1)p if i1 = 1 and i2 = n− 1 is odd.

(3.2.23)

â (3.1.2) ±9 (3.2.23) 1ª, ·k

|f(z)| = |zDi − aDii |(−1)n−i−p |z|(−1)n−pd1

i−1∏j=1

|z|(−1)n−j−pDj

n−1∏j=i+1

|aj|(−1)n−j−pDj ·Qi(z)

= |1− (z/ai)Di |(−1)n−i−p |z/ai|(−1)n−i−p+1di |an−1|(−1)1−pdn u

(−1)n−i−p−(−1)1−p2 ·Qi(z)

= v(−1)1−p u(−1)n−i−p−(−1)1−p

2 |(ai/z)di − (z/ai)di+1|(−1)n−i−p ·Qi(z) ≤ v(−1)1−p u

1−(−1)1−p2 (|ai/z|di + |z/ai|di+1)Qi(z) XJ n− i− p óê

≥ v(−1)1−p u−1−(−1)1−p

2 (|ai/z|di + |z/ai|di+1)−1Qi(z) XJ n− i− p Ûê,

(3.2.24)

Ù¥

Qi(z) =i−1∏j=1

∣∣1− (aj/z)Dj∣∣(−1)n−j−p

n−1∏j=i+1

∣∣1− (z/aj)Dj∣∣(−1)n−j−p

. (3.2.25)

é 1 ≤ i ≤ n− 1, Ä z ∈ Ai. XJ 1 ≤ j < i, â (3.2.20), ·k

|aj/z|Dj ≤ |ai/z|5|aj/ai|5 < 7K ε5(i−j)/2/10 < K−9. (3.2.26)

XJ i < j ≤ n− 1, Kd (3.2.21)

|z/aj|Dj ≤ |z/ai|5|ai/aj|5 < 7K ε5(i−j)/2/10 < K−9. (3.2.27)

du 0 < x ≤ 1 k ex < 1 + 2x ε ≤ K−4, â (3.2.25)–(3.2.27), ·k

Qi(z) <∞∏k=1

(1 + 7K ε5k/2/5

)2 ≤ exp

(14K ε5/2/5

1− ε5/2

)< 1 +K−5 < 1.01. (3.2.28)

Qi(z) >∞∏k=1

(1 + 7K ε5k/2/5

)−2> 1/1.01 > 0.99. (3.2.29)

§3.2. .: ±9V/ 27

éu p = 1, â 3.2.2(2)(3a), éz 1 ≤ i ≤ n− 1, XJ |z| ≤ s, ·k

|zDi/aDii | ≤ |s/a1|Di |a1/ai|Di ≤ (sK−3/2)5K u

5(i−1)K . (3.2.30)

XJ·5¿Ún 3.2.2(1), K

n−1∑i=1

|zDi/aDii | ≤(sK−3/2)

5K

1− u 5K

≤ K10K−10

1−K−10< 1/200. (3.2.31)

XJ p = 0, âÚn 3.2.2(2)(4b), éz 1 ≤ i ≤ n− 1, |z| ≤ s , ·k

|zDi/aDii | ≤ |s/a1|Di |a1/ai|Di ≤ (u1/2/2)5K u

5(i−1)K . (3.2.32)

âÚn 3.2.2(1), K

n−1∑i=1

|zDi/aDii | ≤(u1/2/2)

5K

1− u 5K

≤ K−5

1−K−10< 1/200. (3.2.33)

du 0 ≤ |a| ≤ 1/2 , (1 + 2|a|)−1 ≤ |1 + a|±1 ≤ 1 + 2|a|. â (3.2.31) Ú

(3.2.33), ·

n−1∏i=1

∣∣1− zDi/aDii ∣∣(−1)n−i−p ≤n−1∏i=1

(1 + 2|z/ai|Di

)< e1/100 < K. (3.2.34)

Ïd,

n−1∏i=1

∣∣1− zDi/aDii ∣∣(−1)n−i−p ≥n−1∏i=1

(1 + 2|z/ai|Di

)−1> e−1/100 > 1/K. (3.2.35)

(1) ·ÄkÄ¹ p = 1. XJ n− i ´Ûê, â (3.2.22), (3.2.24) Ú (3.2.28),

z ∈ Ai ·k|f(z)| ≤ v · (7K/5) · 1.01 < 2Kv < s. (3.2.36)

XJ n− i ´óê, â (3.2.22), (3.2.24) Ú (3.2.29), z ∈ Ai ·k

|f(z)| ≥ (v/u) · (7K/5)−1 · 0.99 > v/(2Ku) > K. (3.2.37)

XJ n ´Ûê, âÚn 3.2.2(3a), (3.2.23) Ú (3.2.34), éz z ¦ |z| ≤ s, ·

k

|f(z)| = |z|d1n−1∏i=1

|ai|Di(−1)n−i−1n−1∏i=1

∣∣∣∣1− zDi

aDii

∣∣∣∣(−1)n−i−1

< |s/a1|d1vu−1 · 1.02 < s.


ùL²XJ n ´Ûê, K f(Ds) ⊂ Ds. XJ n ´óê |z| ≤ s, âÚn 3.2.2(3b),

(3.2.23) Ú (3.2.35), ·k

|f(z)| = |a1/z|d1vn−1∏i=1

∣∣∣∣1− zDi

aDii

∣∣∣∣(−1)n−i−1

> |a1/s|d1v/1.02 > K.

Ïdéu n óê, f(Ds) ⊂ C \ DK .

5¿ f 3 DK ¡ “~C” z 7→ zdn ,ù´Ï |ai|Di 'u@ |z| ≥ K

z ó~, Ù¥ 1 ≤ i ≤ n − 1. ¯¢þ, daqu (3.2.34)–(3.2.35) ?Ø,

XJ |z| ≥ K, K

|f(z)| ≥ |z|dnn−1∏i=1

(1 + 2

|ai|Di|z|Di

)−1

> K. (3.2.38)

ùL² f(C \ DK) ⊂ C \ DK . l éz n ≥ 2, k f−1(As,K) ⊂ As,K (ã 3.1).

(2) y3·Ä¹ p = 0. XJ n− i ´óê, â (3.2.22), (3.2.24), (3.2.28) Ú

Ún 3.2.2(4a), XJ z ∈ Ai ·k

|f(z)| ≤ v−1u · (7K/5) · 1.01 < 2Ku/v < s. (3.2.39)

XJ n− i ´Ûê, â (3.2.22), (3.2.24), (3.2.29) ÚÚn 3.2.2(4a), éu z ∈ Ai ·k

|f(z)| ≥ v−1 · (7K/5)−1 · 0.99 > 1/(2Kv) > M, (3.2.40)

Ù¥ M = (2/s)1/dn .

XJ n ´óê, âÚn 3.2.2(4b), (3.2.23) Ú (3.2.34), éz¦ |z| ≤ s z, ·

k

|f(z)| = |z|d1n−1∏i=1

|ai|Di(−1)n−in−1∏i=1

∣∣∣∣1− zDi

aDii

∣∣∣∣(−1)n−i

< |s/a1|d1v−1 · e1/100 < s.

ùL²XJ n óê, K f(Ds) ⊂ Ds. XJ n Ûê |z| ≤ s, âÚn 3.2.2(4c),

(3.2.23) Ú (3.2.35), ·k

|f(z)| = |a1/z|d1uv−1

n−1∏i=1

∣∣∣∣1− zDi

aDii

∣∣∣∣(−1)n−i

≥ |a1/s|d1uv−1 · e−1/100 > M.

ÏdXJ n Ûê, K f(Ds) ⊂ C \ DM .

XJ |z| ≥M , K

|f(z)| = |z|−dnn−1∏i=1

∣∣∣∣1− aDiizDi

∣∣∣∣(−1)n−i

≤M−dnn−1∏i=1

(1 +

2|ai|Di|z|Di

)< 2M−dn = s. (3.2.41)

ùL² f(C \ DM) ⊂ Ds. l éz n ≥ 2, ·k f−1(As,M) ⊂ As,M .

§3.2. .: ±9V/ 29

0

0 ∞

∞

0∞

0∞

< s s |a1| |an−1|1

K> K

< s s |a1| |an/2| |an−1|1

K> K

VnV1

Un−1U1

VnV1

Un−1U1

ã 3.1 f1,d1,··· ,dn N'X«¿ã,lm©OéAX n´ÛêÚóê

/. Ê(L.:Ú., ã¡.àêâLCqI.

½n 3.1.1 y². ·=y²¹ p = 1 ÏâÚn 3.2.4(2) aq?Ø±

ûy² p = 0. üå, ·Ó^ f L« f1,d1,··· ,dn . - Ui f−1(D) ¹ ai

ëÏ©|. XJ n − i ´Ûê, K D = Ds; XJ n − i ´óê, K D = C \ DK . âÚ

n 3.2.4(1), .:8 CPi ⊂ Ui Ui ´¹ Ai «. d, âÚn

3.2.4(1) k f(Ui) ∩ f(Ui+1) = ∅, Ù¥ 1 ≤ i < n − 2, u´ Ui ∩ Ui+1 = ∅. ùL²éuØÓ i, j k Ui ∩ Uj = ∅. b Ui k mi >.ëÏ©|. Ï Ui ¥Tk Di .:

f : Ui → D ŃÝu Di ©|CX, l â Riemann-Hurwitz úª

χUi = 2 −mi = DiχD − Di = 0, Ù¥ χ L«î.«5ê. ùL² mi = 2. Ïdéz

1 ≤ i ≤ n− 1, Ui ´7X:.

é 1 ≤ i ≤ n−2,- Vi+1 UiÚ Ui+1m«.N´wÑ f : Vi+1 → As,K ´

NÝu di+1 CXN. 5¿ As,K ´ëÏØ¹., f−1(As,K)

zëÏ©|´. ùL²3ü V1 Ú Vn, ©Oá3 0 Ú U1, Un−1 Ú

∞ m, ¦ f : V1, Vn → As,K ŃÝ©Ou d1 Ú dn CXN. ¯¢þ, f 3

∂U1 Ú ∂Un−1 þ©OäkNÝ d1 Ú dn V1 Ú Vn ¥Ø¹.: (ã 3.1).

¼ê f Julia 8´ J =⋂k≥0 f

−k(As,K). âE, éz 0 ≤ j ≤ n, f−1 :

As,K → Vj z_©|Ñ´/. ùL² J zëÏ©|Ñí@3 0 Ú ∞ m;8. Ï J ©|ØU´ü: f ´V, â [59] ¥½n 1.2, J

zëÏ©|Ñ´^ Jordan (¯¢þ´[). ¼ê f 3Ù Julia8©|þÄå


XÚÓu n ÎÒ Σn := 0, 1, · · · , n− 1N ü>²£ (one-side shift). AO/, J Ó

u Σn × S1, ù´·¤I Cantor ± (~Xã 1.1). ùÒ¤½n 3.1.1

y².

5 3.2.5. du f ´V, XJ·éëê ai 6Ä, f Julia 8E,´

Cantor ±, Ù¥ 1 ≤ i ≤ n− 1.

½n 3.2.6. éu n ≥ 3, ½n 3.1.1 ¥@ÀJ ai ¦ fp,d1,··· ,dn Julia 8´

Cantor ±, K fp,d1,··· ,dn Ú McMullen N3§éA Julia 8þØUÿÀÝ.

y². Ï fp,d1,··· ,dn 3 Julia©|þÄåXÚÝu nÎÒ Σn := 0, 1, · · · , n−1N ü>²£, McMullen N3Ù Julia ©|þÄåXÚÝu 2 ÎÒ Σ2 :=

0, 1N ü>²£. ùL² n ≥ 3, fp,d1,··· ,dn Ú McMullenN3§éA Julia

8þØUÿÀÝ.

§3.3 Cantor ±. Julia 8mÿÀÝ

ù!¥, ·y²éu?¿½ Julia 8 Cantor ±kn¼ê, Ñ±3

(3.1.2) ¥éN fp,d1,··· ,dn ¦ùükn¼ê3§g Julia 8þ´ÿÀ

Ý.

Ún 3.3.1. XJ f ´ Julia 8 Cantor ±kn¼ê. K J(f) þØ¹

.:.

y². b f 3 Julia ©| J0 ¹ f ê d .: c0. K f 3

c0 NCØ´é. ®² f(J0) ´¹ f(c0) Julia ©| [6, Ún

5.7.2]. ÀJ f(c0)ÿÀ U ¦ U ∩ f(J0)´^ü f−1(U)

¹ c0 ©|´ d + 1 é 1 /N÷ U . 5¿ f−1(U ∩ f(J0)) ¹ c0 ©| J ′

´ëÏ¹3 J0 ¥. , , J ′ äk(G(Ï Ø´^ü. ùb J(f)

´ Cantor ±gñ.

·¡;8 A ⊂ C ©l 0 Ú ∞ XJ 0 Ú ∞ ©Oá3 C \A üØÓ©|¥. - A Ú B üØ;8¿©O©l 0 Ú ∞. XJ A á3 C \B ¹ 0

©|p, KP A ≺ B. - A ©l 0 Ú ∞ , ·^ ∂−A Ú ∂+A L« A

ü>.©|¿¦ ∂−A ≺ ∂+A.

½n 3.3.2. f ´ Julia 8 Cantor ±kn¼ê. K3 p ∈ 0, 1,n ≥ 2 ±9÷v

∑ni=1

1di< 1 ê d1, · · · , dn ¦ f Ú fp,d1,··· ,dn 3§ Julia 8

þ´ÿÀÝ.

§3.3. CANTOR ±. JULIA 8mÿÀÝ 31

y². f Julia 8 J(f) ´ Cantor ±, KâÚn 3.3.1 , f z

±Ï Fatou ©|o´á5o´Ô. ·=[/y²á5 (éA f V)

¹ |^ Cui ó5)ºÔ/ [16].

e5, ·b f ´V. âb f Fatou 8düüëÏ Fatou

©|ÚÙ§©||¤. - D Ú A ©O f üëÏÚëÏ Fatou ©|8

Ü. ·äó f(D) ⊂ D éz A ∈ A Ñ3ê k ≥ 1 ¦ f k(A) ∈ D. (

Ø f(D) ⊂ D ´w,, ù´ÏüëÏ Fatou ©|E,´üëÏ. XJ

f(A1) = A2, Ù¥ A1, A2 ∈ A, Kâ Riemann-Hurwitz úª A1 Ø¹.:. ùL

²z A ∈ AØU´±ÏÏz±ÏáÚ Fatou©|Ì7¹.:.

,¡, â Sullivan ½n, kn¼êvki Fatou . ùäóÒy².

3 Mobius C¿Âe, ·±b 0 Ú ∞ ©Oáu f üüë

Ï Fatou ©| D0 Ú D∞. =, D = D0, D∞. Ï f(D) ⊂ D, Ø5, ·b

f(D0) = D0 Ú f(D∞) = D∞. - f−1(D0) = D0 ∪A1 ∪ · · · ∪Am, Ù¥ A1, · · · , Am m

©l 0 Ú ∞ ¿¦éz 1 ≤ i ≤ m− 1 k Ai ≺ Ai+1. N´wÑ m ≥ 1. Ä

K, D0 ´ØC, K J(f) = ∂D0, ùÚb J(f) ´ Cantor ±gñ.

b deg(f |D0 : D0 → D0) = d1, deg(f |∂−Ai : ∂−Ai → ∂D0) = d2i é 1 ≤ i ≤ m,

k deg(f |∂+Ai : ∂+Ai → ∂D0) = d2i+1. ùL² deg(f) =∑2m+1

j=1 dj. - W1 0u D0

A1 m Wi 0u Ai−1 Ú Ai m«, Ù¥ 2 ≤ i ≤ m. ·k

f(Wi) = C \D0 deg(f |Wi: Wi → C \D0) = d2i−1 + d2i. ùL²3 Fatou

©| Bi ( Wi ¦ f(Bi) = D∞. XJ3 B′i 6= Bi ¦ B′i ( Wi f(B′i) = D∞, K

73 f−1(D0) ©|á3 Wi ¥, ùb A1 ∪ · · · ∪ Am ´ f−1(D0) ¤k

©|¤8Ügñ. Ï TÐ3 Fatou ©| Bi ( Wi ¦ f(Bi) = D∞

deg(f |Bi : Bi → D∞) = d2i−1 + d2i. aq?Ø±ûy² D∞ ´ f−1(D∞) á

3 C \Am Ã.©|±N÷ D∞ ©|. Ïd, f−1(D∞) = B1 ∪ · · · ∪Bm ∪D∞d deg(f) =

∑2m+1j=1 dj deg(f |D∞) = d2m+1. P C \ (D0 ∪D∞) E. c f−1(E)

d 2m+ 1 ©| E1, · · · , E2m+1 ¤, §÷véu 1 ≤ i ≤ 2m k Ei ≺ Ei+1. N

f : Ei → E ŃÝ di ÃÜCX, Ù¥ 1 ≤ i ≤ 2m+ 1 (ã 3.2).

- n = 2m+1 p = 1. Ïz EiÑ/¹3 E¥mod(Ei) = mod(E)/di,

l â Grotzsch Øª∑n

i=1 1/di < 1. e5, ·òE[/N

φ : C→ C ò f Julia 8þÄåXÚÝ f1,d1,··· ,dn Julia 8þ.

üå, ·P f1,d1,··· ,dn F . 5¿ F (0) = 0 Ú F (∞) =∞. 3üüëÏ

Fatou ©| D′0 Ú D′∞, ¦§3 F êØC 0 ∈ D′0 Ú ∞ ∈ D′∞. â½n

3.1.1y²,· F−1(D′0) = D′0∪A′1∪· · ·∪A′m,Ù¥ A′1, · · · , A′m m©l 0Ú

∞ ¦éz 1 ≤ i ≤ m− 1 k A′i ≺ A′i+1. d, deg(F |D′0 : D′0 → D′0) = d1,


D∞

E2m+1

Am

∞

D0 0

E2m

E2

E3

E1

B1

E B2

A1

D0 0

D∞ ∞

d1 d1

d2

d2

d3

d3

d2m+1

d2m+1

d2m

d2m

W2

W1

ã 3.2 f N'X«¿ã,Ù¥ di, 1 ≤ i ≤ 2m+ 1L« f 3ã¥¤« Fatou

©|>.þNÝ.

deg(F |∂−A′i : ∂−A′i → ∂D′0) = d2i, éu 1 ≤ i ≤ m k deg(F |∂+A′i : ∂+A

′i → ∂D′0) =

d2i+1. - W ′1 0u D′0 ÚA

′1 m« W ′

i 0u A′i−1 Ú A′i m

«, Ù¥ 2 ≤ i ≤ m. KTÐ3 Fatou ©| B′i ( W ′i ¦ F (B′i) = D′∞

deg(F |B′i : B′i → D′∞) = d2i−1 + d2i. u´·k F−1(D′∞) = B′1 ∪ · · · ∪ B′m ∪ D′∞ deg(F |D′∞) = d2m+1. aq/, - E ′ := C \ (D′0 ∪D′∞). 3 F−1(E ′) 2m + 1

©| E ′1, · · · , E ′2m+1 ¦éu 1 ≤ i ≤ 2m k E ′i ≺ E ′i+1. N F : E ′i → E ′ Ń

Ý di CX, Ù¥ 1 ≤ i ≤ 2m+ 1.

â[/Ãâ, ± ∂D0, ∂D∞, ∂D′0, ∂D

′∞ Ú§cÑ´[ û

k½~ê. ùL²3[/N φ0 : C → C ¦ φ0(D0) = D′0

φ0(D∞) = D′∞, Ï φ0(∂D0) = ∂D′0 φ0(∂D∞) = ∂D′∞. d, À φ0 ¦3

∂D0 ∪ ∂D∞ þk φ0 f = F φ0.

y3·E φ0 : E → E ′ J, φE1 : E1 → E ′1 Xe. éz z ∈ E1 \ ∂−E1,

·ÀJü γ : [0, 1] → E ¦ γ(1) = f(z) γ(0) = w ∈ ∂−E. Ï

f : E1 → E ĆXN, 3 γ J, γ : [0, 1] → E1 ¦ γ(1) = z

w := γ(0) ∈ ∂−E1. aq/,Ï F : E ′1 → E ′ĆXN,3 φ0(γ) : [0, 1]→ E ′

J, α : [0, 1]→ E ′1¦ α(0) = φ0(w),ù´Ï3 ∂D0 = ∂−E1þ, φ0f = Fφ0.

½Â φE1(z) := α(1). 5¿ f, F ÑŃÝ d1 XCXN φ0 : E → E ′ ´[

/, u´3 E1 þk φ0 f = F φE1 φE1 : E1 → E ′1 ´[/. y3·é

φ1 : C→ C 3/®²k½Â: φ1|D0= φ0|D0

, φ1|D∞ = φ0|D∞ φ1|E1 = φE1 . ,

, 3 ∂E1 þk φ1 f = F φ1. aq/, 3[/N φE2m+1 : E2m+1 → E ′2m+1,

§3.3. CANTOR ±. JULIA 8mÿÀÝ 33

φ0 : E → E ′ J,¦3 E2m+1 þk φ0 f = F φE2m+1 . ½Â φ1|E2m+1 = φE2m+1 .

K, 3 ∂E2m+1 þk φ1 f = F φ1.

ØÓu E1 Ú E2m+1 ¹, éu 2 ≤ i ≤ 2m, φ0 : E → E ′ J, φEi : Ei → E ′i

3´Ø. ·Äk`² φEi 35. Ø5, b i ´óê. Ï f :

∂−Ei → ∂D∞ Ú F : ∂−E′i → ∂D′∞ ÑŃÝ di CXN, 3 φ0 : ∂D∞ → ∂D′∞

J, (Ø) φEi : ∂−Ei → ∂−E′i ¦3 ∂−Ei þk φ0 f = F φEi . |^½

Â φE1 Ó, 3l Ei E ′i φ0 : E → E ′ J,, ·ÓP¤ φEi ,

¦3 Ei þk φ0 f = F φEi . 5¿ φEi : Ei → E ′i ´[/. ½Â φ1|Ei = φEi .

K, 3⋃2m+1i=1 Ei k φ0 f = F φ1 3

⋃2m+1i=1 ∂Ei þk φ1 f = F φ1.

ÚÎÒ, é 1 ≤ i ≤ m, - D2i−1 := Bi Ú D2i := Ai. Ké 1 ≤ i < j ≤ 2m, ·

k Di ≺ Dj. ·I3⋃2mi=1Di þ½Â φ1. éz Di,Ù¥ 1 ≤ i ≤ 2m,§ü>.

©| ∂+Ei Ú ∂−Ei+1 Ñ´[. Ï φEi Ú φEi+1Ñ´[/N, N φ1|∂+Ei∪∂−Ei+1

k¦ φDi(Di) = D′i [/*Ü φDi : Di → D′i. y3·[/N

φ1 : C→ C, §½Â φ1|Ei := φEi , φ1|Dj = φDj φ1|D0∪D∞ = φ0, Ù¥ 1 ≤ i ≤ 2m+ 1

1 ≤ j ≤ 2m.

e5, ·½Â φ2. Äk, é j ∈ 0, 1, · · · , 2m,∞, - φ2|Dj = φ1. ,·±·

ªJ, φ1 : E → E ′ φ2 : Ei → E ′i, Ù¥ 1 ≤ i ≤ 2m+ 1. , ·u

N φ2 : C→ C ëY5. e¡´[`². y φ2 3 D0 ∪E1 þëY5, ·

Ik φ2|∂−E1 = φ1. u´é φ1 : E → E ′ 5`, =3«J,ª φ2 : E1 → E ′1.

5¿ φ2|D1 = φ1, ·Iu φ2 3>. ∂+E1 þëY5. ¯¢þ, φ0|E Ú φ1|E *dÓÔ φ1|∂E = φ0|∂E, ùL² φ2|∂+E1 = φ1|∂+E1 , ù´Ï φ2|∂−E1 = φ1|∂−E1 . ù`²

φ2 3 ∂+E1 þ´ëY. aq/, é 2 ≤ i ≤ 2m+ 1, ·±J, φ1 : E → E ′ 5

φ2 : Ei → E ′i y φ2 ëY5. nþ, N φ2 : C → C ÷v (1) φ2 ´[/Ù

û K(φ2) = K(φ1); (2) φ2|f−1(D0∪D∞) = φ1; (3) 3⋃2m+1i=1 Ei þk φ1 f = F φ2

Ïd3 f−2(∂D0 ∪ ∂D∞) þk φ2 f = F φ2.

bé, k ≥ 1 ·®² φk, K φk+1 daqu φ1 φ2

½Â. 8B/, ·±[/N φkk≥0 ¦ (1) éu k ≥ 1 k K(φk) =

K(φ1) ≥ K(φ0); (2) é z ∈ f−k(D0 ∪D∞) k φk+1(z) = φk(z); (3) 3 f−k(∂D0 ∪ ∂D∞)

þk φk f = F φk. ù¿X φkk≥0 ´5x. À φkk≥0 Âñ

f, Ù4·P φ∞, K φ∞ ´[/N3⋃k≥0 f

−k(∂D0 ∪ ∂D∞) þk

φ∞ f = F φ∞. d, K(φ∞) ≤ K(φ1). Ï φ∞ ´ëY, φ∞ f = F φ∞ 3⋃k≥0 f

−k(∂D0 ∪ ∂D∞) 4þ¤á, ù4Ò´ f Julia 8. Ïd φ = φ∞ ´·

é[/N, §ò f Julia 8þÄåXÚÝ F þ. ùÒ(å¹

f(D0) = D0 Ú f(D∞) = D∞ y².


Ù§n«¹: (1) f(D0) = D∞, f(D∞) = D∞; (2) f(D0) = D∞, f(D∞) = D0; Ú

(3) f(D0) = D0, f(D∞) = D0 ±aq/y².

XJ D0 Ú D∞ ¥k½ü©|´Ô, K3 f 6Ä fε ¦ fε ´

V3éA Julia 8þ, fε Ú f ´ÿÀÝ [16]. duâþ¡y² fε

3 (3.1.2) ¥´k “.” éA, ùL² f 3 (3.1.2) ¥k.. ùÒ(å

½n 3.3.2 Ú 3.1.2 y².

â½n 3.3.2 3V¹ey², ·á=kXeíØ.

íØ 3.3.3. XJëê ai ½n 3.1.1 ¥@ÀJ, Ù¥ 1 ≤ i ≤ n− 1, K fp,d1,··· ,dn

z Julia ©|Ñ´[.

§3.4 Julia 8þÿÀd Cantor ±kn¼êÝa

ù!8ÓNÝ d ≥ 5 Cantor ±kn¼ê Julia 8þØÓ

ÿÀÝaê.

- RatCCd ¤kNÝu d ≥ 2 Julia 8 Cantor ±kn¼ê|¤8

Ü. â½n 3.1.2, ·éuzkn¼ê f ∈ RatCCd , 3 (n + 1)-|

(p, d1, · · · , dn) Ú f éA, Ù¥ p ∈ 0, 1, n ≥ 2,∑n

i=1 di = d ∑n

i=11di< 1. u´·

ke¡íØ.

íØ 3.4.1. XJ d ≤ 4, K RatCCd 8.

y². XJ d ≤ 4, K§|∑n

i=1 di = d∑ni=1

1di< 1

n ≥ 2, di ≥ 2 1 ≤ i ≤ n

Ã).

XJ f ∈ RatCCd , Ù¥ d ≥ 5, ½Â f I8

Index(f) := (p, d1, · · · , dn) : f Ú fp,d1,··· ,dn 3§ Julia 8þÿÀÝ.

8Ü Index(f) ¥¡*dd. ½kn¼ê f ∈ RatCCd ±kõ

I, ù´du f Ú§ “.” fp,d1,··· ,dn üëÏ Fatou ©|UkõéA. ·^

]A 5L«k8 A ¥ê.

§3.4. JULIA 8þÿÀd CANTOR ±kn¼êÝa 35


Ýaê

N(d) =∑n≥2

](d1, · · · , dn) :n∑i=1

di = d n∑i=1

1

di< 1

+∑n≥3

](d1, · · · , dn) :n∑i=1

di = d,

n∑i=1

1

di< 1, (d1, · · · , dn) = (dn, · · · , d1) n Ûê .

y². é (3.1.2) ¥z, ®² 0 Ú ∞ áu fp,d1,··· ,dn Fatou 8. - D0

Ú D∞ ©O¹ 0 Ú ∞ üëÏ Fatou ©|. ùpk 4 «¹:

(1) XJ p = 1 n Ûê, K f(D0) = D0 f(D∞) = D∞;

(2) XJ p = 1 n óê, K f(D0) = D∞ f(D∞) = D∞;

(3) XJ p = 0 n Ûê, K f(D0) = D∞ f(D∞) = D0;

(4) XJ p = 0 n óê, K f(D0) = D0 f(D∞) = D0.

·òdþ¡ 4«¹íÑ N(d)Oúª. éu (1)Ú (3) nÛê¹,

(p, d1, · · · , dn) du (p, dn, · · · , d1) (p, d1, · · · , dn) Ødu (1− p, d1, · · · , dn), Ù¥

p ∈ 0, 1. éu (2) Ú (4) n óê¹, (p, d1, · · · , dn) du (1− p, dn, · · · , d1)

XJ (d1, · · · , dn) 6= (d′1, · · · , d′n), Kz (p, d1, · · · , dn) Ødu (p, d′1, · · · , d′n), Ù

¥ p ∈ 0, 1. w,, XJ n + m Ûê, (p, d1, · · · , dn) Ødu (p′, d′1, · · · , d′m), Ù¥

p, p′ ∈ 0, 1.éz½ d ≥ 5 p ∈ 0, 1, ½Â

Np1 = (p, d1, · · · , dn) :

n∑i=1

di = d,n∑i=1

1

di< 1, (d1, · · · , dn) = (dn, · · · , d1) n Ûê,

Np2 = (p, d1, · · · , dn) :

n∑i=1

di = d,

n∑i=1

1

di< 1, (d1, · · · , dn) 6= (dn, · · · , d1) n Ûê,

Np3 = (p, d1, · · · , dn) :

n∑i=1

di = d,n∑i=1

1

di< 1 n óê.

5¿ ]N01 = ]N1

1 ]N02 = ]N1

2 , l NÝ d ≥ 5 Cantor ±kn¼ê Julia 8

þØÓÿÀÝaê

N(d) = (]N11 + ]N1

2/2) + (]N01 + ]N0

2/2) + ]N03 = (]N0

1 + ]N02 + ]N0

3 ) + ]N01 ,

ù½n 3.4.2 Òy².

éuz½ d ≥ 5, â½n 3.4.2 ±ÏLqÞéN´/OÑ N(d).

e!L 3.1 Ñ 5 ≤ d ≤ 36 N(d). d, ·ò3e!Ñ N(d)

þe.O.


§3.5 Cantor ±kn¼ê Julia 8þÿÀÝaêO

éz n ≥ 1,- P n ⊂ Rnäk n+1º: (0, 0, · · · , 0), (1, 0, · · · , 0), (0, 1, · · · , 0)

Ú (0, 0, · · · , 1) ü I. =,

P n = (x1, · · · , xn) :n∑i=1

xi < 1 xi > 0, Ù¥ 1 ≤ i ≤ n. (3.5.1)

®² P n î¼NÈ´ Vol(P n) = 1/n!. - a ∈ R l > 0. ½Â#I

P na,l = (a+ lx1, · · · , a+ lxn) : (x1, · · · , xn) ∈ P n. (3.5.2)

N´wÑ P na,l NÈ

Vol(P na,l) = ln/n!. (3.5.3)

e¡Ún±â²þØª.

Ún 3.5.1. (1) - x, y, α, β 4 ¢ê, K

1

x+

1

y≥ (α + β)2

α2x+ β2y. (3.5.4)

d, Ò¤á= αx = βy.

(2) - xi, 1 ≤ i ≤ n n ¢ê, K

n∑i=1

1

xi≥ n2∑n

i=1 xi. (3.5.5)

d, Ò¤á=éz 1 ≤ i, j ≤ n k xi = xj.

½ê d ≥ 5. éz n ≥ 1, ½Â Rn+ f8

Dn = (x1, · · · , xn) ∈ Rn+ :

n∑i=1

1

xi+

1

d−∑n

i=1 xi< 1, xi > 0 d−

n∑i=1

xi > 0. (3.5.6)

Ún 3.5.2. 8Ü Dn = (n+ 1)2 ≥ d.

y². |^Ún 3.5.1, ·k

n∑i=1

1

xi+

1

d−∑n

i=1 xi≥ 1

(∑n

i=1 xi)/n2

+1

d−∑n

i=1 xi≥ (n+ 1)2

d. (3.5.7)

ùL²XJ (n+ 1)2 ≥ d, K Dn = ∅.b (n + 1)2 ≤ d − 1. ª (3.5.7) ¥Ò¤á=éz 1 ≤ i ≤ n k

xi = dn+1

. AO/, k ( dn+1

, · · · , dn+1

) ∈ Dn. Úny..

§3.5. CANTOR ±kn¼ê JULIA 8þÿÀÝaêO 37

e5, 3vkAO`²¹e, ù!¥·o´b

(n+ 1)2 ≤ d− 1. (3.5.8)

Ún 3.5.3. (1) 8Ü Dn ´à.

(2) 8Ü Dn ¹I P nan,bn−an, Ù¥

an =d+ n2 − 1−

√∆

2n, bn =

d− n2 + 1 +√

∆

2(3.5.9)

∆ = (d− n2 − 1)2 − 4n2.

y². (1) - (x1, · · · , xn), (y1, · · · , yn) ∈ Dn t ∈ [0, 1], Ï x 7→ 1/x 3 (0,∞) þ

´à, ·k

n∑i=1

1

txi + (1− t)yi+

1

d−∑n

i=1(txi + (1− t)yi)

=n∑i=1

1

txi + (1− t)yi+

1

t(d−∑n

i=1 xi) + (1− t)(d−∑n

i=1 yi)

<n∑i=1

t

xi+

n∑i=1

1− tyi

+t

d−∑n

i=1 xi+

1− td−

∑ni=1 yi

< 1.

ùL² Dn ´à.

(2) 5¿e¡ n+ 1 :

(an, an, · · · , an), (bn, an, · · · , an), (an, bn, · · · , an), · · · , (an, an, · · · , bn)

Ñá3 Dn >.þ. Ï Dn ´à,ùL² Dn ¹I P nan,bn−an . ã 3.3`² n = 2

¹.

- (x1, · · · , xn) ∈ Rn. XJz xi Ñ´ê, K¡ù:´ê:, Ù¥

1 ≤ i ≤ n.

Ún 3.5.4. - a ∈ R l ≥ n. K P na,l ¥ê:ê Num(P n

a,l) ÷v

Num(P na,l) ≥

(l − n)n

n!. (3.5.10)

y². y²©¤ a ∈ Z Ú a 6∈ Z ü«¹?Ø. ·3dÑ[!.


O an nan bnx1

xn

DnPnan,bn−an

x1 = · · · = xn

ã 3.3 « Dn 3 Rn+ ¥ , Ù¥ n = 2.

íØ 3.5.5. bn − an ≥ n , Dn ¥ê:ê Num(Dn) ÷v

Num(Dn) ≥ (bn − an − n)n

n!. (3.5.11)

Ù¥^ bn − an ≥ n du

1 + 3n+ (1 + n)√

1 + 4n2

2≤ d. (3.5.12)

y². ª (3.5.11) Ún 3.5.4 íØ. ·=y² (3.5.12). â (3.5.9), bn −an ≥ n du

d− n2 + 1 +√

∆

2− d+ n2 − 1−

√∆

2n− n ≥ 0. (3.5.13)

O,

d ≥ 1 + 3n+ (1 + n)√

1 + 4n2

2½ö d ≤ 1 + 3n− (1 + n)

√1 + 4n2

2. (3.5.14)

5¿â (3.5.8) k d ≥ (n+ 1)2 + 1. ùL² (3.5.12) ¤á.

§ ∑n+1

i=1 di = d∑n+1i=1

1di< 1

n ≥ 1, di ≥ 2 1 ≤ i ≤ n+ 1

ê) (d1, · · · , dn, dn+1) ê S(n+ 1) u Dn ¥ê:ê Num(Dn).

- [a] ¢ê a êÜ©. âíØ 3.5.5, ·k

§3.5. CANTOR ±kn¼ê JULIA 8þÿÀÝaêO 39

½n 3.5.6.∑n≥1

S(n+ 1) ≥n0∑n=1

(bn − an − n)n

n!≥

n0∑n=1

(d− n2 − 3n− 1)n

n!, (3.5.15)

Ù¥ n0 = [(√

4d+ 5− 3)/2].

y². â (3.5.9), ·k

√∆ =

√(d− n2 − 1)2 − 4n2 ≥ d− (n+ 1)2.

Ïd, ·k

bn − an − n =d(n− 1) + n+ 1− 3n2 − n3 + (n+ 1)

√∆

2n

≥ d− n2 − 3n− 1.

(3.5.16)

5¿ n ≥ 1 ,

1 + 3n+ (1 + n)√

1 + 4n2

2< n2 + 3n+ 1 (n+ 1)2 < n2 + 3n+ 1. (3.5.17)

Ïd, n ≤ (√

4d+ 5 − 3)/2 , bn − an − n ≥ 0. ª (3.5.15) >Øª´díØ

3.5.12 Ñ(Ø.

â (3.5.16), ·k

n0∑n=1

(bn − an − n)n

n!≥

n0∑n=1

(d− n2 − 3n− 1)n

n!,

Ù¥ n0 = [(√

4d+ 5− 3)/2].

5 3.5.7. XJ·Ä¤k¦∑n+1

i=1 di = d Ñ^∑n+1

i=1 1/di < 1

Uê| (d1, · · · , dn, dn+1), Ù¥ di ≥ 2, N´wÑ∑n≥1

S(n+ 1) ≤n1∑n=1

C nd−n−2, (3.5.18)

Ù¥ n1 = [√d− 1]− 1 C n

m = m!/(n!(m− n)!) |Üê.


Ýaê N(d) ÷v

n0∑n=1

(d− n2 − 3n− 1)n

n!≤ N(d) < 2

n1∑n=1

C nd−n−2, (3.5.19)

Ù¥ n0 = [(√

4d+ 5− 3)/2] n1 = [√d− 1]− 1.


y². â½n 3.5.6, 5 3.5.7 Ú½n 3.4.2 ¥ã N(d) úª½n 3.5.8

¤á.

¯¢þ, â½n 3.5.6 Ú½n 3.4.2, N(d) e.Úþ.±de¡°(ê

O:

n0∑n=1

(bn − an − n)n

n!Ú

∑1 ≤ n ≤ n1

n is odd

C nd−n−2 + 2

∑1 ≤ n ≤ n1

n is even

C nd−n−2. (3.5.20)

·Ñe¡L5' 5 ≤ d ≤ 36 , N(d) °(Úd (3.5.20) ª¤(

½ N(d) þe..

d 5 6 7 8 9 10 11 12

LB(d) 2 3 4 5 6 7 10 14

N(d) 2 3 4 5 6 11 22 37

UB(d) 2 3 4 5 6 37 50 65

d 13 14 15 16 17 18 19 20

LB(d) 19 25 33 41 50 61 75 93

N(d) 46 57 68 81 110 159 228 290

UB(d) 82 101 122 145 390 483 590 712

d 21 22 23 24 25 26 27 28

LB(d) 117 147 186 233 290 358 438 531

N(d) 410 519 716 872 1070 1323 1722 2258

UB(d) 850 1005 1178 1370 1582 11505 14040 16978

d 29 30 31 32 33 34 35 36

LB(d) 642 774 935 1130 1370 1662 2018 2450

N(d) 3066 4227 5566 6950 8604 10483 12916 15838

UB(d) 20360 24229 28630 33610 39218 45505 52524 60330

L 3.1 NÝ d ≥ 5 Cantor ±kn¼ê Julia 8þØÓÿÀ

Ýaê N(d) ±9d (3.5.20) ¤ N(d) þe. UB(d) Ú LB(d), Ù¥

5 ≤ d ≤ 36.

§3.6. JULIA 8 CANTOR ±Vkn¼ê 41

§3.6 Julia 8 Cantor ±Vkn¼ê

m,n ≥ 2 ü÷v 1/m+ 1/n < 1 ê, -

Pλ(z) =1n((1 + z)n − 1) + λm+nzm+n

1− λm+nzm+n, (3.6.1)

Ù¥ λ ∈ C∗ = C \ 0. λ ¿©, Pλ ±w¤é¡Ôõª6Ä:

P (z) =(1 + z)n − 1

n. (3.6.2)

5¿ P 3:k¦fu 1 ÔØÄ: −1 ´Ùêu n − 1 .:.

ùL² P küëÏ Fatou ©|, Ù¥¤k:Ñ 0 ¤áÚ. AO/, P

Julia 8´äkÃ¡õk: Jordan .

λ éÿ, ·F"þ¡ã P ,5é Pλ Ó¤á. éw,,

Pλ Ú P m´3éõÉ. ¼ê Pλ NÝ´ m + n Pλ(∞) = −1. §k

2(m+ n)− 2 .:: m− 1 3 ∞, n− 1 ~C −1 e m+ n .:á

3 Tr0/|λ| NC, Ù¥ r0 = m+n√n/m (Ún 3.6.3). ¯¢þ, 3ù!"·ò¬w

Pλ ±w¤´ “Ô” McMullen N, ù´Ï3éA Julia 8þ Pλ Ý

u McMullen N fλ.

Äk, ·y²éz n ≥ 2, P ØÄÔ Fatou ©|¹î¼ D(−34, 3

4)

XJ λ ¿©, K Pλ ò D(−34, 3

4) N\gC.

Ún 3.6.1. (1) éz n ≥ 2, P (D(−34, 3

4)) ⊂ D(−3

4, 3

4) ∪ 0.

(2) XJ 0 < |λ| < 1/(3n), K Pλ(D(−34, 3

4)) ⊂ D(−3

4, 3

4) ∪ 0. AO/, D(−3

4, 3

4) á

3 Pλ ¹ØÄ: 0 Ô Fatou ©|¥.

y². XJ z ∈ D(−34, 3

4), K |P (z) + 1/n| = |1 + z|n/n ≤ 1/n. AO/, ØÒ±

UÒ= z = 0. ùÒy² (1).

(2) y²ò©¤ü«¹: |z| é±9 |z| Ø´é. éz z = −34

+ 34eiθ ∈

∂D(−34, 3

4), Ù¥ −π < θ ≤ π, â (1), ·k |1 + P (z)| ≤ 5/2 |λz|m+n < 1/2, ù´

Ï |λ| < 1/(3n). ùL²

|Pλ(z)− P (z)| =

∣∣∣∣∣λm+nzm+n(1 + P (z))

1− λm+nzm+n

∣∣∣∣∣ ≤ 5|λz|m+n. (3.6.3)

Ï |z| = 34|1− eiθ| = 3

4|e−iθ/2 − eiθ/2| = 3

2| sin θ

2| ≤ 3

4|θ| |λ| < 1/(3n), ·k

|Pλ(z)− P (z)| ≤ 5 (|θ|/(4n))m+n. (3.6.4)


,¡, ÏXJ |θ| ≤ π2, K | sin θ| ≥ 2

π|θ|, ·k

|P (z) + 3/4| =∣∣∣∣(1

4+ 3

4eiθ)n − 1

n+

3

4

∣∣∣∣ ≤∣∣1

4+ 3

4eiθ∣∣n − 1

n+

3

4

=(1− 3

4sin2 θ

2)n/2 − 1

n+

3

4≤

(1− 3θ2

4π2 )n/2 − 1

n+

3

4.

(3.6.5)

XJ |θ| < 2π/n, K 3θ2

4π2 <2n. âÚn 3.2.1(3), ·k

|P (z) + 3/4| ≤ −n2· 3θ2

4π2

3n+

3

4=

3

4− θ2

8π2. (3.6.6)

Ïd, (Ü (3.6.4) Ú (3.6.6), XJ |θ| < 2π/n, K

|Pλ(z) + 3/4| ≤ |P (z) + 3/4|+ |Pλ(z)− P (z)| ≤ 3

4− θ2

8π2+ 5

(|θ|4n

)m+n

≤ 3/4. (3.6.7)

XJ 2π/n ≤ |θ| ≤ π, â (3.6.5) Ú (3.6.6), ·

|P (z) + 3/4| ≤ 3

4− 1

2n2. (3.6.8)

â (3.6.4) Ú (3.6.8), ±XJ 2π/n ≤ |θ| ≤ π, K

|Pλ(z) + 3/4| ≤ 3

4− 1

2n2+ 5 (

|θ|4n

)m+n < 3/4. (3.6.9)

ÃØN, ·®²y²éz z ∈ ∂D(−34, 3

4)k |Pλ(z) + 3

4| ≤ 3

4 |Pλ(z) + 3

4| = 3

4

= z = 0. y²..

§3.2L§, ·5(½ Pλ gd.: .ÏLO, Pλ k

. m+ 2n− 1 .:é¡§):

(1 + z)n−1 + λm+nzm+n−1(1 +m/n)[(1 + z)n + n− 1]− z(1 + z)n−1 = 0. (3.6.10)

Ún 3.6.2. XJ 0 < |λ| < 1/(3n), K Pλ k n − 1 .:á3 D(−1, |λ|) (D(−3

4, 3

4) ¥.

y². XJ |z + 1| ≤ |λ| < 13n

, K |z| · |1 + z|n−1 ≤ (1 + |λ|)|λ|n−1 < 1

(1 +m/n) |(1 + z)n + n− 1| ≤ (1 +m/n)(|λ|n + n− 1) < m+ n. (3.6.11)

ùL²XJ |z + 1| ≤ |λ|, K∣∣λm+nzm+n−1(1 +m/n)[(1 + z)n + n− 1]− z(1 + z)n−1∣∣

< |λ|n−1 · |λz|m−1|λ|2|z|n(m+ n+ 1) < |λ|n−1 · (2n)1−m(9n2)−1e1/3(m+ n+ 1)

< |λ|n−1 · (m+ n− 1)/(2n)m+1 < |λ|n−1.

(3.6.12)

â Rouche ½nÚ (3.6.10), ½ny.


- CP := wj = r0λ

exp(πi 2j−1m+n

) : 1 ≤ j ≤ m+ n mλm+nzm+n + n = 0 ":8,

Ù¥ r0 = m+n√n/m. Ï h(x) = x1/x, x > 0 3 x = e ? e1/e < 3/2, ·k

2/3 < 1/ m√m < r0 <

n√n < 3/2. (3.6.13)

e¡ÚnL« Pλ m+ n .:Ñ “~C” CP .

Ún 3.6.3. XJ0 < |λ| < 1/(2mn2), K (3.6.10) k) wj ¦ |wj − wj| <2(m+ n)/m, Ù¥ 1 ≤ j ≤ m+ n. d, wi = wj = i = j.

y². 3 (3.6.10) ü>ÓØ± (1 + z)n−1, ·k

1 + λm+nzm+n−1

(m

nz +

m+ n

n

(1 +

n− 1

(1 + z)n−1

))= 0. (3.6.14)

½ö, ¤k^/ª

n

mλm+n+ zm+n +

(m+ n)zm+n−1

m

(1 +

n− 1

(1 + z)n−1

)= 0. (3.6.15)

- Ω = z : |zm+n + nmλ−(m+n)| ≤ β|λ| · n

m|λ|−(m+n), Ù¥ β = 2(m+n)

mr0< 3(m+n)

m. X

J z ∈ Ω, K |λm+nzm+n + nm| < β|λ| · n

mâÚn 3.2.1(2) |z − wj| < βr0 é,

1 ≤ j ≤ 2n ¤á. XJ z ∈ Ω 0 < |λ| < 1/(2mn2), ·k

n− 1

|1 + z|n−1<

n− 1

((|λ|−1 − β)r0 − 1)n−1<

n− 1

(2m+1n2/3− 3− 2n/m)n−1<

1

15(3.6.16)

β|λ| ≤ 2(m+ n)

2mn2 ·mr0

<3

2mn

(1

m+

1

n

)<

1

4, ¤±

1 + β|λ|2(1− β|λ|)

<5

6. (3.6.17)

Ïd, XJ z ∈ Ω 0 < |λ| < 1/(2mn2), â (3.6.16) Ú (3.6.17), ·k∣∣∣∣(m+ n)zm+n−1

m

(1 +

n− 1

(1 + z)n−1

)∣∣∣∣ =m+ n

m|λ|m+n

∣∣∣∣λm+nzm+n

z

(1 +

n− 1

(1 + z)n−1

)∣∣∣∣<

m+ n

m|λ|m+n

(β|λ|+ 1)n/m

r0(1/|λ| − β)· 16

15=

nβ|λ|m|λ|m+n

1 + β|λ|2(1− β|λ|)

· 16

15<

nβ|λ|m|λ|m+n

.

(3.6.18)

é (3.6.15) A^ Rouche ½n,âÚn 3.2.1(2), Ún1Ü©y²Ò(å.

aqu (3.2.15) ?Ø, XJ 0 < |λ| < 1/(2mn2), ·k

(r0/|λ|) · sin(π/(m+ n))

2(m+ n)/m≥ mr0

(m+ n)2|λ|>

2m+1m

3(m/n+ 1)2> 1. (3.6.19)

ùL² wi = wj = i = j. Úny..


- CP := wj : 1 ≤ j ≤ m + n Pλ C± Tr0/|λ| m + n .:P

CV := Pλ(wj) : 1 ≤ j ≤ m+ n. - CP−1 Pλ C −1 n− 1 .: (Ún

3.6.2) P CV−1 = Pλ(z) : z ∈ CP−1.P T0 Pλ ¹:á5s Fatou ©|- U := D(−3

4, 3

4). âÚn

3.6.1(2) Ú 3.6.2, · CP−1 ∪ CV−1 ⊂ U ⊂ T0. du Pλ(∞) = −1, ùL²3 ∞¦ Pλ òùN −1 . - T∞ ¦ ∞ ∈ T∞ Fatou

©| U0, U∞ ©O¦ 0 ∈ U0 ∞ ∈ U∞ P−1λ (U) ëÏ©|. éw,, ·k

U ⊂ U0 ⊂ T0 U∞ ⊂ T∞.

Ún 3.6.4. XJ 0 < |λ| ≤ 1/(210mn3), K3¹ T1/|λ| ∪ CP A1, ¦

Pλ(A1) ⊂ U ′∞ ⊂ U∞, Ù¥ U ′∞ ´ ∞ .

y². âÚn 3.6.3 CP “A” þ!/á3± Tr0/|λ| þ¿ Pλ ¤kk

4:Ñá3± T1/|λ| þ. ½Â

A1 = z : 1/(2|λ|) < |z| < 2/|λ|. (3.6.20)

5¿r0

|λ|+

2(m+ n)

m<

3

2|λ|+ 2 +

2n

m<

2

|λ|(3.6.21)

r0

|λ|− 2(m+ n)

m>

2

3|λ|− 2− 2n

m>

1

2|λ|. (3.6.22)

âÚn 3.6.3, ·k T1/|λ| ∪ CP ⊂ A1. XJ z ∈ A1 |λ| ≤ 1210mn3 , K

|Pλ(z) + 1| ≥ (|z| − 1)n

n(|λz|m+n + 1)≥

( 12|λ| − 1)n

n(2m+n + 1)=

(1− 2|λ|)n

2nn|λ|n(2m+n + 1)>

2

|λ|1+ nm

+ 1.

(3.6.23)

¯¢þ,

(1− 2|λ|)n

2m+n + 1>

(1− 2210mn3 )n

2m+n + 1>

0.9

2m+n + 1>

1

2m+n+1+ 2nn|λ|n. (3.6.24)

Ï |λ| ≤ 1210mn3 , ùL² (3.6.23) ±deª:

2m+2n+2 n |λ|n ≤ |λ|1+n/m. (3.6.25)

y3·®²y²: XJ z ∈ A1 |λ| ≤ 1210mn3 , K |Pλ(z)| > 2

|λ|1+n/m .

,¡, XJ |z| ≥ 2|λ|1+n/m , K

|Pλ(z) + 1| ≤ (|z|+ 1)n + 1

|λz|m+n − 1≤ (1 + |z|−1)n + |z|−n

2m − |z|−n<

1

2. (3.6.26)


U ′∞

T∞

U∞

T0

U0−1

0

∞ ∞U ′∞

T∞

T0

−1

0

U

ã 3.4 Pλ N'X«¿ã. Ê(L«.:.

ùL² Pλ(z) ∈ D(−1, 12) ⊂ U . - U ′∞ P−1

λ (D(−1, 12)) ¹ z : |z| ≥ 2

|λ|1+n/m ©|, ù`² Pλ(A1) ⊂ U ′∞ ⊂ U∞ (ã 3.4).

½n 3.1.3 y². éz¦ 0 < |λ| ≤ 1/(210mn3) λ. - A := C \ (U ∪ U ′∞).

Ï Pλ : U ′∞ → D(−1, 12) ŃÝu m _;N, ùL² U ′∞ ´üëÏ A ´

.Ï P−1λ (U ′∞)¹ m+n.:¿ Pλ 3þ¡NÝ´ m+n,¤±

P−1λ (U ′∞) ´. ùL² P−1

λ (A) düØ I1 Ú I2 ¤ I1 ∪ I2 ⊂ A.

Pλ 3 I1 Ú I2 NÝ©O m Ú n.

e¡?Ø~aqu½n 3.1.1 y². Pλ Julia 8´ Jλ =⋂k≥0 P

−kλ (A).

âE,éu j = 1½ 2, P−1λ : A→ Ij ´/, l Jn ëÏ©|í@3 −1Ú∞

m;8. du Jn ëÏ©|ØU´ü:, [59] ¥½n 1.2 y²ÓéAÛk

kn¼ê¤á ( [59, §9] Ú [73]), · Jn zëÏ©|Ñ´ Jordan .

¼ê Pλ 3Ù Julia 8©|þÄåXÚÓu 2 ÎÒ Σ2 := 0, 1N ü>²£. A

O/, Jλ Óu Σ2 × S1, ù´ Cantor ±.

5: â½n 3.1.3 y²¿(Ü½n 3.3.2, · Pλ 3 Julia 8þÄåXÚ

Ýu, fλ Julia 8þÄåXÚ, Ù¥ fλ (3.1.1) ¥½Â McMullen N. Ï

d, ·±ò Pλ w¤´ “Ô” McMullen N, ÏOÒ´ gη

áÚ9Ùcd Pλ ÔÚAcO (ã §3.6).


ã 3.5 Pλ Julia 8, Ù¥ m = 3, n = 2 λ v¦ Jλ ´ Cantor

±. Pλ ¤k Fatou©|ÑSÙ¥äkÔØÄ: 1ØC

Ô Fatou ©|¥ (ã¥mÜ©¤« “ès”).

§3.7 õV Cantor ±. Julia 8~f

ù!, ·òEõVkn¼ê, ¦¦ Julia 8´ Cantor ±.

É½n 3.1.1 éu, éz n ≥ 2, ·½Â

Pn(z) = An(n+ 1)z(−1)n+1(n+1)

nzn+1 + 1

n−1∏i=1

(z2n+2 − b2n+2i )(−1)i−1

+Bn, (3.7.1)

Ù¥ |bi| = si, 0 < s ≤ 1/(25n2)

An =1

1 + (2n+ 2)Cn

n−1∏i=1

(1−b2n+2i )(−1)i , Bn =

(2n+ 2)Cn1 + (2n+ 2)Cn

Cn =n−1∑i=1

(−1)i−1b2n+2i

1− b2n+2i

.

(3.7.2)

Ún 3.7.1. (1) Pn(1) = 1 P ′n(1) = 1.

(2) 1− s2n+1/(n+ 1) < |An| < 1 + s2n+1/(n+ 1) |Bn| < s2n+1/(3n+ 3).

y². ÏLO± Pn(1) = 1. 5¿

Fn(z) :=zP ′n(z)

Pn(z)−Bn

=n−1∑i=1

(−1)i−1(2n+ 2)z2n+2

z2n+2 − b2n+2i

+ (−1)n+1(n+ 1)− n(n+ 1)zn+1

nzn+1 + 1.

(3.7.3)

§3.7. õV CANTOR ±. JULIA 8~f 47

ùL²

P ′n(1)

Pn(1)−Bn

= (2n+ 2)n−1∑i=1

(−1)i−1b2n+2i

1− b2n+2i

+ (2n+ 2)n−1∑i=1

(−1)i−1 + (−1)n+1(n+ 1)− n

= (2n+ 2)n−1∑i=1

(−1)i−1b2n+2i

1− b2n+2i

+ 1 := (2n+ 2)Cn + 1.

(3.7.4)

Ïd, ·k

P ′n(1) = (1−Bn)((2n+ 2)Cn + 1) = 1. (3.7.5)

ùL² 1 ´ Pn ÔØÄ:. ùÒ¤ (1) y².

éu (2),Ïé 1 ≤ i ≤ n−1k |1−b2n+2i |−1 ≤ 1+2|b1|2n+2 0 < s ≤ 1/(25n2) ≤

1/100, u´

(2n+ 2)|Cn| < (2n+ 2) (1 + 2|b1|2n+2)n−1∑i=1

|bi|2n+2 ≤ (2n+ 2)(1 + 2s2n+2)s2n+2

1− s2n+2<

s2n+1

4n+ 4.

(3.7.6)

·k

|Bn| =∣∣∣∣ (2n+ 2)Cn1 + (2n+ 2)Cn

∣∣∣∣ < (2n+ 2)|Cn|(1 + (4n+ 4)|Cn|) <s2n+1

3n+ 3(3.7.7)

|An| < (1+(4n+4)|Cn|)n−1∏i=1

(1+2|bi|2n+2) < (1+s2n+1

2n+ 2)(1+5s2n+2) < 1+

s2n+1

n+ 1. (3.7.8)

d, ·k

|An| > (1− (2n+ 2)|Cn|)n−1∏i=1

(1− |bi|2n+2) > (1− s2n+1

4n+ 4)(1− s2n+2

1− s2n+2) > 1− s2n+1

n+ 1.

(3.7.9)

Úny..

·Äk)ºeE¡g. éu n ≥ 2, ½Â Q(z) = (zn+1 + n)/(n+ 1)

ϕ(z) = 1/z, K Q(z) := ϕ Q ϕ−1(z) = (n + 1)zn+1/(nzn+1 + 1) ÷v: ∞ ´ Q ê

n .:, T.:ÔØÄ: 1 ¤áÚ. Ï bi1≤i≤n−1 Ñé, kn¼ê

Pn ±w´ Q 6Ä. An Ú Bn y 1 o´ Pn ÔØÄ: (Ú

n 3.7.1). ±y² Pn ò T|bi| N\ T0 ½ö T∞, ùûu i´Ûê


´óê, Ù¥ T0 Ú T∞ ©OL«¹ 0 Ú ∞ Fatou ©| (Ún 3.7.5). Fatou ©|

T∞ o´Ô, T0 ´á5½öS T∞ ûu n ´Ûê´óê. ½n 3.1.4

y²òæ^cü!aq.

XJ |z| ≤ 1, K |Q(z)| ≤ 1. ùL²éz n ≥ 2, Q ØÄÔ Fatou ©|¹

ü . Ïd, Q Ô Fatou ©|¹4ü Ü C \D. ¦+õª Q

®²6Ä Pn, ·E,ke¡Ún.

Ún 3.7.2. Pn(C \D) ⊂ (C \D) ∪ 1. AO/, C \D á3 Pn >.þ¹k

ÔØÄ: 1 Ô Fatou ©|p.

Ún 3.7.2 y²'æ, ·3e!.

Ún 3.7.3. é n ≥ 2 Ú 1 ≤ i ≤ n− 1, K

∑1≤j<i

(−1)j +∑

i<j≤n−1

(−1)j−1 +1 + (−1)n+1

2= 0. (3.7.10)

y². ?ØÄuL 3.2 ¥Ñ¹.

∑16j<i(−1)j

∑i<j6n−1(−1)j−1 (1 + (−1)n+1)/2

Ûê n Ûê i 0 −1 1

óê i −1 0 1

óê n Ûê i 0 0 0

óê i −1 1 0

L 3.2 Ún 3.7.3 y².

±c, ·ÄkéÑ Pn .: . 5¿ 0 Ú ∞ Ñ´ Pn ê n

.: Pn NÝ´ n2 + n. Pn 2(n2− 1) .:Ñ´§ Fn(z) = 0

) ( (3.7.3)).

éu 1 ≤ i ≤ n− 1, - CP i := wi,j = bi exp(πi 2j−12n+2

) : 1 ≤ j ≤ 2n+ 2 þ!©Ù3± T|bi| þ 2n+ 2 :8Ü. e¡ÚnaquÚn 3.2.3 Ú 3.6.3.

Ún 3.7.4. éz wi,j ∈ CP i, Ù¥ 1 ≤ i ≤ n − 1 1 ≤ j ≤ 2n + 2, 3

Fn(z) = 0 ) wi,j, ¦ |wi,j − wi,j| < sn+1/2|bi|. d, wi1,j1 = wi2,j2 =

(i1, j1) = (i2, j2).


y². 5¿ Fn(z) = 0 du

n−1∑i=1

(−1)i−1 z2n+2 + b2n+2

i

z2n+2 − b2n+2i

+1 + (−1)n+1

2− nzn+1

nzn+1 + 1= 0. (3.7.11)

3 (3.7.11) ü>Ó¦± z2n+2 − b2n+2i , Ù¥ 1 ≤ i ≤ n− 1, ·k

(−1)i−1(z2n+2 + b2n+2i ) + (z2n+2 − b2n+2

i )Gi(z) = 0, (3.7.12)

Ù¥

Gi(z) =∑

1≤j≤n−1, j 6=i

(−1)j−1z2n+2 + b2n+2

j

z2n+2 − b2n+2j

+1 + (−1)n+1

2− nzn+1

nzn+1 + 1. (3.7.13)

- Ωi = z : |z2n+2 + b2n+2i | ≤ sn+1/2|bi|2n+2, Ù¥ 1 ≤ i ≤ n − 1. XJ z ∈ Ωi,

KâÚn 3.2.1(2) k |z|n+1 ≤ (1 + sn+1/2)|bi|n+1 ≤ (1 + sn+1/2)sn+1. Ïd, d s ≤1/(25n2) ≤ 1/100 ∣∣∣∣ nzn+1

nzn+1 + 1

∣∣∣∣ ≤ n(1 + sn+1/2)sn+1

1− n(1 + sn+1/2)sn+1≤ (1 + 100−5/2)sn+1/2/5

1− (1 + 100−5/2)100−5/2/5< 0.3 sn+1/2.

éz z ∈ Ωi, XJ 1 ≤ j < i, ·k

|z/bj|2n+2 = |z/bi|2n+2|bi/bj|2n+2 < (1 + sn+1/2) s(2n+2)(i−j). (3.7.14)

XJ i < j ≤ n− 1, âÚn 3.2.1(2) 1ã, ·k

|bj/z|2n+2 = |bi/z|2n+2|bj/bi|2n+2 ≤ (1 + 2 · sn+1/2) s(2n+2)(j−i). (3.7.15)

â (3.7.14), (3.7.15) ÚÚn 3.7.3, ·k∣∣∣∣Gi(z) +nzn+1

nzn+1 + 1

∣∣∣∣=

∣∣∣∣∣ ∑1≤j<i

(−1)j1 + (z/bj)

2n+2

1− (z/bj)2n+2+

∑i<j≤n−1

(−1)j−1 1 + (bj/z)2n+2

1− (bj/z)2n+2+

1 + (−1)n+1

2

∣∣∣∣∣< 3 · (1 + 2 · sn+1/2)

(∑1≤j<i

s(2n+2)(i−j) +∑

i<j≤n−1

s(2n+2)(j−i)

)< 6 · (1 + 2 · sn+1/2)2 s2n+2.

(3.7.16)

(3.7.16) ¥1Øª¤á´ÏXJ x < 1/3 Kk 2x/(1 − x) ≤ 3x (ùp x ≤(1 + 2 · sn+1/2) s2n+2 < 10−10). Ïd·k

|Gi(z)| < 6 · (1 + 2 · sn+1/2)2 s2n+2 + 0.3 sn+1/2 < 0.4 sn+1/2. (3.7.17)


l , XJ z ∈ Ωi, K

|z2n+2 − b2n+2i | · |Gi(z)| < (2 + sn+1/2)|bi|2n+2 · 0.4 sn+1/2 < sn+1/2|bi|2n+2. (3.7.18)

â (3.7.12) Ú Rouche ½n, 3 Fn(z) = 0 ) wi,j ¦éz 1 ≤ j ≤ 2n+ 2

k wi,j ∈ Ωi. AO/,âÚn 3.2.1(2)1ã |wi,j− wi,j| < sn+1/2|bi|. d,

aqu (3.2.14) Ú (3.2.15) y, wi1,j1 = wi2,j2 = (i1, j1) = (i2, j2).

é 1 ≤ i ≤ n− 1, - CPi := wi,j : 1 ≤ j ≤ 2n+ 2 Pn “~C” ± T|bi|.:¤8Ü.

Ún 3.7.5. 3÷v An−1 ≺ · · · ≺ A1 n − 1 Ain−1i=1 ±9©O¹ 0

Ú ∞ üüëÏ« U0 Ú U∞, ¦

(1) U∞ ⊃ C \ D Pn(U∞) ⊂ U∞ ∪ 1;(2) Ai ⊃ T|bi| ∪ CPi, i Ûê Pn(Ai) ⊂ U0, i óê Pn(Ai) ⊂ U∞;

(3) n óêk Pn(U0) ⊂ U∞ n Ûêk Pn(U0) ⊂ U0.

y². - U∞ := C \D 4ü Ü. XJ·5¿Ún 3.7.2, K (1) ´w

,. - ε = sn+1/2 Ai = A|bi|(1−2ε),|bi|(1+2ε). â (3.7.1), ·

|Rn(z)| :=∣∣∣∣Pn(z)−Bn

An· nz

n+1 + 1

n+ 1

∣∣∣∣ = |z|(−1)n+1(n+1) |z2n+2− b2n+2i |(−1)i−1

Hi(z), (3.7.19)

Ù¥

Hi(z) =i−1∏j=1

|bj|(2n+2)(−1)j−1n−1∏j=i+1

|z|(2n+2)(−1)j−1 ·Qi(z) (3.7.20)

Qi(z) =i−1∏j=1

∣∣1− (z/bj)2n+2

∣∣(−1)j−1n−1∏j=i+1

∣∣1− (bj/z)2n+2∣∣(−1)j−1

. (3.7.21)

XJ z ∈ Ai, Ù¥ 1 ≤ i ≤ n− 1, ·k

Qi(z) <i−1∏j=1

(1 + 3 |bi/bj|2n+2

) n−1∏j=i+1

(1 + 3 |bj/bi|2n+2

)< (1 + 6s2n+2)2 (3.7.22)

Qi(z) >i−1∏j=1

(1 + 3 |bi/bj|2n+2

)−1n−1∏j=i+1

(1 + 3 |bj/bi|2n+2

)−1> (1 + 6s2n+2)−2. (3.7.23)


5¿ ε = sn+1/2 ≤ (5n)−2n−1 ≤ 10−5. XJ n ´óê 1 ≤ i ≤ n − 1 ´Ûê, K

é z ∈ Ai, ·k

|Rn(z)| = |z2n+2 − b2n+2

i ||z|n+1

1

s(i−1)(n+1)Qi(z) <

|bi|n+1(1 + (1 + 2ε)2n+2)

(1− 2ε)n+1

(1 + 6s2n+2)2

s(i−1)(n+1)

=1 + (1 + 2ε)2n+2

(1− 2ε)n+1(1 + 6s2n+2)2sn+1 < 2.1 · sn+1.

XJ n Ú 1 ≤ i ≤ n− 1 Ñ´óê, Ké z ∈ Ai, ·k

|Rn(z)| = |bi−1|2n+2|z|2n+2

|z|n+1|z2n+2 − b2n+2i |

1

s(i−2)(n+1)Qi(z) >

(1− 2ε)n+1

1 + (1 + 2ε)2n+2(1− 6s2n+2)2 > 0.49.

ùL²XJ n ´óê 1 ≤ i ≤ n− 1 ´Ûê, éu z ∈ Ai, âÚn 3.7.1(2), ·k

|Pn(z)| <∣∣∣∣2.1 · sn+1 · (n+ 1)An

nzn+1 + 1

∣∣∣∣+ |Bn|

≤ 2.1 (sn+1/2/5) · (1 + s2n+1/(n+ 1))

1− n(1 + 2ε)sn+1+

s2n+1

3n+ 3< sn+1/2.

XJ n Ú 1 ≤ i ≤ n− 1 Ñ´óê, Ké z ∈ Ai, ·k

|Pn(z)| >∣∣∣∣0.49(n+ 1)An

nzn+1 + 1

∣∣∣∣−|Bn| ≥0.49(n+ 1)(1− s2n+1/(n+ 1))

1 + n(1 + 2ε)sn+1− s2n+1

3n+ 3>n+ 1

3≥ 1.

|âq?Ø, ±y²XJ n ´Ûê, éu z ∈ Ai, ·k

éÛê i k |Pn(z)| < sn+1/2 éóê i k |Pn(z)| > 1. (3.7.24)

- U0 = Dr, Ù¥ r = sn+1/2. ù (2) y².

XJ n ´Ûê, éz¦ |z| ≤ sn+1/2 z, ·k

|Pn(z)| ≤∣∣∣∣(n+ 1)Annzn+1 + 1

∣∣∣∣ |z|n+1

n−1∏i=1

|bi|(2n+2)(−1)i−1n−1∏i=1

∣∣∣∣1− z2n+2

b2n+2i

∣∣∣∣(−1)i−1

+ |Bn|

≤(n+ 1)(1 + s2n+1/(n+ 1))

1− nsn2+n/2s3(n+1)/2

n−1∏i=1

(1 + 2

|z|2n+2

|bi|2n+2

)+

s2n+1

3n+ 3< sn+1/2.

ùL²XJ n Ûê, K Pn(Dr) ⊂ Dr, Ù¥ r = sn+1/2.

XJ n óê, K Pn ò 0 N ∞ . éz¦ |z| ≤ sn+1/2

z, ·k

|Pn(z)| ≥ (n+ 1) s−(n+1)/2 (1− s2n+1/(n+ 1))

1 + nsn2+n/2

n−1∏i=1

(1− 2

|z|2n+2

|bi|2n+2

)− s2n+1

3n+ 3> n > 1.

(3.7.25)

ù (3) Òy². Úny..


ã 3.6 P3 Cantor ±. Julia 8. ëê s À¿©. ã¥ÚÜ©

L@ªS¹:@ Fatou ©|, xÚÜ©KL«@ªS

>.¹ØÄ: 1 Ô Fatou ©|. Ô Fatou ©|Ú§c¥

Fatou I³xÑ5. ã¡: [−1.6, 1.6]× [−1.2, 1.2].

½n 3.1.4 y². - A := C \ (U0 ∪ U∞). Pn Julia 8u⋂k≥0 P

−kn (A). 5¿

Pn ÁÛk. ?Øaqu½n 3.1.1 Ú 3.1.3 y². Pn Julia ©|¤

8ÜÓu n ÎÒ Σn := 0, 1, · · · , n− 1N ü>²£. AO/, Pn Julia 8Ó

u Σn × S1, ù´·@ Cantor ± (ã 3.6). [L§3dÑ.

§3.8 Ún 3.7.2 y²

ù!·y²Ún 3.7.2, TÚný²Ún 3.7.5 Ú½n 3.1.4 '.

y². - R(z) = 1/Pn(1/z),KÚn 3.7.28(y² R(D) ⊂ D∪1. - w = zn+1,

ÏLO, ·k

R(w) := R(z) =w + n

n+ 1· 1

S(w), (3.8.1)

Ù¥

S(w) = An

n−1∏i=1

(1− b2n+2i w2)(−1)i−1

+w + n

n+ 1Bn = 1 +

w − 1

1 + (2n+ 2)Cn

(H(w)− 1

w − 1+ 2Cn

)(3.8.2)

§3.8. Ún 3.7.2 y² 53

H(w) =n−1∏i=1

(1− b2n+2i )(−1)i

n−1∏i=1

(1− b2n+2i w2)(−1)i−1

. (3.8.3)

du H(1) = 1, ùL² H ′(1) ´kê. ¯¢þ,

I(w) :=H ′(w)

H(w)= −2w

n−1∑i=1

(−1)i−1b2n+2i

1− b2n+2i w2

. (3.8.4)

l I(1) = H ′(1) = −2Cn. é?Û¿© w − 1, ·±ò S(w) ¤

S(w) = 1 +(w − 1)2

1 + (2n+ 2)Cn·H(w)−1w−1

+ 2Cn

w − 1=: 1 +

(w − 1)2

1 + (2n+ 2)Cn· Φ(w), (3.8.5)

Ù¥

Φ(w) =∑k≥2

H(k)(1)

k!(w − 1)k−2. (3.8.6)

eÚ´é?Û k ≥ 2 O H(k)(1).

éz k ≥ 1, -

Yk(w) =n−1∑i=1

(−1)i−1

(b2n+2i

1− b2n+2i w2

)k. (3.8.7)

AO/, Y1(1) = Cn

Y ′k(w) = 2kw Yk+1(w). (3.8.8)

XJ |w| = 1, ·k

|Yk(w)| ≤∣∣∣∣ b2n+2

1

1− b2n+21

∣∣∣∣k(

1 +n−1∑i=2

∣∣∣∣b2n+2i (1− b2n+2

1 )

b2n+21 (1− b2n+2

i )

∣∣∣∣k)≤ 11

10

∣∣∣∣ b2n+21

1− b2n+21

∣∣∣∣k . (3.8.9)

aq/, ·k |Yk(w)| ≥ 910|b2n+2

1 /(1− b2n+21 )|k. ùL²∣∣∣∣Yk+1(w)

Yk(w)

∣∣∣∣ ≤ 11

9

∣∣∣∣ b2n+21

1− b2n+21

∣∣∣∣ ≤ 2s2n+2 < 1/2. (3.8.10)

·Äkäóéz k ≥ 0 k |I(k)(1)| ≤ 2k+1k!|Cn|. du I(0)(w) = −2wY1(w)

I(1)(w) = −2Y1(w)− 4w2Y2(w), ±8B/y² I(k)(w) ±¤

I(k)(w) =2k∑j=1

Qk,j(w) =2k∑j=1

Pk,j(w)Yk,j(w), (3.8.11)

Ù¥ Pk,j(w)ŃÝõ k+ 1õªé, 1 ≤ l ≤ k+ 1k Yk,j = Yl. 5

¿, Qk,j Uu 0 (éAõª Pk,j NÝw¤´ −∞)Lª (3.8.11)


±z, ·IÒ´ù “” L«. AO/, Ø5, é 1 ≤ j ≤ 2k, ·

?Ú¦

Pk+1,2j−1(w)Yk+1,2j−1(w) = P ′k,j(w)Yk,j(w) Pk+1,2j(w)Yk+1,2j(w) = Pk,j(w)Y ′k,j(w).

(3.8.12)

5¿ deg(Pk,j) ≤ k + 1 é, 1 ≤ l ≤ k + 1 k Yk,j = Yl. ,, d (3.8.10), éz

l ≥ 1 k |Yl+1(1)/Yl(1)| ≤ 1/2. ùL²

|Pk+1,2j−1(1)Yk+1,2j−1(1)|+ |Pk+1,2j(1)Yk+1,2j(1)|

= |P ′k,j(1)Yl(1)|+ |Pk,j(1)Y ′l (1)|

≤ (k + 1)|Pk,j(1)Yl(1)|+ 2(k + 1)|Pk,j(1)Yl+1(1)|

≤ 2(k + 1)|Pk,j(1)Yk,j(1)|.

(3.8.13)

P ||I(k)(1)|| :=∑2k

j=1 |Pk,j(1)Yk,j(1)|, ·k ||I(k)(1)|| ≤ 2k||I(k−1)(1)||. ùL²

|I(k)(1)| ≤ ||I(k)(1)|| ≤ 2kk!||I(0)(1)|| = 2k+1k!|Cn|. (3.8.14)

ùÒy²éz k ≥ 0, äó |I(k)(1)| ≤ 2k+1k!|Cn| ¤á.

Ùg, ·8B/yé k ≥ 1 k |H(k)(1)| ≤ 4kk!|Cn|. é k = 1, ·k |H ′(1)| =2|Cn| < 4|Cn|. béz 1 ≤ i ≤ k k |H(i)(1)| ≤ 4ii!|Cn|. â (3.8.4), ·

k H ′(w) = H(w)I(w). Ï |I(k−i)(1)| ≤ 2k−i+1(k − i)!|Cn| éz 1 ≤ i ≤ k k

|H(i)(1)| ≤ 4ii!|Cn|, l

|H(k+1)(1)| ≤ |I(k)(1)|+k∑i=1

k!

i!(k − i)!|H(i)(1)| · |I(k−i)(1)|

≤ 2k+1k!|Cn|(1 + 2k+1|Cn|) ≤ 4k+1(k + 1)!|Cn|.

(3.8.15)

XJéu |θ| ≤ 1/20 k w = eiθ, K |w − 1| < |θ| ≤ 1/20. â (3.8.6) Ú (3.8.15),

·k

|Φ(w)| ≤∑k≥2

4k|Cn|(1/20)k−2 ≤ 16|Cn|∑k≥0

5−k = 20|Cn|. (3.8.16)

â (3.8.5) Ú (3.8.16), k

|S(w)| ≥ 1− θ2

1− (2n+ 2)|Cn|20|Cn| ≥ 1− s2n+1

n+ 1θ2. (3.8.17)

ù´du n ≥ 2 â (3.7.6) k |Cn| < s2n+1/(8(n+ 1)2).

§3.8. Ún 3.7.2 y² 55

,¡, XJé 0 ≤ |θ| ≤ π k w = eiθ, K∣∣∣∣w + n

n+ 1

∣∣∣∣ =

(1− 4n

(n+ 1)2sin2 θ

2

)1/2

≤(

1− 4n

π2(n+ 1)2θ2

)1/2

≤ 1− 2n

(n+ 1)2π2θ2.

(3.8.18)

ù´Ïé 0 ≤ x < 1 k (1− x)1/2 ≤ 1− x/2. ùL²XJé |θ| ≤ 1/20 - w = eiθ, K

|R(w)| ≤ (1− 2n

(n+ 1)2π2θ2)(1− s2n+1

n+ 1θ2)−1 ≤ 1. (3.8.19)

d, |R(w)| = 1 = w = 1.

XJé |θ| > 1/20 k w = eiθ, â (3.8.2) ÚÚn 3.7.1(2), ·k

|S(w)| ≥ (1− s2n+1

n+ 1)n−1∏i=1

(1− |bi|2n+2)− s2n+1

3n+ 3≥ 1− 3s2n+1

n+ 1. (3.8.20)

â (3.8.18) Ú (3.8.20), ·k

|R(w)| ≤ (1− 2

202(n+ 1)π2)(1− 3s2n+1

n+ 1)−1 < 1. (3.8.21)

ùL²Ø w = 1ù:, R(w)òü >.Nü . ÏXJ |w| ≤ 1

K R(w) 6= ∞, · R(D) ⊂ D ∪ 1. Ïd, R(D) ⊂ D ∪ 1 R òk8

z ∈ C : zn+1 = 1 N÷ 1. ùÚn 3.7.2 Òy².

1oÙ McMullen N Julia 8[é¡x

éukn¼ê Julia 8ÿÀ5, éõö®²ïÄ. éukn¼ê Julia

8[é¡AÛ5, Kék<9. ¯¢þ, =B´éu~AÏ/, 8cé

k<é Julia 8[é¡AÛ5?1x. ùÙ¥, ·òÄ McMullen N

Julia 8[é¡AÛ5. ·Ñgd.;<º¹e[é¡üz©

a, ¿égd.:;<º¹?1?Ø. AO/, ·y²3Vkn

¼ê, Ù Julia 8[é¡du/#.

§4.1 cóÚÄ½Â

- l,m ≥ 2 ü÷v 1/l + 1/m < 1 ê, I0 = [0, 1] ⊂ R ¢¶þ4ü «m. ·ò I0 lm©¤ 3 f«m, z«mÝ 1/l, 1 − 1/l − 1/m, 1/m,

,ò¥m«mSÜK.#8Ü I1 ü«m¿ [0, 1/l]∪ [1−1/m, 1].

- g0(x) = (1− x)/l g1(x) = 1 + (x− 1)/m. ·k I1 = g0([0, 1]) ∪ g1([0, 1]). éz

n ≥ 1,- In =⋃i1,··· ,in∈0,1 gi1 · · · gin([0, 1]). Kz In Ñ´;k In ⊇ In+1. ·

¡¤k In ⋂n≥0 In ÍO Cantor 8,¿P Cl,m. AO/, C3,3 IOn

© Cantor 8. ½ÂIO Cantor ± Al,m Cl,m×S1, Ù¥ S1 := z ∈ C : |z| = 1´ü ±.

S ⊂ C ´ Riemann ¥¡þëÏÛÜëÏ;8. XJ S SÜ´8,

C \ S z©|>.Ñ´ Jordan C \ S ØÓ©|4*dpØ, K

¡ S ´ Sierpinski /# (k¡/#) [74]. d/, XJ S SÜ´8

±¤ S = C \⋃i∈NDi, Ù¥ Di ´÷vXJ i 6= j K ∂Di ∩ ∂Dj = ∅ Jordan

«, d, i → ∞ ¥¡» diam(Di) → 0, K S ´ Sierpinski /#. ¤k

Sierpinski /#*dÑ´Ó. XJéz i ∈ N, Di ¥¡ (½öî¼) ±, K

ù/#¡´/#.

XJ®²üÝþm (X, dX) Ú (Y, dY ) ´Ó, Ä¯K´wùü

m´Ä´*d[é¡d. - (X, dX) Ú (Y, dY ) üÝþm. XJ3 X

Y Ó f : X → Y Ú η : [0,∞)→ [0,∞) ¦éuzØÓ: x, y, z ∈ X k

dY (f(x), f(y))

dY (f(x), f(z))≤ η

(dX(x, y)

dX(x, z)

),

K (X, dX) Ú (Y, dY ) ¡´*d[é¡d. XJ X = Y = C, K X Y Ó

´[é¡=§´[/ ( [36, ½n 11.14]).

56

§4.1. cóÚÄ½Â 57

ùÙ¥, ·ïÄ McMullen N

fλ(z) = zm + λ/zl, (4.1.1)

Julia 8[é¡AÛ, Ù¥ λ ∈ C∗ := C \ 0 l,m ≥ 2 ü÷v 1/l+ 1/m < 1

ê. N´wÑ fλ 3 ∞ ká5ØÄ:, Ù_ 0 Ú§gC. k, XJ·

rNê l Ú m Ø´ λ, K^ f l,m Ú J l,m 5©OL« fλ Ú§éA Julia 8

Jλ. Riemann ¥¡ C ;f8¡´ Cantor ±XJ3ù;8ÚIOn

© Cantor 8 A3,3 mkÓ.

éu Jλ ÿÀ5, Devaney, Look Ú Uminsky y²<ºn©½n. ù½

n`XJ fλ ¤k.:Ñ ∞ áÚ, K Jλ o´ Cantor 8, Cantor ±

½ö´ Sierpinski /# ( [24, ½n 0.1] ½ö½n 4.2.1). éu Jλ AÛ5, ·

ke¡½n.

½n 4.1.1. b fλ ¤k.:Ñ ∞ áÚ. Ke¡n«¹k=k«u

):

(1) Jλ ´ Cantor 8, K Jλ [é¡duIOn© Cantor 8 C3,3;

(2) Jλ ´ Cantor ±, K Jλ [é¡duIO Cantor ± Al,m, Ù¥ l

Ú m (4.1.1) ¥ê;

(3) Jλ ´ëÏ, K Jλ [é¡du/#.

Ø 0 Ú ∞, N fλ e l + m .:¡´gd.:. ùgd.

:þkÓ; ( §4.2). §oÓáÚ ∞ o;Ók.. XJgd

.:vk<º ∞, K Jλ ´ëÏ ( [25]). 3ù«¹e, üg,¯KÑy:

(1) Jλ oÿ´ Sierpinski /#? (2) §´Ä[é¡du/#?

XJgd.;k.á5±Ï;¤áÚ, § Fatou ©|÷v,«

“ìv” 5, K Jλ ´ Sierpinski /# ( [23]). éõ λ ¦ Jλ ´

Sierpinski /#, ·Ä fλ ëêm. ½Âùx¼ê<º,

Λ = λ ∈ C∗ : fλ gd.;´k.. (4.1.2)

®² Λ ´ëÏ¹ Mandelbrot 8éõÓ, ù Mandelbrot 8Ó

¥:éAuëê ( [20, 69]).

P Mandelbrot 8 M := c ∈ C : Pc(z) = z2 + c Julia 8´ëÏ. M V©|´ M ëÏmf8, 3p¡ 0 Pc á5±Ï;¤áÚ.

- H M V©|. éz c ∈ H, õª Pc k±Ï k á5±Ï;.

d H ¡´ M ±Ï k V©|, ½öò H ±ÏP p(H). 3

58 1oÙ McMullen N Julia 8[é¡x

ëê rH á3 H >.þ¦õª PrH k¹.: 0 ±Ï k Ô

Fatou©|. ù: rH ¡´ H .AO/,- H♥ M ±Ï 1ëÏ©|. e

I ♥ x H♥ /G, §>.´^%9. M V©|±©¤üa: satellite

Ú primitive ( §4.3 ¥½Â). */, ùüa©|Ä«O´: primitive V

©|>.d%9¤, satellite V©|KØ´.

Ï Λ¥zV©| H Ñá3 Mandelbrot8,Óp ( [69,½

n 10] Ú½Â 4.3.2), é H, ·±½ÂéA rH ¿±ò Λ ¥V©|©

satellite Ú primitive ùü«a. ( §4.3).

éu Jλ oÿ´ Sierpinski /#¯K, [62] ¥é λ > 0 /Ä.

P Kλ = C \ A(∞) fλ “W¿” Julia 8, Ù¥ A(∞) ∞ áÚ.

½n 4.1.2 ([62, 76]). 3ü¢ê λ0 < λ1 ¦XJ λ ∈ [λ0, λ1], K fλ Julia

8 Jλ ´ÛÜëÏ. d, Jλ ´ Sierpinski /#=e¡÷v:

(I) Kλ SÜ Int(Kλ) = ∅ λ 6= λ0;

(II) λ ∈ H ∪ rH, Ù¥ H ´ M 3 Λ ¥Ó primitive V©|¿

H Ø´ H♥ 3ÓNe.

l Mandelbrot 8 Λ ÓN°(½Âò3 §4.3 ¥Ñ. [62] ¥=éþ

¡ã½n3 l = m ≥ 3 ¹y². Xie òù(J3¨Æ¬Ø©¥í

2/, y²gáq ( [76]). ùÙ¥, ·òy²

½n 4.1.3. XJ λ ∈ [λ0, λ1], K Jλ [é¡du/#= λ ÷v (I)

½ö (II) λ 6= rH. AO/, XJ λ ∈ (λ0, λ1) ¦ fλ Ù¥.:´îý±Ï

, K Jλ [é¡du/#.

á=íØ, ·k

íØ 4.1.4. 3Vkn¼ê, Ù Julia 8[é¡du/#.

¯¢þ, â½n 4.1.3 y², ·$3Ã¡kn¼ê, Ù Julia

8[é¡du Sierpinski /#.

·òÄ/¦ Jλ ´ Sierpinski /#. AO/, ëê λ ±3E

²¡þCz Ø==´3¢¶þ. , , ù¦·b l = m ≥ 3.

½n 4.1.5. b l = m ≥ 3 fλ gd.;´k.. XJ λ ÷ve¡^

, K Jλ ´ Sierpinski /#.

(a) SÜ Int(Kλ) = ∅ gd.:;4 ∞ áÚ>.Ø;

§4.2. gd.:<º/ 59

(b) λ ∈ H ∪ rH, Ù¥ H ´ M 3 Λ ¥Ó primitive V©|¿

H Ø´ H♥ 3ÓNe.

d, XJ λ ÷v (a) ½ö (b) λ 6= rH, K Jλ [é¡du/#.

ùÙ´ùSü: 3 §4.2 ¥, ·ã McMullen NÄ5¿y²

½n 4.1.1. 3 §4.3 ¥, ·y²½n 4.1.3, íØ 4.1.4 Ú½n 4.1.5. 3 §4.4 ¥, ·

Ä Cantor ±[é¡AÛ.

§4.2 gd.:<º/

N fλ 3:kê l − 1 .:, 3 ∞ Ã¡kê m− 1

.:, ωj = e2πij/(l+m)(λl/m)1/(l+m) ´ü.:, Ù¥ 0 ≤ j ≤ l + m − 1. ù.

: ωjl+m−1j=0 ¡´ fλ gd.:. ÏL/O, |f nλ (ωj)| = |f nλ (ωk)| é

u 0 ≤ j, k ≤ l +m− 1 Ú n ≥ 0 ¤á. Ïdù l +m gd.:;þdÙ¥

:;û½.

§4.2.1 <ºn©½nÚëêm©)

- Bλ Ú Tλ fλ ©O¹∞Ú 0 Fatou©|. K Bλ∩Tλ = ∅½ö Bλ = Tλ.

Devaney, Look Ú Uminsky 3e¡½n¥Ñ McMullen N Julia 8ÿÀa

.©a½n.

½n 4.2.1 ([24]). b fλ gd.:Ñ<º ∞. Ke¡n«¹k=k

«u):

(1) 3, j ¦ fλ(ωj) ∈ Bλ, K Jλ ´ Cantor 8;

(2) 3, j ¦ fλ(ωj) ∈ Tλ 6= Bλ, K Jλ ´ Cantor ±;

(3) 3, j Ú k ≥ 1 ¦ f kλ (ωj) ∈ Tλ 6= Bλ, K Jλ ´ Sierpinski /#.

¯¢þ, fλ ëêm C∗ ±©¤oÜ© Λ t Λ∞ t Λ0 t ΛS ( [69, §7]), Ù

¥ Λ (4.1.2) ¥½Â<º,; Λ∞ = λ ∈ C∗ : Jλ ´ Cantor 8 ´Ã.«, ¡ Cantor ,; Λ0 := λ ∈ C∗ : Jλ ´ Cantor ± ´ 0 B

, ¡´ McMullen , Ï McMullen kïÄù«¹ ( [49, §7]);

ΛS = λ ∈ C∗ : Jλ ´ Sierpinski /# \ Λ ¡´ Sierpinski É, §dêõü

ëÏ©||¤. d, Λ ¥zV©|Ñ¹3 Mandelbrot 8,Ó¥ (

ã 4.2 ¥ã).


§4.2.2 Cantor 8[é¡üz

David Ú Semmes ·K [17, ·K 15.11] y²: XJVkn¼ê

Julia 8ÓuIOn© Cantor 8 C3,3, Kù Julia 8[é¡du C3,3. â½n

4.2.1(1), ½n 4.1.1(1) ¤á.

§4.2.3 Cantor ±[é¡üz

·ò^[/Ãâ5y²½n 4.1.1(2). éü÷v 1/l + 1/m < 1 ê

l,m ≥ 2, ·3þ½ÂS¼êXÚ (P IFS), ¦ùS¼êXÚ

áÚf/duIO Cantor ± Al,m. ,·òùS¼êXÚ_ò

ÿl C g[KN. , 3é C þ3T[KNeØC Beltrami Xê¿nAE(, ·y²3[/NòùIO

Cantor ± Al,m N McMullen N f l,mλ Julia 8, Ù¥ëê λ 6= 0 ~.

½ 0 < r0 < 1,½Â I = [r0, 1]. - I0 = [r0, r0+(1−r0)/l] I1 = [1−(1−r0)/m, 1]

I üf«m. ½Â g0 : I → I0 g0(r) = (1− r)/l + r0 g1 : I → I1 g1(r) =

1− (1− r)/m. N´wÑS¼êXÚ g0, g1áÚfIO Cantor8 Cl,m 3

Ce. - A := reit : r ∈ I 0 ≤ t < 2π, Aj := reit : r ∈ Ij 0 ≤ t < 2πn, Ù¥ j = 0, 1. ½Â F : A0 t A1 → A ¦

F |A0(reit) = g−1

0 (r) · e−ilt F |A1(reit) = g−1

1 (r) · eimt.

5¿ F |A0 : A0 → A Ú F |A1 : A1 → A ©ONÝu l Ú m CXN. S¼

êXÚ (F |A0)−1, (F |A1)

−1 áÚf/duIO Cantor ± Al,m.

½Â F (z) = rl0/zl, §ò Dr0 := z : |z| < r0 N÷4ü Ü C \ D

F (z) = zm ò C \ D N÷gC. e5, ·òÿ F ¦ F ò Ar1,r2 := z :

r1 < |z| < r2 N÷ Dr0 , Ù¥ r1 = r0 + (1− r0)/l r2 = 1− (1− r0)/m.

XJ©|CX f : C → C ±¤ ψ R ϕ, Ù¥ R ´knN ψ, ϕ

þ[/Ó, K¡ f ´[K. - Tr = z : |z| = r ±:%» r > 0

±.

Ún 4.2.2. - f1(z) = r0rl1/z

l Ú f2(z) = r0zm/rm2 ü©O½Â3 Tr1 Ú Tr2

þCXN, ¦ f1(Tr1) = Tr0 f2(Tr2) = Tr0. K3ëYòÿ F : Ar1,r2 →Dr0 ¦ (1) F |Tri = fi, Ù¥ i = 1, 2; (2) F : Ar1,r2 → Dr0 ´[K; (3)

F (e2πi/(l+m)z) = e2mπi/(l+m)F (z).

y². Ù/ãEL§, ·=é m = 3 Ú l = 2 ¹y². Ù§¹

±aq/?Ø. ·Äkb½ r0 = r2 = 1 Ú r1 = ε ~, ÏÚn¥éòÿ

±3·EÐùbeòÿ2»?n=.

§4.2. gd.:<º/ 61

- A D ¥±:%Ê(«, ¿¦ A ⊃ Dε, Ù¥ xi, 0 ≤ i ≤ 9

ùÊ(U_ü 10º: x0 Ë 0. éu 1 ≤ j ≤ 5,- Bj ¹

> [x2j−2, x2j−1] Ú [x2j−1, x2j] Ê>/, Ù¥ x10 = x0. Ï ε v, ·±ÀJ

· A ¦ Dε ⊂ A∪⋃5j=1 Bj ⊂ D. éu 1 ≤ j ≤ 5, - y2j−2, y2j−1 Bj ,

üvkIPU_üº:. éu 1 ≤ j ≤ 5 Ú 2j − 2 ≤ k ≤ 2j − 1, - γk+j =

[yk, e2π(k+j)i/15] ë yk Ú e2π(k+j)i/15 ã. éu 0 ≤ n < 5, - γ3n = [x2n, e

2πni/5].

ùL²·k 15 ^ã γs ò E := D \ (A ∪⋃5j=1 Bj) U_^Sy©¤ 15 ÿÀ

o>/ Es ¿¦ γs Ú γs+1 Es ü^>, Ù¥ 0 ≤ s < 15 γ15 := γ0. aq/, -

ηt = [xt, εe2πti/10] ë xt Ú e2πti/10 ã, Ù¥ 0 ≤ t < 10. Kù 10 ^ã ηt ò

D := A \ Dε U_^Sy©¤ 10 ÿÀo>/ Dt ¿¦ ηt Ú ηt+1 Dt ü^

>, Ù¥ 0 ≤ t < 10 η10 := η0.

- B D ¥±:%Ê>/, Ù¥ zn, 0 ≤ n < 5 B ±_^S

ü 5 º:, Ù¥ z0 Ë 0. - ζn = [zn, e2πni/5] ë zn Ú e2πni/5 ã,

Ù¥ 0 ≤ n < 5. u´ù 5 ^ã ζn ò D \ B U_^Sy© 5 ÿÀo>/ Gn

¿¦ ζn Ú ζn+1 Gn ü^>, Ù¥ 0 ≤ n < 5 ζ5 = ζ0 (ã 4.1).

x0 γ0

E0

γ1y0

E1

B1

y1

γ2E2

x1

D0

D1

x2

B2

z0

z1

z2

z3 z4

ζ0

ζ1

ζ2

ζ3

ζ4

G0

G1

G2

G3

G4

B

ã 4.1 lNÝu 5©|CXN«¿ã,Ù¥ m =

3 l = 2. ã¥ÑaqÎÒ.

·Äkòÿ F : E0 → G0. E0 >.d γ0, [x0, y0], γ1 Úl [1, e2πi/15] |¤,

G0 >.d ζ0, [z0, z1], ζ1 Úl [1, e2πi/5] |¤. ®² F 3l [1, e2πi/15] þ

½Â F (z) = z3. ·½Â F (γ0, [x0, y0], γ1) = (ζ0, [z0, z1], ζ1) n±éAº

:N. K F ±[/òÿ E0 SÜ¿¦ F (E0) = G0. aq/, ½

Â[/òÿ F (E1) = G1 F (E2) = G2 ¦ F (γ1, [y0, y1], γ2) = (ζ1, [z1, z2], ζ2)

F (γ2, [y1, x2], γ3) = (ζ2, [z2, z3], ζ3). â½Âé¡5, éu z ∈ Es, Ù¥ 3 ≤ s < 15,


½Â F (z) = e6πi/5F (e−2πi/5z).

·±3 A \ Dε þ|âqòÿ¦ F (D0) = G4 Ú F (D1) = G3. é

u 2 ≤ t < 10 Ú z ∈ Dt, ½Â F (z) = e−4πi/5F (e−2πi/5z). éu 1 ≤ j ≤ 5, Ï Bj Ú

B ÑÊ>/ F 3 Bj >.þŃ, Ï 3 Bi B g,

/òÿ. y3, ·©|CX F : D \ Dε → D, Ù¥ x0, x2, x4, x6 Ú x8

©Ü:. âE, F A??´ÛÜ[/, l F ´[KN. d,

F (e2πi/5z) = e6πi/5F (z).

y3·Kb r0 = r2 = 1 Ú r1 = ε ~. éz z = reiθ ∈ Ar1,r2 , Ù¥

r ∈ [r1, r2] θ ∈ [0, 2π), ½Â

F (z) =1

r0

F ([ε+1− εr2 − r1

(r − r1)]eiθ).

K F ´Ún¥Iòÿ. y²..

íØ 4.2.3. òÿN F : C→ C ŃÝ l + m [KNé¤

k z ∈ C ÷v F (e2πi/(l+m)z) = e2mπi/(l+m)F (z).

y². â F ½Â±y.

·K 4.2.4. 3[/N ϕ : C → C ¦ ϕ F ϕ−1 = fλ, Ù¥ fλ ´

Julia 8 Cantor ± McMullen N. AO/, ½n 4.1.1(2) ¤á.

y². 3 C \ D þ½ÂIOE( σ(z) = σ0. âE, C ¥A¤k:3f êÑS C \ D. XJé¤k n ≥ 0 k F n(z) 6∈ C \ D, ½Â σ(z) = σ0. Ä

K, 3 n ¦ F n(z) ∈ C \ D. u´·½Â σ(z) = (F n)∗(σ0)(z), ùL

«IOE(3 C \D þ.£. Ï¤k;²L Ar1,r2 õg, ùL² σ

´ C þ F−ØCE(.

- ϕ : (C, σ) → (C, σ0) È[/Ó¦ ϕ(0) = 0 z → ∞ ϕ(z)/z → 1. â F E, F ò ∞ ±NÝ m N ∞ 0 4:, ê

l. Ïd ϕ F ϕ−1 ŃÝu l +m kn¼ê. ùL² ϕ F ϕ−1 äk/ª

ϕ F ϕ−1 = (z − a1)(z − a2) · · · (z − al+m)/zl,

Ù¥ ajl+mj=1 F ":3 ϕ e.

âíØ 4.2.3 ¥ãé¡5, ajl+mj=1 þ!/á3±:%±

þ. Ïd ϕ F ϕ−1 = fλ, Ù¥ fλ(z) = zm + λ/zl ´ McMullen N λ =

(−1)l+ma1 · · · al+m é. y²..

§4.2. gd.:<º/ 63

§4.2.4 Sierpinski /#[é¡üz

é?Ûf8 X ⊂ C, X ¥¡»½Â diam(X) = supx,y∈X |x− y|, Ù¥ | · | L«¥¡Ýþ. é?Ûf8 X, Y ⊂ C, - dist(X, Y ) := infx∈X,y∈Y |x − y| X, Y m

¥¡ål.

½Â 4.2.5. γ ⊂ C ´^ Jordan , XJ3~ê k ≥ 1 ¦

diam(I) ≤ k |x− y| (4.2.1)

é¤kØÓ: x, y ∈ γ ¤á, Ù¥ I γ \ x, y äká¥¡»©|. @où

γ ¡´[. d, ¡ γ ´ k k.ò=.

γ ´^ Jordan , XJ§´ü ±3l C g[/N,

K γ ´[. XJù[/N ûØL K, @où^ γ ¡´

K–[. d, ~ê K Ú k *dp6.

½Â 4.2.6. XJéz i ∈ N, γi ´ K–[, Ù¥ K ´Ø6u i ~

ê, Kùx Jordan γii∈N ¡´x[. XJ3~ê s > 0 ¦

dist(γi, γj)

mindiam(γi), diam(γj)≥ s, (4.2.2)

Ù¥ i, j ∈ N i 6= j, Kùx Jordan γii∈N ¡´©.

½n 4.2.7 ([9, ½n 1.1]). b Γ := γii∈N ´ C ¥x Jordan , §©

OxpØ4 Jordan «>.. XJ Γ ´dx[¤¿Ù¥

*d´©, K3[/N f : C → C ¦é?¿ i ∈ N, f(γi)

´¥¡±.

y²½n 4.1.1(3),·Iy²XJgd.:´<º Jλ ´ Sierpinski

/#,K fλ Fatou©|>.¤8Ü´x[§÷vé©^

.

Ún 4.2.8 ([50, ½n 2.5]). - A ⊂ C äkØ γ, Ù mod(A) >

m > 0. - D C \ A k.©|. K3î¼Ýþe,

dist(D, γ) > C(m) diam(γ),

Ù¥ C(m) > 0 ´=6u m ~ê.


þãÚnL²XJ Ae.±,@o dist(D,C\A ∪D)/diam(D)

e.±.

½n 4.2.9. b fλ gd.:´<º Jλ ´ Sierpinski/#. - γii∈NL« fλ ¤k Fatou ©|>.¤8Ü. K3~ê K ≥ 1 ¦z γi ´

K–[ γii∈N ÷v (4.2.2).

y². XJ fλ ¤kgd.:Ñ<º Jλ ´ëÏ, ÏL[/Ãâ,

± ∞ áÚ>. ∂Bλ ´[. Ïd ∂Bλ eZg_, z

γi Ñ´[ (5¿d¤k Fatou ©|Ñ´üëÏ).

3 C \ Bλ ¥ÀJ¿©C ∂Bλ Jordan ζ, ¦± ζ Ú ∂Bλ >.

« A0 Ø¹ fλ .;. ù´±Ï¤k.:Ñ<º ∞. u

´ A0 c A1 := f−1λ (A0) ´, Ùk.Ö©| Tλ.

/, é?¿ i ≥ 0, ·^ γi,l1≤l≤ki 5L« f−iλ (∂Bλ) ¤këÏ©|, Ù¥

ki 6u i ê. AO/, k0 = k1 = 1, γ0,1 = ∂Bλ γ1,1 = ∂Tλ. du A0 Ø

¹ fλ .;, é?¿ i ≥ 1, f−iλ (A0) ?ÛëÏ©|Ñ´. ·^ Ai,l

5L« f−iλ (A0) ¹ γi,l ëÏ©|.

â½n 4.2.1(3),3 k ≥ 1¦¤kgd.:ÑS Tλ. Ï e i ≥ k+1,

K fλ 3 Ai,l Ú§k.Ö©|þ´/. u´3~ê C1 ≤ C2 ±9 ρ < 1,

¦é? i ≥ 0 Ú 1 ≤ l ≤ ki, k

C1ρi ≤ diam(γi,l) ≤ C2ρ

i.

- N ≥ 1 ¦ C2ρN < C1 ¤áê. À ζ ¿©C ∂Bλ, yXJ

(i, l1) 6= (j, l2), K Ai,l1 ∩ Aj,l2 = ∅, Ù¥ 0 ≤ i, j ≤ N , 1 ≤ l1 ≤ ni, 1 ≤ l2 ≤ nj. 5¿

ù(Øé p ≤ i, j ≤ p+N ¤á, Ù¥ p ≥ 0.

du i ≥ k + 1 , fλ 3 Ai,l Ú§k.Ö©|þ´/. Ïd, 3~

ê m > 0 ¦é¤k i ≥ 0 Ú 1 ≤ l ≤ ni k mod(Ai,l) ≥ m. âíØ 2.4.2, ùL²

γii∈N [.

Ai,l1 Ú Aj,l2 A0 ÏL¦eZg_ØÓ.Ø5,b i ≥ j.

e i− j ≤ N , K Ai,l1 ∩ Aj,l2 = ∅. - βi,l1 Ú βj,l2 ©O Ai,l1 Ú Aj,l2 Ø. âÚ

n 4.2.8 , 3~ê C(m) ¦

dist(γi,l1 , γj,l2)

mindiam(γi,l1), diam(γj,l2)≥ min

dist(γi,l1 , βi,l1)

diam(βi,l1),dist(γj,l2 , βj,l2)

diam(βj,l2)

≥ C(m).

§4.3. gd.:<º/ 65

e i− j > N , Kd N À diam(γi,l1) < diam(γj,l2). 5¿d Ai,l1 Aj,l2

U, , ·%k Ai,l1 ∩ γj,l2 = ∅. âÚn 4.2.8


mindiam(γi,l1), diam(γj,l2)=


diam(γi,l1)≥ dist(γi,l1 , βi,l1)

diam(βi,l1)≥ C(m).

ùL² γii∈N ÷vé©^.

½n 4.1.1 y². (1) ´â David Ú Semmes ·K ( [17, ·K 15.11]). (2)

¤á´Ï·K 4.2.4. (3) (Ü½n 4.2.7 Ú 4.2.9.

§4.3 gd.:<º/

P M Mandelbrot 8. gõª Pc(z) = z2 + c, c ∈M ¡´V=§kk.áÚ±Ï;. é?Û½ M V©| H Ú§ rH , PrH k

±Ï m Ô±Ï;. ®²3,ê v ≥ 1 ¦ p(H) = vm (

[52, Ún 6.3]), Ù¥ p(H) V©| H ±Ï. XJ v = 1, K H ¡´ primitive

, XJ v ≥ 2, K§¡´ satellite . */, M V©|´ primitive

=Ø3Ù§V©|N3 H þ. ½ö`, H primitive =§

>.´%9.

§4.3.1 nØ

·Äk£Ák'nØ, §±3 [29, 50] ¥é. b U, V Cþ÷v U ⊂ V üëÏ«. XJ g : U → V ´X_;N, K¡n| (g, U, V )

áõª. (g, U, V ) W¿ Julia 8½Â K(g) :=⋂n≥0 g

−n(U). 5¿3W¿

Julia 8þ, g ?ÛgSÑk¿Â. gõª Pc ¡´XJ3¹ 0

U, V ¦ (P nc , U, V ) áõª, Ù¥ n > 1 ±Ï. - Kj := P jc (K0),

K Pc(Kj) = Kj+1, Ù¥ j = 0, 1, · · · , n− 1 Kn = K0. z Kj Ñ¡´

Julia 8§þ Pc Julia 8 JPc f8. XJ i 6= j, K Ki Ú Kj SÜ´Ø

, Ù¥ 0 ≤ i, j < n. Ý 0 3 JPc þXº:´½5½öÔØÄ

:, ¡ Pc β– ØÄ:. ®² β–ØÄ: Julia 8´Ø [50, ½n 7.10].

XJé 0 ≤ i, j < n k Ki ∩Kj = ∅, Kù«¡´Ø..

e¡(J´¬m½n, ±l [52, ½n 2.4, Ún 2.7Ú §6]íÑ.5¿ H♥

M ±Ï 1 V©|.

½n 4.3.1. - H M V©| H 6= H♥. é? c ∈ H, K

(1) Pc ´, ±Ï p(H).


(2) ´Ø.= H ´ primitive . AO/, Pc k. Fatou ©|

4*dpØ= H ´ primitive , Ù¥ c ∈ H ∪ rH rH H

.

§4.3.2 ÓN

£½Â3 (4.1.2) ¥ Λ fλ <º,. d [69, §7] Λ zV©|

M V©|3 Λ ¥Ó. - ΦM : M →M Mandelbrot 8Ú§3 Λ p¡

mÓ, ¦ ΦM ò M zV©| H N÷ Λ ,V©|. ù

Ó¦÷ve¡ü^: (1) XJ c ∈ H ¦ Pc k¦f ζ ∈ D áÚ±Ï;, K Φ(c) ∈ H ´¦ McMullen N fΦ(c) k¦f ζ á5±Ï;

ëê, Ù¥ H = ΦM(H) M V©|. H ½Â rH = ΦM(rH). (2) Ó

ΦM : M →M ´. äN/, XJ ΦM′ : M →M′ ´÷v^ (1) Ó,

KM′ ⊂M.

½Â 4.3.2. ÓM ¡´ Mandelbrot 83 Λ ¥E (copy)

N ΦM ¡´ÓN.

é Λ¥zV©|H,3 Λ¥MandelbrotE¹H. V

©| H ¡´ satellite½ primitiveXJ H 3ÓNe_´ satellite

½ö primitive .

éu λ ∈ C∗, ½Â fλ “W¿” Julia 8

Kλ := z ∈ C : f kλ (z)k≥0k..

XJ λ ∈ C∗ \ Λ, K Kλ = Jλ. XJ λ ∈ Λ, KU3¹3 Kλ ¥ Fatou ©|

ªØ¬SÃ¡áÚ Bλ ¥. ½n 4.3.1 íØ, e¡Únd

[29] ¥a Mandelbrot 8nØíÑ.

Ún 4.3.3. - H Λ ¥E M = ΦM(M) ¦ H 6= ΦM(H♥) V

©|. é?Û λ ∈ H∪ rH, Ù¥ rH H , W¿ Julia 8SÜ Int(Kλ) ©|

4*dØ= H ´ primitive .

§4.3.3 <º¹e Sierpinski /#[é¡üz

P λ0 = supλ : λ ∈ Λ0 ∩ R+ λ1 = infλ : λ ∈ Λ∞ ∩ R+, Ù¥ Λ0 Ú Λ∞ ©O

L« McMullen Ú Cantor ,.

½n 4.3.4. XJ λ ∈ [λ0, λ1], K Jλ [é¡du/#=e¡ö

¤á:

§4.3. gd.:<º/ 67

(I) W¿ Julia 8SÜ Int(Kλ) = ∅ Ú λ 6= λ0;

(II) λ ∈ H, Ù¥ H ´ M 3 Λ ¥Ó primitive V©|¿ HØ´ H♥ 3ÓNe.

y². â½n 4.1.2, ·Iy²¿©Ü©`²XJ λ = rH, K Jλ ØU[é

¡du/#, Ù¥ H ´ Λ primitiveV©|. XJ H ´ primitive

V©|, @o JrH ØU[é¡du/#´w,Ï frH Fatou 8¹

kk:üëÏÔ Fatou ©|, ù©|>.½Ø´[.

â [62,Ún 4.1]½ö [76,Ún 4.15],3lMandelbrot8N Λ,f8

ÓN Φ : M →M,¦M∩R+ = [λ0, λ1]. AO/, Φ([−2, 1/4]) = [λ0, λ1].

d, éz c ∈ [−2, 1/4], 3½Â3 Pc Julia 8 JPc S[

/Ó Ψc ¦ Ψc(JPc) ⊂ KΦ(c) Ψc(JPc) ∩ R = [pΦ(c), qΦ(c)], Ù¥ 0 < pλ < qλ ü

¢ê¦ λ ∈ [λ0, λ1] fλ(pλ) = fλ(qλ) = qλ. ¯¢þ, pλ = supz : z ∈ Tλ ∩ R+ qλ = infz : z ∈ Bλ ∩ R+ (ã 4.2).

- P (fλ) := Closuref kλ (ωj) : 0 ≤ j < l +m, k > 0 fλ (gd) postcritical 8

4. dþã©Û, ·éz λ ∈ (λ0, λ1], Tλ _zëÏ©|4

¢¶Ø. ùL²éu f−1λ (T λ) zëÏ©| U ′, Ñ3 U ′ üëÏm

U , ¦ P (fλ) ∩ U = ∅.XJ Int(Kλ) = ∅, KØ Bλ , ¤k Fatou ©|ÑS Tλ þ. Ï ∂Bλ

´[ ( [62, Ún 3.8] ½ [76, Ún 4.14]), ÏLaqu½n 4.2.9 ?

Ø, ±`² Jλ ÷v[Ú©^. ùL²XJ Int(Kλ) = ∅ λ 6= λ0, K

Jλ [é¡du/#.

XJ λ ∈ H ¦ Jλ ´ Sierpinski /#, K fλ ´V. aqu½n 4.2.9

?Ø±y² Jλ [é¡du/#. y²..

½n 4.1.3 ÚíØ 4.1.4 y². §´½n 4.3.4íØ (ã 4.2m>ã

).

e5, ·o´b½ n := l = m ¦ fλ NÝ´ 2n, Ù¥ n ≥ 3 ´½

ê. e¡½n´ [61] ¥½n 1.3 f.

½n 4.3.5 ([61]). b Julia 8 Jλ ´ëÏ fλ .;Ø±>. ∂Bλ þ

:à:, K Jλ ´ÛÜëÏ.

- P Ú Q ©OL«¹3 C \Kλ Ú Kλ ¥ Fatou ©|¤8Ü.


ã 4.2 N fλ(z) = z3 + λ/z3 ëêmÚÄåXÚ²¡. ÀJëê λ =

0.0258324407020185 · · · ¦ fλ ¤kgd.:Ñ´ý±Ï Julia 8 Jλ

´ Sierpinski /#¿[é¡du/#.

Ún 4.3.6. - U ∈ Q ¹3 Kλ ¥ Fatou ©|. XJ λ ∈ H ∪ rH, Ù

¥ H ´ Λ ¥ primitive V©|, H Ø´ H♥ 3ÓNe, K

U ∩Bλ = ∅.

y². b U ∩Bλ 6= ∅. XJI, Ä fλ eZgEÜ, ·±b½ U 3

fλ e´ØC. Äk, ·äó U ∩ Bλ = z0 ´ fλ ØÄ:. du ∂Bλ ´

^ Jordan [61, ½n 1.1], 3Ó γ : S1 → ∂Bλ ¦ γ(eint) = fλ(γ(eit)). XJ

](U ∩ Bλ) ≥ 2, K3 t1 < t2 ¦ γ(eit1), γ(eit2) ∈ U ∩ Bλ U ∩ Bλ ⊂ γ([eit1 , eit2 ]),

Ù¥ 0 ≤ t2 − t1 ≤ 2π/n, ù´Ï fλ Fatou ©|äké¡5. Ï U ´ØC, K

fλ(γ([eit1 , eit2 ])) = γ([eit1 , eit2 ]). , ,é?Ûfl I ⊂ Bλ,3 k ≥ 0¦ f kλ (I) = ∂Bλ,

ù´Ï ∂Bλ ´^ Jordan fλ 3 ∂Bλ þÝu z 7→ zn. ù´gñ, l

äóy².

â [69, §7], 3 Mandelbrot 8E M, ¦ λ ∈ H ⊂ M. -

Φ : M →M ÓN. K z0 = U ∩ Bλ PΦ−1(λ) β–ØÄ:[/Ó

. Ï β–ØÄ: Julia 8Ø, ù H 6= Φ(H♥) gñ. Úny.

½n 4.1.5 y². ·Äky²XJ λ÷v (a)½ (b),K Jλ ´ Sierpinski/

#. XJ Int(Kλ) = ∅,K Qλ = ∅. â½n 4.3.5,·Iy²ézØÓ V1, V2 ∈ Pk V 1 ∩ V 2 = ∅. ·Äk`² Bλ ∩ T λ = ∅. Ø,, aqu [24, ·K 4.3] y²,

±íÑ z0 ∈ ∂Bλ∩∂Tλ ´ fλ .:,ù (a)¥bgñ. b Vi f−kiλ (Bλ)

§4.4. CANTOR ±[é¡AÛ 69

©|,Ù¥ i = 1, 2 0 ≤ k1 ≤ k2. ·©ü«¹?Ø.XJ k1 = k2 V1 6= V2,

K V 1 ∩ V 2 = ∅. ÄK, aqu [24, ·K 4.3] y², ±íÑ ∂V1 ∩ ∂V2 c;¥

,:´ fλ .:, ù (a) ¥bgñ. XJ k1 < k2, K Bλ = f k2−1λ (V1)

Tλ = f k2−1λ (V2). d Bλ ∩ T λ = ∅ V 1 ∩ V 2 = ∅. Ïd, XJ Int(Kλ) = ∅ gd

.;4 ∂Bλ Ø, K Jλ ´ Sierpinski /#.

XJ λ ∈ H ∪ rH, Ù¥ H ´ M 3 Λ ¥ primitive V©|, ¿ H Ø´H♥ 3ÓNe.âÚn 4.3.3 Q¥ Fatou©|4pØ. d

, aquþã?Ø, P ¥ Fatou ©|4pØ. ·Iy² P¥ Fatou ©|4Ú Q ¥ Fatou ©|4pØ=.

- U ∈ Q á3 Kλ ¥ Fatou ©|. âÚn 4.3.6, U ∩Bλ = ∅, u´éz V ∈ P k U ∩ V = ∅. l fλ ¤k Fatou ©|4ÑpØ. Ïd, Jλ 3

ù«¹e´ Sierpinski /#.

XJ λ á3 Λ V©|¥, K Jλ ´ëÏ fλ .;4ØU

∂Bλ . Ï>. ∂Bλ ´[ ( [61, ½n 1.2]), |^½n 4.2.9 ¥

Ó?Ø, ±y² Jλ ÷v[Úé©l^. XJ rH ´ primitive V

©|, K frH Fatou 8¹kk:üëÏÔ Fatou ©|, ù©

|>.½Ø´[. Ïd JrH ØU[é¡du/#. y²..

§4.4 Cantor ±[é¡AÛ

lÿÀÝ5w,¤k Cantor±Ñ´,ù´Ï§ÑÿÀdu “I

O” Cantor ± C × S1, Ù¥ C n© Cantor 8 S1 ü ±. Ïd, ¤

k Cantor±¤8Ü´L(, ·±^[é¡AÛÝ5w. ¯¢þ,

[é¡AÛp¡Ä¯KÒ´äü½ÓÝþm´Ä´*d[é¡

d.

Ýþm X /ê confdim(X) ¤k X [é¡dÝþm Haus-

dorff êe(..

Ún 4.4.1. XJ n ≥ 3, K x = log(n)/ log(n+ 1) Øé§

l−x +m−x = 1, (4.4.1)

Ù¥ l,m ≥ 2 ü¦ 1/l + 1/m < 1 ê.

y². Ø5,·b 2 ≤ l ≤ m,K 1/lx ≥ 1/mx,Ù¥ x = log(n)/ log(n+1).

XJ n ≥ 3, du log(n− 1)/ log(n) < log(n)/ log(n+ 1), K

1

lx+

1

mx≤ 1

2log 3/ log 4+

1

3log 3/ log 4= 0.9960381127 · · · < 1.


ùÒ¤Ún 4.4.1 y².

éz n ≥ 2, - Jp,d1,··· ,dn (3.1.2) ¥½Â fp,d1,··· ,dn Julia 8. e5, ·

o´b½ ai ®²½n 3.1.1 ¥@ÀÐ¦ Jp,d1,··· ,dn ´ Cantor ±, Ï·

=éù«¹a,. Ó, ·b½ λ v¦ (4.1.1) ¥½Â McMullen N

fλ Julia 8 Jl,m Cantor ±, Ù¥ 1/l + 1/m < 1. XJéz 1 ≤ i ≤ n k

di = n+ 1, ·^ gn 5L« fp,n+1,··· ,n+1 - Jn ÙéA Julia 8.

½n 4.4.2. Jp,d1,··· ,dn /êu 1 + αd1,··· ,dn, Ù¥ αd1,··· ,dn e¡§

:n∑i=1

d−αd1,··· ,dni = 1.

AO/, XJéz 1 ≤ i ≤ nk di = n+1, K αn := αd1,··· ,dn = log(n)/ log(n+1). XJ

k 6= n, K αk 6= αn. XJ n ≥ 3, Kéz¦ 1/l + 1/m < 1 l,m ≥ 2 k αn 6= αl,m.

y². â½n 3.1.1 y², · fp,d1,··· ,dn |Ü3 Haıssinsky Ú Pilgrim

¿Âedêâ D := (d1, · · · , dn) ∈ Nn û½ [35, §2]. â [35] ¥·K 1.1 Ú 2.2,

fp,d1,··· ,dn Julia 8/ê´ confdim(Jp,d1,··· ,dn) = 1 + αd1,··· ,dn , Ù¥ αd1,··· ,dn ´

§∑n

i=1 d−αd1,··· ,dni = 1 . AO/, XJéz 1 ≤ i ≤ n k di = n + 1, K

αn := αd1,··· ,dn = log(n)/ log(n + 1). ùL² m 6= n du αm 6= αn. âÚn 4.4.1

, ½n¤á.

½n 4.4.2 ÑäN~f5y3ùVkn¼ê, Ù Julia 8

Cantor±/ê?¿Cu 2 ( [35,½n 2]). â½n 3.1.1y²,·

¤k fp,d1,··· ,dn Ñ´V. 5¿/ê´Ýþm[é¡ØCþ ( [47])

?¿Vkn¼ê Julia 8 Hausdorff êîu 2 ( [70, ½n 4 ÚíØ]).

Ïd, ½n 4.4.2 á=ke¡üíØ.

íØ 4.4.3. é?¿ k, n ≥ 2, Julia 8 Jk Ú Jn *d[é¡d= k = n.

d, XJ n ≥ 3, K Jn Ø[é¡du?Û Jl,m, Ù¥ 1/l + 1/m < 1.

íØ 4.4.4. Jn Hausdorff ê Hdim(Jn) ÷v

1 + log(n)/ log(n+ 1) ≤ Hdim(Jn) < 2.

íØ 4.4.3 Ú 4.4.4 y². XJ·5¿/ê´[é¡ØCþ, @o

ùüíØÑ´½n 4.4.2 íØ.

1ÊÙ x¼êÄåXÚ

éX¼ê Siegel >.ÿÀÚAÛ5ïÄ´~¯K. é

^=ê\þ½â^, Douady ß Siegel >.½´ Jordan . 3

ù¯Kþ, éõöÑLz. 8cÐ(Jáu Zhang Ú Zakeri. ^

=ê θ k.., ¦©Oy²NÝØ$u 2 kn¼ê±9äk/ª f(z) =

P (z) exp(Q(z)) 5¼ê Siegel >.Ñ´²L.:[, Ù¥ f(0) = 0

f ′(0) = e2πiθ, P Ú Q Ñ´õª.

3ùÙ¥, ·ò?Ø±Ï¼êx. §Ø¹3 Zakeri ¤Äx

¥. ·Óy², XJ^=êk.., @oT¼êx¥z±:% Siegel

>.½´²L.:[.

§5.1 cóÚy²g´Vã

- f 3:äkÃn¥5ØÄ:5X¼ê. =, f(0) = 0 f ′(0) =

e2πiθ, Ù¥ 0 < θ < 1 ´Ãnê. XJ3X¼êò f Ýf5^=

Rθ : z 7→ e2πiθz, K¡ f 3ØÄ: 0 ?´ÛÜ5z. ¼ê f Ýu Rθ «

´üëÏ« ∆f , ¡´ f ±:% Siegel . Ï f 3 Siegel ∆f

p¡ÄåXÚ´ü. éõÿ, ·'%>. ∂∆f AÛÚÿÀ5. ~X, §´

Ä´^ Jordan ? ½ö?Ú, §´Ä´[?

éuX¼ê, ù¯Kvk)û. 3½^, éõöé

X¼êx)ûù¯K.AO´3^=ê θk..¹e,kéõ'(J.

5¿XJÃnê 0 < θ < 1ë©ªÐm θ = [a1, a2, · · · , an, · · · ]÷v supan <∞,

K¡ θ ´k... 3ùê, Douady, Zakeri Ú Shishikura, ©Oy²gõ

ª, ngõª±9¤kNÝØu 2 õªk.. Siegel >.´[

[26, 77, 66]. éõöé¼ê/Äù¯K. Geyer, Keen Ú Zhang ©

OÄ z 7→ e2πiθzez Ú z 7→ (e2πiθz +αz2)ez [32, 41]. Ó, Cheritat Ä“

ü”¼ê [14].

C, Zakeri Ú Zhang ©O(J. éu5¼ê f(z) =

P (z) exp(Q(z)), Ù¥ f(0) = 0 f ′(0) = e2πiθ, P Ú Q Ñ´õª, Zakeri y²XJ

0 < θ < 1 ´k..Ãnê, @o f ±:% Siegel >.´[

¹ f .: [78]. 3 [80] ¥, Zhang y²ù(Øé¤kNÝØ$

u 2 kn¼ê¤á.

Zhang ÓéÙ§±Ï¼êÄù¯K. ¦e¡(J.

71

72 1ÊÙ x¼êÄåXÚ

½n (Zhang, [79]). - 0 < θ < 1 k..Ãnê. K¼ê f(z) =

e2πiθ sin(z) Siegel >.´TÐ²Lü.: π/2 Ú −π/2 [.

3ùÙ¥, ·ò?Ø±Ï¼êx. - 0 < θ < 1 k..Ãn

ê. P

F = fα(z) = e2πiθ sin(z) + α sin3(z) : α ∈ C \ 0. (5.1.1)

éz fα ∈ F , ·y²e¡

½n 5.1.1. ¼ê fα ±:% Siegel >.´[, 3Pê¿

Âe²L fα 2 , 4 ½ö 6 .:.

½n 5.1.1 y²´É Keen Ú Zhang [41, 79] óéu. ·|^[/N

E|5Ex¼êx,ùx¼êx¥z¼êÑäk±:% Siegel,

ù Siegel ´dx3:äkAÛá5ØÄ:¼êxÏLé5z«Ã

â=z5. ùpEI±±Ï5, aq/EkÑy3 [79] ¥.

·`²e½n 5.1.1 y² Keen-Zhang [41] y²ØÓ?. ¦Ä

¼êx¥z¼ê=küIP.:, ¤±§éAëêmkg,y

©. äN/, §ÄkÚ\x Σ′1/2 = gα(z) = (z/2 + αz2)ez : α ∈ C \ 0 ,é^ Jordan C ⊂ Σ′1/2 ò Σ′1/2 ©¤üÜ© Ai, Ù¥ i = 1, 2. ,¦`²XJ

α ∈ Ai, K±:%5z«>. ∂Dα TÐ=²L gα .: ciα,

Ù¥ i = 1, 2. XJ α ∈ C, K ∂Dα ²Lü.:. ·Ä¼êxkÃ¡õ

.:. ¦+ù¼êÑ´±Ï, ´éJ(½=.:±á3±:%

5z«>.þ. , , |^, “f5” ?Ø, ·±ëêm

aqy© (½n 5.1.2 Ú §5.3).

·ÄkVãe½n 5.1.1 y²g. 3ùÙÜ©, ·½k.

.Ãnê 0 < θ < 1 - λ = e2πiθ. N´/IP.:, ·½ÂBf

Λ = c : |Re(c)| ≤ π/2 \ 0. (5.1.2)

éz½ t 6= 0, Äe¡ëêm:

Σt =

fc(z) = t

(sin(z)− sin3(z)

3 sin2(c)

): c ∈ Λ

. (5.1.3)

OL²éz fc ∈ Σt k f ′c(z) = t sin(z − π/2) sin(z + c) sin(z − c)/ sin2(c). l

fc .:8 Critc = π/2 + kπ, ±c + kπ : k ∈ Z. AO/, XJ c = π/2 ½ö

−π/2, K fc äkê 3 .: π/2 + kπ, Ù¥ k ∈ Z. Ï fc ´±Ï 2π

§5.1. cóÚy²g´Vã 73

Û¼ê, XJ c 6= π/2 Ú −π/2, K fc þkgd.:. Ï éu fc, ·

IÄ.: c Ú π/2 ÄåXÚ5=.

,, ·ÄüAÏëêm Σ1/2 Ú Σλ. ëêm Σ1/2 d@3:äk

¦f 1/2 á5ØÄ:¼ê¤, Σλ K¹½n 5.1.1 ¥¤k¼ê (Ún

5.2.3). ùp, Σ1/2 ¥eI 1/2 Ø´, §±d?Û÷v 0 < |t| < 1 t O.

éz fc ∈ Σ1/2, - Ωc fc ±:%5z«.·òy² Ωc ´

k.üëÏ«. éu Λ ¥^©l 0 Ú ∞ ü4 γ, ·^ γint Ú γext

5©OL« Λ \ γ k.ÚÃ.©| (5¿ Λ \ γ Ã.©|UØ). ÏLïÄ

ëêm Σ1/2 ¥¼êÄåXÚ5, ·kXe©a½n.

½n 5.1.2. 3²L π/2 Ú −π/2 ©l 0 Ú ∞ ü4 γ ⊂ Σ1/2 ¦

(1) XJ c ∈ γint, K ∂Ωc TÐ²L 2 .: c Ú −c.(2) XJ c ∈ γext, K ∂Ωc TÐ²L 2 .: π/2 Ú −π/2.

(3) XJ c ∈ γ \ ±π/2, K ∂Ωc TÐ²L 4 .: ±π/2 Ú ±c.(4) XJ c = π/2 ½ −π/2, K ∂Ωc ²L 2 ê 3 .: π/2 Ú −π/2.

Ø5, éu c ∈ γext ∪ γ, ·^ fc ∈ Σ1/2 5E.N Fc. E

L§Ì´ò5z« Ωc =z¤ Fc ^=. ,, é Fc–ØC

Beltrami Xê νc. ÏL)äkXê νc Beltrami §, ·± 0, 2π Ú ∞ØÄ[/N wc. Ïd Tc = wc Fc w−1

c 3 C þ´)Û. ÏL?Ø Tc ":©

Ù±9Ù§&E, ±y² Tc ∈ Σλ (Ún 5.2.5 ÚíØ 5.5.9).

ÏLù«ª, ·ÃâN S : γext ∪ γ → Σλ, ½Â S(fc) = Tc. ,,

S ±aq/l γint ∪ γ Σλ ½Â. ,, ·y² S ´ëY S(γ) k½n

5.1.2 ¥ãaq5.

!, ·y²ÃâN S : Σ1/2 → Σλ ´÷. ùÄuy² S : γint ∪ γ →S(γ)int ∪ S(γ) Ú S : γext ∪ γ → S(γ)ext ∪ S(γ) Ñ´÷ (·K 5.6.4). ù·B

½n 5.1.1 ÚXe½n.

½n 5.1.3. - F = fα : α ∈ C \ 0 (5.1.1) ¥½Â¼êx. K3ü

H : C \ 0 → Λ Ú^²L −λ/3 ©l 0 Ú ∞ ü4 Γ ⊂ C \ 0 ¦(1) XJ α ∈ Γint, K ∂∆fα TÐ²L 2 .: π/2 Ú −π/2.

(2) XJ α ∈ Γext, K ∂∆fα TÐ²L 2 .: H(α) Ú −H(α).

(3) XJ α ∈ Γ \ −λ/3, K ∂∆fα TÐ²L 4 .: ±π/2 Ú ±H(α).

(4) XJ α = −λ/3, K ∂∆fα TÐ²L 2 ê 3 .: π/2 Ú −π/2.

½n 5.1.3 Ñaqu Zakeri ng Siegel õª [77] ±9 Keen-Zhang ¼ê

x [41] ëêmaqy©.


§5.2 Ä(J

- f : R→ R ¢¶Ó. XJ3~ê k ¦

1

k≤ f(x+ t)− f(x)

f(x)− f(x− t)≤ k (5.2.1)

é¤k x ∈ R Ú t > 0 ¤á, KN f ¡´[é¡. ~ê k ¡´[é¡~ê.

â Beurling Ú Ahlfors ½n [7], ^ (5.2.1) du f kl R þ²¡H [/*Ü. AO/, ¦y²½Â

fBA(x+ iy) =1 + i

2

(∫ 1

0

f(x+ ty)dt− i∫ 1

0

f(x− ty)dt

)(5.2.2)

N fBA : H→ H´ f K–[/*Ü,Ù¥K ≤ k2. ù*Ü¡ Beurling-

Ahlfors *Ü.

- f : S1 → S1 Ó F : R → R f 3 τ(x) = e2πix eJ,, Ù¥

τ ´l R S1 CXN. 5¿ F ÷v F (x + 1) = F (x) + 1 3\

~ê¿Âe´. XJé F k (5.2.1) ¤á, KN f : S1 → S1 ¡´ k-[é¡

. u´, F k Beurling-Ahlfors *Ü FBA, ù*Ü´l H ½ÂgC. 3CXN

τ(z) = e2πiz e, du FBA(z + 1) = FBA(z) + 1, FBA g,pl D NgC[/N. ·^ fBA L«ùN, §´[/N±:ØÄ. Ó, N

fBA : D→ D ¡´ f : S1 → S1 Beurling-Ahlfors *Ü.

- γ ^ü4, XJ3[/N fγ : C→ C ¦ γ = fγ(S1), K

γ ¡´[. AO/, XJ fγ ûØL K, K γ ¡´ K–[.

- f : C → C, g : S1 → S1 ±9 γ ©O[/N, [é¡NÚ

[. ·^ K(f), K(g) Ú K(γ) 5©OL« f , gBA ±9 fγ û.

^ (½4l) ¡´¢)ÛXJ§´± (½ö4ã) 3½Â3T

± (½ö4ã) NC/N. ^ml¡´¢)ÛXJ§¤k4flÑ

´¢)Û ( [43, p.20]). e¡Ún´ Schwarz n/ª ( [1]).

Ún 5.2.1. D «, γ ¹3 ∂D ¥m¢)Ûã. b f ´

½Â3 D þX¼ê, ¦ f ±ëY/òÿ γ þ, f(γ) ´¢)Û

ã. @o, f ±X/òÿ¹ D «þ, ¦ γ ¹3T«

SÜ.

b f : S1 → S1 ´Ó f 3 S1 þ^=ê´Ãnê θ. XJ3

½Â3ü ±þ[é¡N ϕ ¦ ϕ f ϕ−1 = Rθ 3 S1 ¤á, KN f ¡

§5.2. Ä(J 75

´±[é¡5z, Ù¥ Rθ L«f5^= z 7→ e2πiθz. 5¿XJ\þ5z^

ϕ(1) = 1, K ϕ ´.

e¡[é¡5z½nw·¢)Û±ÓÛ±[é¡5z.

½n 5.2.2 (Herman-Swiatek, [37, 72]). - f : S1 → S1 ^=ê θ ¢)Û

.±Ó. K f [é¡Ýu Rθ = θ k... d, [é¡~ê=6u

θ Ú f ±)Ûòÿ¹ü ±.

éz t ∈ C\0,- ΛÚ Σt (5.1.2)Ú (5.1.3)¥½Â8Ü.5¿ c 7→ sin2(c)

´±Ï π ó¼ê. ·k

Ún 5.2.3. ¼ê τ(c) = −λ/(3 sin2(c)) ´l Λ C \ 0 ÷.

ùÚnL² Σλ ¹½Â3 (5.1.1) ¥ F ¥¤k. |^ü?

Ø, N´wÑ τ : Λ→ C \ 0 “Øõ” ŃÝ 2 XCX.

ëêm Σt ¥¼êäkkõ.vkkì? (Ún 5.3.2). Ï

§´±Ï, § Fatou ©|©a~ü. ~X, ®²§vkiÚ

Baker [33]. § Fatou8õkü;. XJ t = 1/2½ö λ, 3 Fatou

©|;ªÑS¹: Fatou ©|þ, ù Fatou ©|©OéAáÚ

½ Siegel . d, U3Ù§ Fatou ©|;, ù¿Ø.

½Â 5.2.4. f : C→ C ´¼ê, K f O ρ(f) ½Â

ρ(f) = lim supr→+∞

log logM(f, r)

log r, (5.2.3)

Ù¥ M(f, r) := sup|z|=r |f(z)|. XJ ρ(f) < +∞, K¼ê f ¡´k.

Ný, ~X, ëêm Σt ¥z¼êÑ´k, Ù¥ t 6= 0.

Ún 5.2.5. b f : C → C ´k¼ê, Ó´±Ï 2π

Û¼ê. d, f ′(0) = t 6= 0 f ":8´ kπ,±a + kπ : k ∈ Z, Ù¥ a 6= 0. K

f ∈ Σt.

y². âÚn 5.2.3, 3 c ∈ Λ, ¦√

3 sin(c) = sin(a). éu fc ∈ Σt, fc Ú f

":8´. K f/fc Û:´3 C þk f/fc 6= 0. ùL² f = fc exp(h),

Ù¥ h ´,¼ê. 5¿ f Ú fc Ñ´Û¼ê, ùL² h Ó´ó¼ê.

Ï fc k, ùL² exp(h) ´Xd. u´ h 7õª. Ï f ±

Ïu 2π ¼ê, K3ê k ¦ h(z + 2π) = h(z) + 2kπi éz z ¤á.

¯¢þ, Ï h ´ z ëY¼ê, ù k ´½e5. P H(z) = h(z) − ikz, ·k


H(z + 2π) = H(z). u´ H ´±Ïõª, l 7~ê. ùL² h(z) = ikz + b

é,~ê b ¤á. Ï h ´óê, ¤± k 7u 0. 5¿ f ′(0) = t, Ïd eb = 1. Ú

ny²..

§5.3 5z«

3Øu· ¹e, ·ò Σt ¥¼ê fc Ú Λ ¥ëê c w¤´. é

z c ∈ Σt, fc Q´Û¼êq´±Ï¼ê. Ïd z Ú −z 3 fc eÄåXÚ1

3==ÎÒ ± ¿Âe´.

3ù«¹e, · fc Fatou 8Ú Julia 8'u:é¡. AÏ:, '

X, .:7L¤éÑy.

Ún 5.3.1. ¼ê fc Fatou 8'u:é¡äk±Ï π.

y². 5¿ fc ´Û¼êäk±Ï 2π. éz n ≥ 1 Ú k ∈ Z, XJ k Ûê,

K f nc (z + kπ) = −f nc (z) = f nc (−z). XJ k óê, K f nc (z + kπ) = f nc (z).

â5x½Â,· f nc n≥1 3: z,−z Ú z+ kπ ?Ó5½ÓØ

5. y²..

XJ3 C ¥^ëY γ(t) : [0,∞) → C ÷v limt→∞ γ(t) = ∞, ¿¦

limt→∞ f(γ(t)) = ζ, K: ζ ¡´¼ê f ìC.

Ún 5.3.2. éz c ∈ Σt, fc vk?ÛkìC.

y². ·^yy². b ζ ´ fc kìC.â½Â,3^ë

Y γ(t) = x(t) + y(t)i : [0,∞)→ C, ¦ γ(t)→∞ t→∞ k fc(γ(t))→ ζ.

Ï fc ´±Ï 2π ëY¼ê, K3 0 < t1 < t2 < · · · < tn < · · ·¦ tn → ∞ n → ∞ k y(tn) → ∞. OL² y(tn) → ∞ kfc(γ(tn))→∞. ù´gñ.

éz c ∈ Σ1/2, fc 3:ká5ØÄ:. â Kœnigs 5z½n, 3½Â

3 0 SX¼ê φc, ò fc ÝØ N L1/2(w) = w/2. ù5z¼ê φc

¡´ Kœnigs ¼ê. â [53, íØ 8.4], 3:áÚþf,

¦3ùfþ, fc XÝu L1/2. ùf¡´ fc ±:%5z

«. ·^ Ωc L«. XJ\þ5^ φ′c(0) > 0, K5zN φc : Ωc → D ´. d, φc ±)Û/òÿ Ωc ;þ.

íØ 5.3.3. 5z« Ωc ´k.'u:é¡. d, φc ´Û¼ê.

§5.3. 5z« 77

y². XJ Ωc ´Ã., K fc(∂Ωc) ¹ fc kìC, ùÚn 5.3.2

gñ. é¡5âÚn 5.3.1 .

éu1(Ø, 5¿ −φc(−z) 3 Ωc þÓò fc Ý L1/2, −φc(−z) 3

:äkê. â5, ùL² φc ´Û¼ê.

Ún 5.3.4. éu¤k c ∈ Σ1/2, >. ∂Ωc ´'u:é¡²Lüé

¡.:[.

y². â [53] ¥Ún 8.5 y²L§, · ∂Ωc ¹ fc .:.

âÚn 5.3.1 ¥¤y²é¡5, ∂Ωc 'u:é¡²Lüé¡.:.

XJ·£ ∂Ωc þ¤k.:, Kekõ^ml, §Ñ´¢)Û. d

, §3.:?±"Ý. ùL² ∂Ωc ´[ ( [43, p.104]).

âÚn 5.3.4, éu?¿ c ∈ Σ1/2, K73,.:á3 ∂Ωc þ. ·

'%§´=.:. 5¿ fc .:¤8Ü´ Critc = π/2 + kπ, ±c + kπ :

k ∈ Z. ù!eÜ©òy²e¡½n.

½n 5.3.5 ((½n). 3²L π/2 Ú −π/2 ü γ ⊂ Σ1/2 ©l 0 Ú ∞,

¦XJ c á3 γ SÜ, K ∂Ωc TÐ²Lü.: ±c; XJ c á3 γ Ü, K

∂Ωc TÐ²Lü.: ±π/2.

¯¢þ, ·Äk±y² cá3:S, ∂Ωc =²Lü.

: cÚ −c. éu Im(c)é/, ·±y² ∂Ωc =¹ü.: π/2Ú −π/2.

ù´Ï3l:é/, fc Ú sin(z)/2 ~C.

éz k ∈ Z, P Ωkc = z + kπ : z ∈ Ωc. 5¿ Ω0

c = Ωc.

Ún 5.3.6. XJ i 6= j, K Ωic ∩ Ωj

c = ∅, Ù¥ i, j ∈ Z. d, éu i 6= j, XJ

Ωi

c ∩ Ωj

c 6= ∅, K3, k ∈ Z ¦ Ωi

c ∩ Ωj

c = π/2 + kπ.

y². Ø5, béu, k0 6= 0 k Ωc ∩ Ωk0c 6= ∅. é?¿áu Ωc ∩ Ωk0

c

z 6= 0, · ±z ±(z− k0π)Ñá3 Ωc ¥. Ïé?¿ k ∈ Zk π/2 + kπ 6∈ Ωc,

ùL² z 6= k0π − z. XJ k0 Ûê, K fc(z) = fc(k0π − z); XJ k0 ´óê, K

fc(z) = fc(z − k0π). Ï fc 3 Ωc þ´ü, ù´gñ. Ï fc 3 Ωc þÓ

´üéu, k0 ∈ Z, z = k0π − z ∈ Ωc ´#N, ùL²1(Ø¤á.

íØ 5.3.7. é?¿ c ∈ Σ1/2, 3, k ∈ Z ¦ ∂Ωc ∩Critc ⊂ π/2,−π/2, c+

kπ,−c− kπ.


y². XJ3 c ∈ Σ1/2 ÚüØÓ k1, k2 ∈ Z, ¦ ∂Ωc ¹ 4 ØÓ.

: ±(c+ k1π) Ú ±(c+ k2π), K Ωc ∩Ωk1−k2c ¹ c+ k1π Ú −(c+ k2π), ùÚn 5.3.6

gñ. ùL² ∂Ωc õ¹é/X ±(c + kπ) .:, Ù¥ k ∈ Z. aq/, ∂Ωc

¹õéäk/ª ±(π/2 + kπ) .:, Ù¥ k ∈ Z.

b3 c ∈ Σ1/2 ¦ ±(π/2 + k0π) ∈ ∂Ωc é, k0 ≥ 1 ¤á. 5¿ φc : Ωc → D\5z^ φ′c(0) > 05zN. - I := [φc(π/2 +k0π),−φc(π/2 +k0π)]

D ¥ë φc(π/2 + k0π) Ú −φc(π/2 + k0π) », K J0 := φ−1c (I) ´^ë

π/2 + k0π Ú −(π/2 + k0π) 'u:é¡ Jordan . P Jn = z + (2k0 + 1)nπ :

z ∈ J0, K J := ∪n∈ZJn ´±Ï (2k0 + 1)π ±Ï. é?¿ x ∈ R, P

Jx = z + x : z ∈ J, K J ∩ Jx 6= ∅. AO/, J ∩ Jπ 6= ∅. â±Ï5Úé¡5, 3,

z ∈ J ∩ Jπ ¦ Re(z) ∈ [0, π/2 + k0π]. d, Ï J0 ∈ Ωc, z ∈ Ωc ∩ Ω1c . âÚ

n 5.3.6 z = π/2 + k0π. ¯¢þ, XJ3 k 6= k0 ¦ z = π/2 + kπ ∈ J ∩ Jπ, K

π/2 + kπ ∈ Ωc ´ØU. y3· ∂Ωc ¹4 ØÓ.: ±(π/2 + k0π)

±(π/2− k0π). ùþã?Øgñ.

5¿íØ 5.3.7 ¥ k U6 c. y²½n 5.3.5, ·I`²éz c Ñ

k k = 0.

- fn : C → C X¼ê, Ù¥ fn(0) = 0, n = 1, 2, · · · . b n ªu

∞ , fn ÛÜÂñ f∞ 0 < |f ′∞(0)| < 1, K f∞ ´±:ØÄ~

X¼ê. ùL² f∞ 3:kAÛá5ØÄ:. u´éu¿© n, ·k

0 < |f ′n(0)| < 1. Ø5, béu¤k n = 1, 2, · · · , fn k±:%5z« Ωn. ?Ú, b f∞ ±:%5z« Ω∞ ´k..

6/, · Kœnigs ¼ê φn : Ωn → C 5z^ φ′n(0) = 1. ¯¢þ,

φn ½Â φn(z) = limk→∞ fkn (z)/(f ′n(0))k, Ù¥ 1 ≤ n ≤ ∞. 3¢ê rn > 0

¦ φn : Ωn → Drn ´/ limn→∞ rn = r∞. ½Â/N Gn : Ω∞ → Ωn

Gn(z) = φ−1n ( rn

r∞φ∞(z)), §±ëY/òÿ ∂Ω∞ þ. - An fn ¹ 0 áÚ

. Ï fn ÛÜ/Âñ f∞, ùL²XJ;8 E∞ ¹3 A∞ S, Kéu¿©

n, k E∞ ¹3 An S. AO/, XJ n ¿©, K3 Ω∞ m Ω′∞, ¦

Ω∞ ⊂⊂ Ω′∞ φn 3 Ω′∞ þ½Â´Ün.

·K 5.3.8. n ªu ∞ , 5z« Ωn Ú§>. ∂Ωn 3 Hausdorff

ÿÀ¿Âe©OÂñ Ω∞ Ú ∂Ω∞.

y². ·Äky² Gn : f∞(Ω∞) → fn(Ωn) ÛÜÂñðN. Ï

φ∞(f∞(Ω∞)) = Dρr∞ φn3 Ω′∞þÛÜÂñ φ∞,3 ε > 0¦ φn(f∞(Ω∞)) ⊂

§5.3. 5z« 79

Dρr∞+ε ⊂ D(ρ+1)rn/2, Ù¥ ρ = |f ′∞(0)| ∈ (0, 1). ùL² φ−1n ±3 φn(f∞(Ω∞)) þ½Â.

éu z ∈ f∞(Ω∞), 3 ξn ∈ φn(f∞(Ω∞)) ¦

|Gn(z)− z| = |Gn(z)− φ−1n φn(z)| ≤ |(φ−1

n )′(ξn)| |rnφ∞(z)/r∞ − φn(z)|. (5.3.1)

5¿ f∞(Ω∞) ´;, f∞(Ω∞) ⊂⊂ Ω∞ φn 3 Ω′∞ þÛÜÂñ φ∞, K3~

ê a > 0 ¦éu¿© n k |φ′n(z)| ≥ a 3 f∞(Ω∞) ¤á. â (5.3.1), ùL²

n ªu ∞ |Gn(z) − z| ≤ 1ar∞

(rn|φ∞(z) − φn(z)| + |φn(z)||rn − r∞|) → 0, ù´Ï

(φ−1n )′(ξn) = 1/φ′n(ξ′n) é, ξ′n ∈ f∞(Ω∞) ¤á. l Gn : f∞(Ω∞)→ fn(Ωn) ÛÜ

ÂñðN.

é?¿ 0 < r < r∞, |^þãaq?Ø±`² Gn : φ−1∞ (Dr) →

φ−1n (Drrn/r∞) ÛÜÂñðN. Kéu?¿ ε > 0, 3¿© N > 0 ¦

n ≥ N , φ−1∞ (Dr∞−2ε) ⊂ Gn(φ−1

∞ (Dr∞−ε)) ⊂ Ω∞. â Gn ½Â, ù¿X

φ−1∞ (Dr∞−2ε) ⊂ Ωn.

À¿© ε > 0 ¦ Ω∞ ⊂⊂ Ωε∞ ⊂ Ω′∞, Ù¥ Ωε

∞ := z : d(z,Ω∞) < ε. Ï Ω∞ ´k., ùL² Ωε

∞ ´;. du φ∞(Ω∞) = Dr∞ , l Dr∞ ⊂⊂ φ∞(Ωε∞). -

δ(ε) > 0 ∂Dr∞ Ú ∂φ∞(Ωε∞) mî¼ål. Ï limn→∞ rn = r∞, 3 N1 > 0, ¦

n ≥ N1 , k rn < r∞ + δ(ε)/2. Ï φn 3 Ωε∞ Ú ∂Ωε

∞ þÛÜÂñ φ∞,

ùL²XJ n ¿©, K Drn ⊂⊂ φn(Ωε∞).

du φn(0) = 0 Ú φ−1∞ (Dr∞−2ε) ⊂ Ωn, 3 r > 0 ¦ φ−1

n ±l Dr

½Â φ−1n (Dr) ⊂ Ωn ∩ Ω∞. Ï Drn Ø¹ φn ., u´ φ−1

n 3 Dr þ

±)Û/òÿ φ−1n |Drn . ¯¢þ, Ï φ−1

n (Dr) ⊂ Ωn, φ−1n (Drn) = Ωn. du

Drn ⊂⊂ φn(Ωε∞), ùL² Ωn ⊂ Ωε

∞. y²..

éu?Û'ê R > 0, P ΛR = c ∈ Σ1/2 : |c| ≥ R. éu?Û'êε > 0, P Λε = c ∈ Σ1/2 : 0 < |c| ≤ ε.

Ún 5.3.9. 3 R > 0 ¦éuz c ∈ ΛR, ∂Ωc TÐ²Lü.: π/2 Ú

−π/2. Ó, 3 ε > 0 ¦éz c ∈ Λε, ∂Ωc TÐ²Lü.: c Ú −c.

y². Ï c3 Σ1/2¥ªu∞, fcÛÜÂñ f∞(z) = sin(z)/2,u´

â·K 5.3.8 Ωc3 HausdorffÿÀeÂñ Ω∞. ùp, Ω∞L« f∞5z«

. §´k., 'u:é¡²L π/2 Ú −π/2 ( [79, Ún 2]). Ï 3 Ω∞

k. Ω′∞, ¦ Im(c) ¿©, k Ωc ⊂⊂ Ω′∞ Ω′∞ ∩Critc = π/2,−π/2.âÚn 5.3.4, Im(c) ¿©, ùL² ∂Ωc TÐ²Lü.: π/2 Ú −π/2.

éu c á3 0 /, 5¿ c ªu 0 , L−1c fc Lc(z) ÛÜ

/Âñngõª f0(z) = z/2 − z3/6, Ù¥ Lc(z) = cz. ùL²3 Hausdorff ÿÀe,


L−1c fc Lc(z) 5z« Ω′c ÛÜÂñ f0 5z« Ω0. §´

k.Ï f0 ´õª. u´3 Ω0 k. Ω′0, ¦ c v

ÿ, k Ω′c ⊂⊂ Ω′0, ùdu cΩ′c ⊂⊂ cΩ′0, Ù¥ cΩ′c := cz : z ∈ Ω′c = Ωc. ùL² c ¿

©ÿ, ∂Ωc ==¹.: c Ú −c. y²..

5¿ Λ = c : |Re(c)| ≤ π/2 \ 0 þ!¥½Âëêm, u´ sin : Int(Λ) →C\(−∞,−1)∪0∪(1,+∞)´/. éz β ∈ (0, π)∪(π, 2π),- ηβ(t) = sin−1(teiβ :

0 < t < +∞). K ηβ ´ Σ1/2¥ë 0Ú∞÷v limt→0 ηβ(t) = 0Ú limt→∞ ηβ(t) =∞^1w.

âÚn 5.3.9, 3 Σ1/2 ©l 0 Ú ∞ ;f8 M , ¦ c á3 Σ1/2 \M k.©| (½Ã.©|) , K ∂Ωc TÐ²Lü.: c Ú −c (½ π/2 Ú −π/2). é

z β ∈ (0, π) ∪ (π, 2π), P

tβ = sup t : éu 0 < s < t, ∂Ωηβ(s) TÐ²Lü.: ± c. (5.3.2)

â tβ ½Â, ·±é sk → t−β ¦éz k k ±c ∈ ∂Ωηβ(sk). â·K

5.3.8, Ï ∂Ωηβ(t) 3 Hausdorff ÿÀ¥ëY6u t, ùL²48 ∂Ωηβ(tβ) Ó¹ ±c.â tβ ½ÂÚíØ 5.3.7, · ∂Ωηβ(tβ) Ó¹ ±π/2.

- γ ±π/2 Ú ηβ(tβ) : β ∈ (0, π) ∪ (π, 2π) ¿, ·k

Ún 5.3.10. éz c ∈ γ \ ±π/2, >. ∂Ωc TÐ²L 4 .: ±π/2 Ú ±c.d, γ ´^ Jordan .

y². ·Iy²1(Ø.½Â χ : (0, π)∪ (π, 2π)→ γ \±π/2 χ(β) =

ηβ(tβ). â ηβ(t) ëY5Ú·K 5.3.8 , é?¿ β0 ∈ (0, π) ∪ (π, 2π) Ú

βn → β0, XJ n ¿©, K |ηβn(tβn) − ηβn(tβ0)| Ú |ηβn(tβ0) − ηβ0(tβ0)| Ñ¿©. ùL

² |χ(βn)− χ(β0)| v χ 3 β0 ?´ëY.

½Â χ(0) = π/2 χ(π) = −π/2. - εn > 0 Âñ 0 S, K c :

3 cn ∈ ηεn ¦ limn→∞ cn = c 3 Σ1/2 ¥à:´ E = (0, π/2) ∪ π/2 + iy :

y ≥ 0. ·äóXJ 0 < c < π/2, K ∂Ωc TÐ²Lü.: c Ú −c, XJc ∈ π/2 + iy : y ≥ 0, K ∂Ωc TÐ²Lü.: π/2 Ú −π/2. ¯¢þ, XJ c ∈ E,

K sin2(c) ∈ R, ùL² Ωc Ó'u¢¶é¡. XJ 0 < c < π/2, Ï Ωc ´üëÏ,

K ∂Ωc ∩ Critc = c,−c. XJ c ∈ π/2 + iy : y ≥ 0, KâÚn 5.3.6 1(Ø

k ∂Ωc ∩ Critc = π/2,−π/2. â tβ ½Â, N´wÑ limn→∞ ηεn(tεn) = π/2. XJ

ε′n < 2π ´Âñ 2π S, K|âq?Øk limn→∞ ηε′n(tε′n) = π/2. ùL²

χ 3 0 ?´ëY. Ó/, ±`² χ 3 π ù:´ëY. y3·këY

V χ : [0, 2π)→ γ, du χ(0) = χ(2π−) = π/2, ùL² γ ´^ Jordan .

§5.3. 5z« 81

e5ò¬y² γ TÐÒ´½n 5.3.5 ¥ãü4. éu?Û½äk

JÜ c ∈ γ, ·Ñ fc |Ü&E. äkKJÜ c ∈ γ ±aq/©Û.

Ø3 ±π/2 Ú ±c ù 4 :, >. ∂Ωc ´¢)Û. 3 f−1c (fc(Ωc)) 4 k

.üëÏ©|N3 Ωc þ, Ù¥ Ω−1c Ú Ω1

c ©ON3 −π/2 Ú π/2 þ, Ù§ü

©| D1 Ú D2 ©ON3 c Ú −c þ. u´ f−1c (fc(Ωc)) =

⋃k∈Z Ωk

c ∪ Dk1 ∪ Dk

2 , Ù¥

Dki := z + kπ : z ∈ Di i = 1, 2. N´wÑ f−1

c (Ωc \ fc(Ωc)) düüëÏ©| E+

Ú E− ¤, §©O uëÏ8 f−1c (fc(Ωc)) þ¡Úe¡, §wå5´ü^Ã

f. ã 5.1.

π/2−π/2

c

−c

ΩcΩ−1c Ω1

c

D1

D2

D−11

D12

E+

E−

ã 5.1 fc ∈ Σ1/2 ÄåXÚ²¡, Ù¥ c = 0.944180 . . . (1 + i) ¦ ∂Ωc TÐ

²Lüéé¡.: ±π/2Ú ±c. ¥mxÚÜ©á3 fc Fatou8¥,

ÚÜ©KL«3 fc Se<º∞@:. Fatou8¥L«@

¦ |φc(z)| = |φc(c)|/2n :, Ù¥ n ∈ Z. ã: [−4.0, 4.0]× [−2.7, 2.7].

5¿ φc : Ωc → D \5^ φ′c(0) > 0 5zN. éu?¿ 0 ≤ r ≤ 1,

½Â ηr(t) := φ−1c (reit), Ù¥ t ∈ R. 3 t0 ∈ [0, 2π) ¦ η1(t0) = c. AO/,

η1(t0 + 2kπ) = c Ú η1(t0 + (2k+ 1)π) = −c, Ù¥ k ∈ Z. éu?¿ r ∈ (1/2, 1], - C+r

Ú C−r ©O ηr 3 E+ Ú E− ¥J,.äN/`,éu t ∈ Rk fc(C±r (t)) = ηr(t).

?Ú,éu½ t,·¦J, C+r (t)Ú C−r (t)ëY6u r ∈ (1/2, 1].

éu t ∈ R, P C+1/2(t) = limr→(1/2)+ C

+r (t) C−1/2(t) = limr→(1/2)+ C

−r (t). N´wÑ

C+1/2 Ú C−1/2 Ñ´ f−1

c (fc(∂Ωc)) = C+1/2 ∪ C

−1/2. - c = C+

1/2(t0) = C+1/2(t0 + 2π)

−c = C−1/2(t0−π) = C−1/2(t0+π). éu?¿ z ∈ E+½ö E−,3 r ∈ (1/2, 1]Ú

t ∈ R¦ z = C+r (t)½ö C−r (t). AO/,éu k ∈ Z,·k c+kπ = C+

1/2(t0−3kπ) =


C+1/2(t0 + 2π − 3kπ) −c+ kπ = C−1/2(t0 − π + 3kπ) = C−1/2(t0 + π + 3kπ).

3 t1 ¦ −π/2 = η1(t1) 0 < t1 − t0 < π. ùL² π/2 = η1(t1 + π). é

u k ∈ Z, P Hk = z : (k − 1)π ≤ Re(z) ≤ (k + 1)π/2πZ, K fc : Hk → C ŃÝu 6 ©|CX. AO/, fc : Hk ∩E+ (½ Hk ∩E−)→ Ωc \ fc(Ωc) ŃÝ

u 3 ©|CX. éu c, −π/2, −c Ú π/2, §3 C+1 ∩Hk Ú C−1 ∩Hk þ©Ok 3

c. ùc±^S. äN/, c, −π/2, −c Ú π/2 3 C±1 þc´ C±1 (t0 + 2kπ),

C±1 (t1 + 2kπ), C±1 (t0 + π + 2kπ) Ú C±1 (t1 + π + 2kπ), Ù¥ k ∈ Z.

P D3 =⋃k<0 Ωk

c ∪ Dk1 ∪ Dk

2 , D4 =⋃k>0 Ωk

c ∪ Dk1 ∪ Dk

2 D =⋃

1≤i≤4Di. ·k

f−1c (D) =

⋃1≤i≤4,k1∈Z(D+

i,k1∪D−i,k1), Ù¥ D±1,k1 , D

±2,k1

, D±3,k1 Ú D±4,k1 ©O f−1c (D) N

3 C±1 (t0 + 2k1π), C±1 (t0 +π+ 2k1π), C±1 (t1 + 2k1π)Ú C±1 (t1 +π+ 2k1π)þ©|. £

E± =⋃

1/2<r≤1C±r , K f−2

c (Ωc \ fc(Ωc)) =⋃k1∈Z(E+

k1∪ E−k1), Ù¥ E±k1 =

⋃1/2<r≤1C

±r,k1

,

fc(C+r,k1

(t) ∪ C−r,k1(t)) = C+r (t) ∪ C−r (t) t ∈ R, E±k1 >.¹ C±1 (t0 + 2k1π). 5¿

f−1c (Ωc)´ëÏ,K C\f−1

c (Ωc)©|Ñ´üëÏ. AO/, C\f−1c (Ωc) = Y +∪Y −,

Ù¥ Y +Ú Y −L« C\f−1c (Ωc)©Oá3 f−1

c (Ωc)þ¡Úe¡©|. N´wÑ f−2c (Ωc)

Ó´üëÏ, Ïd C \ f−2c (Ωc) =

⋃k1∈Z(Y +

k1∪ Y −k1 ), Ù¥ Y +

k1(½ Y −k1 ) E+

k1(½

E−k1)>.. 5¿´ E+ Ú Y + (½ E− Ú Y −) u f−1c (fc(Ωc))þ(½e).

, , E+k1Ú Y +

k1(½ E−k1 Ú Y −k1 ) ±Ø´ù.

éu 1 ≤ i ≤ 4, 1/2 < r ≤ 1, n ≥ 2 Ú k1, · · · , kn ∈ Z, ·±8B/½

Â D±i,k1,··· ,kn , C±r,k1,··· ,kn , E±k1,··· ,kn ±9 Y ±k1,··· ,kn . ¯¢þ, b D±i,k1,··· ,kn−1, C±r,k1,··· ,kn−1

,

E±k1,··· ,kn−1Ú Y ±k1,··· ,kn−1

®²½ÂÐ. Ï fc 3 Y ±k1,··· ,kn−1þ´ü, 3

D±i,k1,··· ,kn , C±r,k1,··· ,kn , E±k1,··· ,kn Ú Y ±k1,··· ,kn ¦

(1) D±i,k1,··· ,kn , C±r,k1,··· ,kn , E±k1,··· ,kn , Y ±k1,··· ,kn Ñ¹3 Y ±k1,··· ,kn−1¥;

(2) fc(⋃k1∈ZD

+i,k1,··· ,kn ∪D

−i,k1,··· ,kn) = D+

i,k2,··· ,kn ∪D−i,k2,··· ,kn ;

(3) fc(⋃k1∈ZC

+r,k1,··· ,kn ∪ C

−r,k1,··· ,kn) = C+

r,k2,··· ,kn ∪ C−r,k2,··· ,kn ;

(4) fc(⋃k1∈ZE

+k1,··· ,kn ∪ E

−k1,··· ,kn) = E+

k2,··· ,kn ∪ E−k2,··· ,kn ;

(5) fc(⋃k1∈Z Y

+k1,··· ,kn ∪ Y

−k1,··· ,kn) = Y +

k2,··· ,kn ∪ Y−k2,··· ,kn .

éu?¿ n ≥ 2, ·k f−nc (Ωc) \ f−n+1c (Ωc) =

⋃1≤i≤4, k1,··· ,kn−1∈ZD

±i,k1,··· ,kn−1

∪E±k1,··· ,kn−1

C \ f−nc (Ωc) =⋃k1,··· ,kn−1∈Z Y

±k1,··· ,kn−1

.

éz c ∈ Σ1/2, 5¿ φc : Ωc → D ¦ φ′c(0) > 0 5zN φc Û

¼ê. éu k 6= 0, P

γk := c ∈ Σ1/2 : ∂Ωc ∩ Critc = π/2,−π/2, c+ kπ,−c− kπ \ ±π/2 (5.3.3)

γ0 := γ. é?¿ c ∈ γk0 (XJù c 3), Ï φc : Ωc → D ±Ó/ò

§5.3. 5z« 83

ÿ φc : Ωc → D, ·½Â π/2 Ú c+ k0π mÝ

A(c) = argφc(c+ k0π)/φc(π/2), (5.3.4)

Ù¥ 0 ≤ arg(z) < 2π Ë¼ê. AO/, XJ c ∈ γ K A(c) = argφc(c)/φc(π/2), A(π/2) = 0, A(−π/2) = π. d, c 3 γ þCz, A(c) ëY6u c. XJ3,

k0 6= 0 ¦ γk0 6= ∅, K3 c ∈ γk0 ¦ A(c) = argφc(c+ k0π)/φc(π/2) 6= 0 Ú π.

âcÓ©Û, c ∈ γ , fc fc kÓ|Ü&E. ½öoÑ/`, fc Äå

XÚaqu fc 3ã 5.1 ¥¤«¹.

·K 5.3.11. XJ A(c1) = A(c2), Ù¥ c1, c2 ∈⋃k∈Z γk, K c1 = c2.

y². XJ c1 Ú c2 ¥ku π/2 ½ −π/2, K?Ø´². e5, ·

b§üÑØu π/2 Ú −π/2. Ï¤k.:Ñ:¤áÚ, Kéz

c ∈⋃k∈Z γk, fc Fatou 8=3;.

Ø5, b c1 Ú c2 Ñáu γ. P φ = φ−1c2 R% φc1 : Ωc1 → Ωc2 , Ù¥

R%(z) = e2πi% L«f5^= % = arg(φc2(π/2)/φc1(π/2))/2π. Ï A(c1) = A(c2), /

N φ3 Ωc1 þò fc1 Ý fc2 ,÷v φ(π/2) = π/2Ú φ(c1) = c2. - ϕ0 : C→ Cφ ¦ ϕ0(∞) = ∞ [/*Ü. ùoÚÏ>. ∂Ωc ´[ (

[43, ½n 8.2]). ·ò^ fc ÄåXÚ8B/½Â[/N ϕn : C→ C.

üå, ·òÎÒ fcj Ú Ωcj ¥eI cj ^ j O, Ù¥ j = 1, 2. âcé

fc |Ü&E©Û, ùL²éz©| Dkj (c1) Ú Ωk

1, 3éA Dkj (c2) Ú Ωk

2,

Ù¥ j = 1, 2 k ∈ Z. 3 Ω1 þ- ϕ1 = ϕ0 = φ, ÏLÀJ f2 ·_, ·±½

Â ϕ1 = f−12 ϕ0 f1, §ò D

k

j (c1) Ú Ωk

1 ©ON Dk

j (c2) Ú Ωk

2.

éu?Û z ∈ f−11 (Ω1 \ f1(Ω1)) = E+(c1) ∪ E−(c1), ½Â ϕ1 = f−1

2 ϕ0 f1, À

J_©| f−12 ¦XJ z = C+

r (t)(c1) ½ C−r (t)(c1), Ù¥ t ∈ R 1/2 < r ≤ 1,

K ϕ1(z) = C+r (t)(c2) ½ C−r (t)(c2). u ϕ1 3>. f−1

1 (f(∂Ω1)) þëY5, ùL²

ϕ1 : f−11 (Ω1)→ f−1

2 (Ω2) ´X.

·Iò ϕ1 òÿ Cþ[/N. Ï C\f−11 (Ω1)ü©| Y +(c1)

Ú Y −(c1) Ñ´üëÏ3 Ω2 ¡Ø3 f2 ., 3 C \ f−11 (Ω1) þ·½Â

ϕ1 ϕ1 = f−12 ϕ0 f1 : Y ±(c1)→ Y ±(c2). N´wÑ ϕ1 : C→ C ´[/N

ϕ1 Ú ϕ0 û÷v ‖µϕ1‖ = ‖µϕ0‖.béz 1 ≤ k ≤ n−1,3[/N ϕk : C→ C¦ f2 ϕk = ϕk−1 f1

ϕk : f−k1 (Ω1)→ f−k2 (Ω2) ´X. y3·Xe8B/½Â ϕn.

3 f−n+11 (Ω1) þ½Â ϕn = ϕn−1. éz D±i,k1,··· ,kn−1

(c1) ⊂ f−n1 (f1(Ω1)) \ f−n+11 (Ω1),

3éAD±i,k1,··· ,kn−1(c2) ⊂ f−n2 (f2(Ω2))\f−n+1

2 (Ω2),Ù¥ 1 ≤ i ≤ 4 k1, · · · , kn−1

∈ Z. ÏLÀJ f2·_,½Â ϕn = f−12 ϕn−1f1 : D

±i,k1,··· ,kn−1

(c1)→ D±i,k1,··· ,kn−1

(c2).


é?Û z ∈ f−n1 (Ω1 \ f1(Ω1)) = E+k1,··· ,kn−1

(c1) ∪ E−k1,··· ,kn−1(c1), ½Â ϕ1 = f−1

2 ϕn−1 f1. ÀJ_ f−1

2 ¦XJ z = C+r,k1,··· ,kn−1

(t)(c1)½ö C−r,k1,··· ,kn−1(t)(c1),Ù¥ t ∈ R

1/2 < r ≤ 1, K ϕn(z) = C+r,k1,··· ,kn−1

(t)(c2) ½ C−r,k1,··· ,kn−1(t)(c2). u ϕ1 3>.

f−n1 (f1(∂Ω1)) þëY5, ùL²·kl f−n1 (Ω1) f−n2 (Ω2) )ÛN ϕn.

5¿ C \ f−nc (Ωc) =⋃k1,··· ,kn−1∈Z Y

+k1,··· ,kn−1

∪ Y −k1,··· ,kn−1. - Y +

k1,··· ,kn−1(c1) C \

f−n1 (Ω1) ©| (Y −k1,··· ,kn−1aq?Ø), 3éA©| Y +

k1,··· ,kn−1(c2) ⊂ C \

f−n2 (Ω2). Ï Y +k1,··· ,kn−1

´üëÏØ¹., 3J,N ϕn : Y +k1,··· ,kn−1

(c1) →Y +k1,··· ,kn−1

(c2) ¦ f2 ϕn = ϕn−1 f1.

3ù«¹e, ·[/N ϕn : C → C ¦ f2 ϕn = ϕn−1 f1

‖µϕn‖ = ‖µϕn−1‖. â8BL§, ·l C NgC[/N ϕnn≥0 ¦

ϕn 3 f−n1 (Ω1) þ´/ ϕn Beltrami Xê÷véz n ≥ 1 k ‖µϕn‖∞ =

‖µϕn−1‖∞. ùL²3~ê 0 < k < 1, ¦ ‖µϕn‖∞ ≤ k < 1 é¤k n ≥ 0 ¤á.

ÀJ ϕnn≥0 Âñfüg, ·ü4[/N ϕ Ú ψ. §± 0,

π/2Ú∞ØÄ,÷v f2ϕ(z) = ψf1(z),ù´Ïéz n ≥ 1k f2ϕn = ϕn−1f1.

âE, 3⋃n≥0 f

−n1 (Ω1) þk ϕ(z) = ψ(z). Ï fc Fatou 83 C þ´È

, ϕ = ψ 3E²¡Èf8þ¤á. âëY5, ùL² ϕ ψ 3 C þ. u´ f1 Ú f2 3 C þ´[/Ý3 Fatou 8þ´/Ý.

â [45, ·K 1.14], éu Julia 8¥A¤k:, Ù ω-48o¹3ÛÉ

c;¥, o<ºÃ¡. y3ÏÛÉc;á3 Fatou 8¥, ùL²

Julia 83"ÿÝ¿Âeu<º8. â [64] ¥½n 1.2, z f ∈ Σ1/2 3

<º8þvkØC. ùL²z f ∈ Σ1/2 3 Julia 8þvkØC.

·âqu [56]¥?Ø5y² f1 3E²¡þ/Ýu f2. - µϕ

[/N ϕEA,Ï ϕò f1Ý f2,Ïd µϕ´ f1–ØC. =, f ∗1 (µϕ) = µϕ.

OL² |µϕ(f1(z))| = |µϕ(z)|. b |µϕ(z)| > 0 3 E þ¤á, ù E ´ f1

äkÿÝf8, u´·k |µϕ(z)/|µϕ(z)‖ = 1, µϕ(z)/|µϕ(z)| 3 E þ

´ f1–ØC. ùL² µϕ(z)

|µϕ(z)|dzdz´ J(f1) þØC, ù´gñ. u´ µϕ = 0

3 C þA??¤á, ϕ 3 C ´/. Ï ϕ ± 0, π/2 Ú ∞ ØÄ, ù`² ϕ

´ðNl f1 = f2.

íØ 5.3.12. γ = c ∈ Σ1/2 : ∂Ωc ∩ Critc = π/2,−π/2, c,−c. é?¿ k 6= 0,

k γk = ∅.

y². âÚn 5.3.10, γ´^²L π/2Ú −π/2 Jordan.Ï A(π/2) = 0,

A(−π/2) = π A(c) ëY6u c, ùL² A : γ → [0, 2π) ´÷. XJ3 c 6∈ γ ¦ ∂Ωc TÐ²L4 .: ±π/2 Ú ±c, K§·K 5.3.11 gñ. 1(Ø±d1

§5.4. äk û[ 85

(ØÚ·K 5.3.11 íÑ.

·K 5.3.11 ÚíØ 5.3.12 w·ëê c ∈ γ dÝ A(c) û½, ùÝ¡

´ π/2 Ú c m “/Ý”.

½n 5.3.5 Ú 5.1.2 y². XJ3 c ∈ Σ1/2 ¦ ∂Ωc TÐ²Lü.:

±(c + k0π), Ù¥ k0 6= 0. ÀJ^ëY η : (0, 1) → Σ1/2 ¦ limt→0 η(t) = c

limt→1 η(t) =∞. âÚn 5.3.9, cá3∞S, ∂ΩcTÐ²Lü.

: π/2Ú −π/2. âëY5,3 c ∈ η ¦ ∂Ωc ²L 4.: ±π/2Ú ±(c+k0π),

ùíØ 5.3.12 1(Øgñ. Ï XJ c ∈ γext, ∂Ωc UTÐ²Lü.:

π/2 Ú −π/2. u´(½n±d γ ½Â, Ún 5.3.10 ÚíØ 5.3.12 íÑ.

§5.4 äk û[

®² ∂Ωc ´[, y²ÃâN´ëY, ·I`²>. ∂Ωc

äk û.

5¿ c 3 Σ1/2 ¥ªu ∞ , = Im(c) → ∞ , ¼ê fc ÛÜÂñu

f∞(z) = sin(z)/2. Ïdéu ∞ 3 Σ1/2 ?Û4, ·±ò ∞ \ù4þ, ¦ ∞ #C¤ Σ1/2 ∪ f∞ ;f8.

XJ c á3 0 S, 5¿ c ªu 0 , L−1c fc Lc(z) ÛÜ

/Âñngõª f0(z) = z/2− z3/6, Ù¥ Lc(z) = cz. |^Óg, ·

0 3 L−1c fc Lc(z) : c ∈ Σ1/2 ∪ f0 ¥;. Ï Ωc ëY6u c, |^k

CX, ·k

Ún 5.4.1. 3~ê M > 0 ¦éu c ∈ Σ1/2 k Ωc ⊂ DM .

e5·y²

½n 5.4.2. >. ∂Ωc ´ K–[, ~ê K Ø6u c ∈ Σ1/2.

y². y²g´Äu [41]. ·=Ié c ∈ Σ1/2\±π/2y² ∂Ωc ´ K–

[. éü¢ê a = a(c)Ú b = b(c),·^ a b5L« a ≤ Cb,Ù¥ C > 0´=

6u c~ê. é?¿½ c ∈ Σ1/2ÚØÓ x, y ∈ ∂Ωc,- L = [x, y]ë xÚ

y ã, I, J ∂Ωc\x, yü¦ |fc(I)| ≤ |fc(J)|ëÏ©|. âÚn 2.4.1

(- r = 1/2) Ú»½Â, ·k |fc(I)| |fc(x)− fc(y)| ≤ Diam(fc(L)) ≤ |fc(L)|.éu½ r > 1 Ú Diam(I)/(8r) ≤ m ≤ (8r − 1)Diam(I)/(8r), ½Â

Ar = z : m ≤ |z − x| ≤ m+ Diam(I)/(8r).


- I1 Ar ∩ I ¦ |I1| ≥ Diam(I)/(8r) ©|. Ï I1 Ú L ;8, 3

u ∈ I1 Ú v ∈ L ¦ |f ′c(u)| = minz∈I1 |f ′c(z)| |f ′c(v)| = maxz∈L |f ′c(z)|. Ïd·k |fc(L)| ≤ |f ′c(v)||L| Ïd Diam(I)|f ′c(u)|/(8r) ≤ |f ′c(u)||I1| ≤ |fc(I1)| ≤ |fc(I)| |fc(L)| ≤ |f ′c(v)||L|, ùL² Diam(I)/|L| |f ′c(v)|/|f ′c(u)|. y²ù½n, ·I

|f ′c(v)|/|f ′c(u)| þ., ½ö*/w, u ATl.:.

5¿ Critc fc.:8Ü.·äó3· rÚm¦é?¿ w ∈ Critc,

k |u − w| ≥ Diam(I)/(8r). ¯¢þ, âÚn 5.4.1, D(x,Diam(I)) ⊂ DM

õ¹kõ, b´ N > 0 .:. - r = N , é k = 1, · · · , 4r, ÄAk = z : (2k − 1) Diam(I)/(8r) ≤ |z − x| ≤ 2kDiam(I)/(8r) . 3 Ak0 ¦

é?¿ u ∈ Ak0 , k |u− w| ≥ Diam(I)/(8r) éz w ∈ Critc ¤á.

y3Ä Ar, Ù¥ r = N m = (2k0 − 1) Diam(I)/(8r). Ï u ∈ I1 ⊂ Ar

v ∈ L, K |u − x| ≤ Diam(I) |v − x| ≤ |L| ≤ Diam(I). Ïd |u − v| ≤ 2 Diam(I) ≤16r |u − w| éz w ∈ Critc ¤á. l |v − w| ≤ |v − u| + |u − w| |u − w|, ùL²|v − w|/|u− w| 1.

3~ê a > 0 ¦é?¿ w ∈ Critc, k | sin(z − w)|/|z − w| ≤ a é

z ∈ DM ¤á, Ù¥ DM ´Ún 5.4.1 ¥½Â. éu k ∈ Z, P Hk = z : kπ ≤Re(z) < (k + 1)π. XJ Re(c) ∈ [π/4, π/2], c 6= π/2 u ∈ Hku , Ù¥ ku ∈ Z 6uu. K Crit′c := Hku ∩ Critc = c + kuπ, π/2 + kuπ,−c + (ku + 1)π. 5¿ Crit′c ⊂ z :

π/4 + kuπ ≤ Re(z) ≤ 3π/4 + kuπ Critc \ Crit′c Hku ålu π/4, ùL²3

~ê b > 0 ¦ | sin(z − w′)|/|z − w′| ≥ b é w′ ∈ Crit′c Ú z ∈ Hku ∩ DM ¤á. Ï

|v − w|/|u− w| 1 é w ∈ Critc ¤á, ·k

| sin(v − w′)|| sin(u− w′)|

=|v − w′||u− w′|

| sin(v − w′)|/|v − w′|| sin(u− w′)|/|u− w′|

≤ |v − w′|

|u− w′|a

b 1. (5.4.1)

5¿ f ′c(z) = sin(z−π/2) sin(z+c) sin(z−c)/(2 sin2(c)) = sin(z−w′1) sin(z−w′2) sin(z−w′3)/(2 sin2(c)), Ù¥ Crit′c = w′1, w′2, w′3, l â (5.4.1) ·k |f ′c(v)|/|f ′c(u)| 1.

ùL² Diam(I)/|L| 1.

XJ Re(c) ∈ [0, π/4), ½Â Hku = z : −3π/8 + kuπ ≤ Re(z) < 5π/8 + kuπ. |^þãaq?Ø, ·Ók Diam(I)/|L| 1. XJ Re(c) ∈ [−π/2, 0), K

Re(−c) ∈ (0, π/2]. 5¿ fc = f−c. ½ny..

§5.5 ÃâNEÚ§ëY5

ù!¥, ·½Âl Σ1/2 Σλ ÃâN S ¿y²§ëY5. Ìg

´ÏLE.Nò5z« Ωc =¤ Siegel . >. ∂Ωc [

§5.5. ÃâNEÚ§ëY5 87

/Ýu Siegel >.. éuEù.N©, [41, 79].

- γint Ú γext ©OL« Σ1/2 \ γ k.ÚÃ.©| (5¿ γext dü©||¤).

â½n 5.3.5 ¥¤ã Σ1/2 (½n, e5, ·=r5¿å8¥3Ã.©| γext

þ, éu γint ±?1aq?Ø.

éz c ∈ γext = γext ∪ γ ∪ f∞, Ù¥ f∞(z) = sin(z)/2, · fc(∂Ωc) ´

¢)Û ∂Ωc ¹üé¡.: ±π/2. P Uc = C \ Ωc, Vc = C \ fc(Ωc)

vc = fc(π/2) ∈ fc(∂Ωc). âÿ Riemann N½n, éz ξ ∈ ∂Ωc, 3

/N hξc : Vc → Uc ¦ hξc(∞) =∞ hξc(vc) = ξ. du Uc Ú Vc 'u:é¡,

âÚn 5.3.3 ¥aq?Ø, hξc ´Û¼ê hξc(−vc) = −ξ. ξ 3 ∂Ωc þ$Ä, N hξc fc|∂Ωc ´l ∂Ωc gCëYÚüN±Ó

x. â [39] ¥·K 11.1.9, 3 ξc ∈ ∂Ωc, ¦hξcc fc|∂Ωc ^=ê´·3

c½Ãnê θ. üå, ·P hξcc hc.

- ϕc : C \D→ Uc ÷v ϕc(∞) =∞ ϕc(1) = π/2 Riemann N. aquk

c?Ø, ϕc ´Û¼ê. P sc = ϕ−1c hc fc ϕc, K sc|S1 : S1 → S1 ´^=ê θ

.±Ó. e¡Úny²~aqu [41] Ún 4.2 Ú [79] ¥Ún 8.

Ún 5.5.1. 3¹ S1 m A, ¦éu?¿ c ∈ γext, sc :

S1 → S1 ±)Ûòÿ A.

y². 5¿ γext ´;. Iy²éuz c0 ∈ γext, 3 c0 3 γext ¥

m D(c0) ±9 S1 m A(c0), ¦éz c ∈ D(c0), sc ±)Ûòÿ

A(c0) =.

Äk·b c ∈ γext ∪ f∞, Ïd ∂Ωc TÐ¹ü.: ±π/2. ?Øò¬©¤

ü«¹.

1«¹, é?Û z ∈ S1 Ú z 6= 1,−1, · sc 3 z l¡N3 Dþ N ′ = D(z, ε) \ D S´X. ùp, ε v¦ sc ò N ′ Ó/N

sc(N′), §Ó´ sc(z) l¡N3 D þ. âÚn 5.2.1, sc ±

X/òÿ z m N þ, ¿¦ sc ò N Ó/N÷ sc(z) ∈ S1

m sc(N) þ.

1«¹,XJ z = 1 (aq?Ø±û z = −1),- sc = (ϕ−1c hc) (fc ϕc).

·Äkl D¡ÀJ 1W ′,, fcϕc òW ′ N÷ÿÀ

fcϕc(W ′),ò)ÛãW ′∩S1 Ó/N ∂Vc þ^)Ûã. ùp, fcϕc|W ′Ø´_;. ¯¢þ, W ′ ±©¤nÜ©,Ù¥üÜ© fc ϕc N fc ϕc(W ′)∩Vc, eÜ©N fc ϕc(W ′) \ Vc. ÓâÚn 5.2.1, fc ϕc|W ′ ±òÿ W ′

m, § 3 : 1 /N fc ϕc(1) m. Ï ∂Vc ´)Û, 3


fc ϕc(1) l¡N3 ∂Vc þ N ′1, ¦ ϕ−1c hc X/òÿ N ′1

m N1, ò N1 Ó/N ϕ−1c hc(N1). ¿© N ′1, ¦ N ′1 ⊂ fc ϕc(W ′).

ùL² sc ±)Û/òÿ± 1 %mþ.

Ï fc, hc Ú ϕc ëY6u c, · sc ëY6u c. Ïdéz c0 ∈γext ∪ f∞ Ú z ∈ S1, 3 γext ¥m D(c0), ±9 z m B(z)

¦éz c ∈ D(c0), sc|B(z)∩S1 ±X/òÿ B(z)þ. Ï S1 ´;, 3 S1

m A(c0) ¦éz c ∈ D(c0), sc|S1 ±X/òÿ A(c0) þ.

XJ c ∈ γ, þ¡òÿL§±aq/?1, ´dIÄüé.: ±1 Ú

±ϕc(c). Ï γext ´;, 3kõ D(ci), i = 1, · · · , n, ¦ γext §¿CX.

- A = A(c1) ∩ · · · ∩ A(cn), Ké?Û c ∈ γext, sc ±)Û/òÿ A þ.

5 5.5.2. N sc ØUòÿ ±1 S.

dÚn 5.5.1, · sc ´äkk..^=ê θ ¢)Û.±Ó. â

½n 5.2.2, ·k

Ún 5.5.3. 3 k1 > 0,¦éz c ∈ γext,k[é¡Ó pc : S1 → S1

÷v p−1c sc pc = Rθ, pc(1) = 1 K(pc) ≤ k1, Ù¥ K(pc) pc [é¡~ê.

Ún 5.5.4. 3~ê K1 ≥ 1, ¦éz c ∈ γext, hc : Vc → Uc, ϕc : C \D→Uc Ú pc : S1 → S1 ©Ok[/*Ü Hc : C → C, Φc : C → C Ú Pc : D → D,

Hc, Φc Ú Pc ûÑØL~ê K1. d,§ÑÛ¼ê. AO/, Hc(0) = 0,

Φc(0) = 0 Pc(0) = 0.

y². 5¿ φc : Ωc → D \5^ φ′c(0) > 0 5z¼ê, §ëY6u c.

P rc = φc hc φ−1c L1/2 : S1 → S1, K rc(−z) = −rc(z), ù´Ï hc Ú φc Ñ´Û¼ê,

Ù¥ La(z) = az ´5N. d, rc ´[é¡Ó, ù´Ï ∂Ωc Ú f(∂Ωc)

Ñ´[.

N rc 3 τ(x) = e2πix eJ, Rc : R → R ÷v Rc(x + 1/2) = Rc(x) + 1/2.

âúª (5.2.2), ùL² Beurling-Ahlfors *Ü (Rc)BA : H → H Ó÷v (Rc)BA(z +

1/2) = (Rc)BA(z) + 1/2. u´ Beurling-Ahlfors *Ü (rc)BA : D → D ÷v (rc)BA(−z) =

−(rc)BA(z).

y3é z ∈ Vc ½Â Hc(z) = hc(z) é z ∈ fc(Ωc) ½Â Hc(z) = φ−1c (rc)BA L2

φc(z). K Hc ´ hc [/*Ü. w,, Hc ´Û¼ê Hc(0) = 0.

â½n 5.4.2, éu c ∈ γext, ∂Ωc K–[, Ù¥ K ´Ø6u c ~ê.

ùL²éu¤k c ∈ γext, N rc [é¡~ê=6u K. ù¿X Hc û

Ø6u c ∈ γext.

§5.5. ÃâNEÚ§ëY5 89

aq?Ø±ûòÿ ϕc Ú pc (|^Ún 5.5.3). y²..

5 5.5.5. Ún 5.5.4 ¥^òÿ¡5z (Beurling-Ahlfors) [/ò

ÿ. k, Douady-Earle òÿ [27] ´ÐÀJ. N´wÑ Hc, Φc Ú Pc ëY6

u c.

½Â

Fc(z) =

Φc Pc Rθ P−1

c Φ−1c (z − kπ) éu z ∈ Ωk

c , k óê,

−Φc Pc Rθ P−1c Φ−1

c (z − kπ) éu z ∈ Ωkc , k Ûê,

Hc fc(z) Ù§¹.

â Fc ½ÂÚÚn 5.5.4, y

Ún 5.5.6. Fc ´ëY, ±Ï 2π ±ÏÛ¼ê÷v Fc(z + π) = −Fc(z). d

, Fc ": kπ,± arcsin(√

3 sin(c)) + kπ : k ∈ Z.

y3·±3 C þ½ÂXe Fc–ØC Beltrami Xê νc(z) :

νc(z) =

(P−1

c Φ−1c )∗σ0 éu z ∈ Ωc,

(F nc )∗(P−1c Φ−1

c )∗σ0 éu z ∈ F−nc (Ωc) \ F−n+1c (Ωc), n ≥ 1,

σ0 Ù§¹.

Ï Fc3 C\⋃k∈Z Ωk

c þ´)Û,âÚn 5.5.4,ùL²3½~ê k ∈ (0, 1),

¦éz c ∈ γext, Beltrami Xê νc ÷v ‖νc‖∞ < k. âÿ Riemann N½n,

3± 0, 2π Ú ∞ ØÄ[/N wc : C → C, ¦§´k Beltrami Xê

νc Beltrami §). d, wc EAØLØ6u c ~ê K2.

Ún 5.5.7. [/N wc ´Û¼ê, wc(z + π) = wc(z) + π wc(π/2) = π/2.

y². Ï Fc ´Û¼ê Fc(z + π) = −Fc(z), â νc ½Â, k νc(z) = νc(−z)

νc(z + π) = νc(z). =, wc(z + π) Ú wc(z) äkÓ Beltrami Xê. Ï§Ñ

± ∞ ØÄ, K3, a Ú b ¦ wc(z + π) = awc(z) + b.

XJ |a| < 1, Ï wc ± 0 ØÄ, ·k wc(kπ) = awc(kπ − π) + b = · · · =

b(1 + a + · · · + ak−1) = b(1 − ak)/(1 − a) éz k ≥ 1 ¤á. u´ k ªu +∞ wc(kπ) ªu b/(1− a). aq/, XJ |a| > 1, Ï wc(z) = (wc(z + π)− b)/a, ·k

wc(−kπ) = b(1− 1/ak)/(1− a), Ù¥ k ≥ 1. ùL² k ªu +∞ wc(−kπ) Óª

u b/(1− a). ùü«¹Ñ´ØUÏ wc ± ∞ ØÄ.

XJ a = e2πix, Ù¥ x = m/n ∈ (0, 1) ´knê m,n *dp, Kéz

k ≥ 0 k wc(kn) = 0. ù´ØU. XJ a = e2πix, Ù¥ 0 < x < 1 ´Ãnê,


âÚn 2.4.3, 3 N f knn≥1, ¦ n ªu ∞ e2πknxi ªu 1. ù

Ó wc(∞) =∞ gñ. Ïd~ê a 7Lu 1. Ï wc(2π) = (1 + a)b = 2π, ùL²

b = π.

P wc(z) = −wc(−z), Ï wc(0) = wc(0) wc Beltrami Xê νc , ùL

²3, d ¦ wc(z) = dwc(z). u´ wc(π) = −wc(−π) = π = dwc(π) = dπ. ùL²

d = 1 wc ´Û¼ê. (Øâ wc(π/2) = wc(−π/2) + π = −wc(π/2) + π

.

Ï νc ´ Fc–ØC, · Tc(z) = wc Fc w−1c (z) ´ C gC)ÛÛ¼

ê, T¼ê3:äk^=ê θ Siegel . âE, ù Siegel >.´

²L Tc é.:[.

`² Tc ∈ Σλ, ·IÄ Tc ":¿|^Ún 5.2.5. âÚn 5.5.6, ·

k

Ún 5.5.8. Tc = wc Fc w−1c ´Û¼ê Tc(z + π) = −Tc(z). d, Tc ":8

´ kπ,±wc(arcsin(√

3 sin(c))) + kπ : k ∈ Z.

íØ 5.5.9. Tc ∈ Σλ.

y². 5¿ Tc k u:^=ê θ Siegel. âÚn 5.2.5Ú 5.5.8

(J, ·I`² Tc ´k.

Ä½Â3:NC[/N 1/wc(1/z). y¿â Mori ½n [43, ½

n 3.2], O |wc(z)| ≤ 16K2 |z|K2 Ú |w−1c (z)| ≤ 16K2|z|K2 , Ù¥ z 3 ∞ SC

z K2 ´ cÃ'~ê, ù´Ïé?¿ c ∈ γext, wc ´ K2–[/N.

âÚn 5.5.4, 3 ∞ S, k |Hc(z)| ≤ 16K1 |z|K1 . â Fc 3 ∞ S½Â, k |Tc(z)| ≤ C(K1, K2)e|z|

K2 , Ù¥ C(K1, K2) ´=6u K1 Ú K2 ~ê.

ùL² Tc ´k.

y3·½ÂÃâN S : γext 7→ Σλ S(fc) = Tc. daq?Ø, S Ó±½Â3 γint þ. AO/, S(f∞) = T∞ S(f0) = T0, Ù¥ T∞(z) = e2πiθ sin(z)

T0(z) = e2πiθ(z − z3/3).

5 5.5.10. /, Ø c = π/2 ½ −π/2, k c 6= wc(c), Ïd·¿ØU Tc =

2e2πiθfc. c = π/2 ½ −π/2 , ÙéA Σλ ¥¼ê z 7→ e2πiθ(sin(z)− sin3(z)/3)

u: Siegel >.´²Lüê 3 .: π/2 Ú −π/2 [.

ù(Jaqu [79] ¥Ì½n.

§5.5. ÃâNEÚ§ëY5 91

y²ÃâN´ëY, âÚn 5.2.5 Ú 5.5.8 ±9íØ 5.5.9, ·Iy²

":N c 7→ wc(arcsin(√

3 sin(c))) ëY6u c. Äk·`²

Ún 5.5.11. N c 7→ νc ´ëY.

y². - cn Âñ c∞ ∈ γext S, ·I`² n→∞ , νcn 3 L∞

¥ÓÂñ νc∞ . e5, üå, ·^ n Ú ∞ ©OOeI cn Ú c∞.

E C¥ Lebesgueÿ8,- Area(E)L« E 3¥¡Ýþe Lebesgue

¡È. éu δ > 0, ½Â

Q δn = z ∈ C : |νn(z)− ν∞(z)| > δ.

- Ekn =

⋃0≤i<k F

−in (Ωn), Ù¥ 1 ≤ n, k ≤ ∞. ·^ En Ú E∞ 5L« E∞n Ú E∞∞ , §

©OéAX Ωn Ú Ω∞ ;. â νc ½Â, k Q δn ⊂ En ∪E∞, ù´Ï3Ù§

/ νn = ν∞ = 0. Iy²éu?¿ δ > 0 Ú ε > 0, 3¿© N ¦éz

n > N , k Area(Q δn) < ε.

½ δ > 0 Ú ε > 0, Ï Ωn ´ Fn–ØC, XJ k v, K Area(En \Ekn) ±

v. Ï Ωc ëY6u c, 3¿© k0 ¦é¤k n,

Area(En \ Ek0n ) < ε/5 and Area(E∞ \ Ek0

∞) < ε/5. (5.5.1)

âëY5,3mÿÀ Ω′ Ú N1 > 0,¦é¤k n > N1, Ω′ ⊂ Ωn∩Ω∞

Area(Ek0n \Dk0

∞) < ε/5 and Area(Ek0∞ \Dk0

∞) < ε/5, (5.5.2)

Ù¥ Dk0∞ =

⋃0≤i<k0 F

−i∞ (Ω′).

âÚn 5.5.4 Ú5 5.5.5, n ªu ∞ , [/N (Φn Pn)−1 3 Ω′ þ

Âñu (Φ∞ P∞)−1. ,, n ªu ∞ , (Φn Pn)−1 F k0n 3 Dk0∞ þÓÂ

ñu (Φ∞ P∞)−1 F k0∞ . ùL² νn 3 Dk0∞ þA??Âñu ν∞. Ïd, 3 N2 > 0

¦é¤k n > N2, ·k

Area(Q δn ∩Dk0

∞) < ε/5. (5.5.3)

- N = maxN1, N2, â (5.5.1), (5.5.2) Ú (5.5.3), ùL²é¤k n > N , k

Area(Q δn) < ε. ùy² c 7→ νc 3 c∞ ?´ëY. â c∞ ?¿5, Úny.

íØ 5.5.12. ÃâN S : γext → Σλ ´ëY.

y². âÚn 5.5.11, · νc ëY6u c. Ï wc ± 0, 2π Ú ∞ ØÄ ‖νc‖∞ ≤ k < 1 éz c ∈ γext ¤á, â Ahlfors Ú Bers ½n, ù`² wc ëY

6u c . âÚn 5.5.8, Tc ":ëY6u c. âÚn 5.2.5, ùL² S ëY6uc ∈ γext.


§5.6 ½n 5.1.1 y²

5¿ γ ëêm Σ1/2 ¥½Â¦éuz c ∈ γ \ ±π/2, 5z«>. ∂Ωc TÐ¹ fc 4 .:8Ü. éz c ∈ γ, - ∆S(c) S(c) ∈ Σλ ±

:% Siegel , Ù¥ S ÃâN. Ø c = ±π/2 , >. ∂∆S(c) ´T

Ð²Lüé.: ±π/2 Ú ±S(c) [.

éu c ∈ Λ, ·^ gc 5P Σλ ¥¼ê. - ϕS(c) : ∆S(c) → D ¦ ϕS(c) gS(c) ϕ−1S(c) = Rθ Ú ϕS(c)(π/2) = 1 5zN. aquÚn 5.3.3 ?Ø, ϕS(c) ´Û¼

ê. ½Â π/2 Ú S(c) mÝ

B(S(c)) = argϕS(c)(S(c)), (5.6.1)

Ù¥ 0 ≤ arg(z) < 2π Ë¼ê. AO/, B(S(π/2)) = B(π/2) = 0 B(S(−π/2)) =

B(−π/2) = π. Ï S ëY6u c, ùL² B(S(c)) ÓXd.

Ún 5.6.1. XJ B(S(c1)) = B(S(c2)), Ù¥ c1, c2 ∈ γ, K S(c1) = S(c2).

ÏÚn 5.6.1y²~aqu·K 5.3.11y²,·=Ñy²V.Äk,

·If5Ún, §aqu [77, íØ 5.2] Ú [41, Ún 5.1].

Ún 5.6.2. z gc 3 Σλ ¥[/Ýa´m8½ö´ü:. XJ

c ∈ γ, K gS(c) [/Ýa´ü:.

y². b c, c′ ∈ Σλ c 6= c′, [/N ϕ : C → C ÷v ϕ gc ϕ−1 = gc′ .

- µϕ L« ϕ Beltrami Xê, §´ gc–ØC. =, µϕ(gc(z))g′c(z)/g′c(z) = µϕ(z), d

/, tµϕ(gc(z))g′c(z)/g′c(z) = tµϕ(z), ùL² Beltrami Xê tµϕ Ó´ gc–ØC, Ù¥

|t| < 1/‖µϕ‖∞. Päk Beltrami Xê tµϕ Beltrami §) ϕt, K ϕt gc ϕ−1t Ñ

´)Û.

aquíØ 5.5.9 ?Ø, z ϕt gc ϕ−1t Ñáu Σλ. P ϕt gc ϕ−1

t

gc(t), K§X6u t, ù´Ï Beltrami Xê tµϕ X6u t. u´ t 7→ c(t) 3

|t| < 1/‖µϕ‖∞ S´X. Ïd[/Ýao´m8oü:.

XJ c ∈ γ \ ±π/2, gS(c) Siegel >.TÐ¹ 4 .:. 3 c ?Û

S, Ñ3 c′ ¦ gS(c′) Siegel >.=¹üé¡.:. ùL² gS(c) Ú

gS(c′) $ÑØ´ÿÀÝ. Ï gS(c) ∈ γ ÿÀÝaü:, Ù¥ c ∈ γ.

Ún 5.6.1 y²üÑ´Äu 5.6.2 ±9Ñy3·K 5.3.11 y²¥/IPE|.

Ún 5.6.1 y². ^/N ϕ = ϕ−1S(c2) ϕS(c1) : ∆S(c1) → ∆S(c2) O·K

5.3.11 ¥ φ : Ωc1 → Ωc2 , Ï B(S(c1)) = B(S(c2)), ¤± ϕ(S(c1)) = S(c2), ù´. Ï

§5.6. ½n 5.1.1 y² 93

∂∆S(ci) ´[, Ù¥ i = 1, 2, K ϕ ±òÿ[/N ϕ0 : C → C ¦ϕ0(∞) =∞ ϕ0 3 ∆S(c1) þ´ ϕ.

üå, ·ò gS(ci) Ú ∆S(ci) ¥eI S(ci) ^ i O, Ù¥ i = 1, 2. e5,

·^ gi ∈ Σλ ÄåXÚ5½Â[/N ϕn. ØÓu·K 5.3.11 y², ùp

òÿN´Ï·ØIÄ/X f−n1 (Ω1 \ f1(Ω1)) «. ·IÄ

Siegel n g_±9 C \ g−ni (∆i) ëÏ©|, Ù¥ i = 1, 2.

béz 1 ≤ k ≤ n−1,3[/N ϕk : C→ C¦ ϕk = g−12 ϕk−1g1

ϕk : g−k1 (∆1)→ g−k2 (∆2) ´X. y3·8B/½Â ϕn Xe.

é g−n1 (∆1) \ g−n+11 (∆1) ?Û©| ∆n

1 , ·oÚé g−n2 (∆2) \ g−n+12 (∆2)

äkÓ/éA©| ∆n2 . y3·3 g−n+1

1 (∆1) þ½Â ϕn = ϕn−1 ½Â

ϕn = g−12 ϕn−1 g1 ¦§ò ∆n

1 N ∆n2 . Ï g−n1 (∆1) ´ëÏ, §Ö?Û©|

Ñ´üëÏ. aqu·K 5.3.11 ¥?Ø, ·±[/N ϕn : C→ C,

§3 g−n1 (∆1) ´X÷v ϕn = g−12 ϕn−1 g1.

aqu·K 5.3.11y²",·ü4[/N h1 Ú h2, §Ñ

± 0, π/2 Ú ∞ ØÄ. Ïé¤k n ≥ 1 k g2 ϕn = ϕn−1 g1, l h1 Ú h2 ÷v

g2 h1(z) = h2 g1(z). du3⋃n≥0 g

−n1 (∆1) þk h1(z) = h2(z), ùL²3E²¡

Èf8þk h1 = h2. dëY5, ùL² h1 = h2. u´ g1 Ú g2 ´*d[/Ý

. âÚn 5.6.2 1(Ø, g1 = g2.

Ún 5.6.3. 8Ü S(γ) ´ Σλ ¥^ Jordan , §d Σλ ¥¤k¦ u

: Siegel >.´[3Pê¿Âe²Lüéé¡.:¼ê g

|¤.

y². Ï S ´ëY, K B(S(c)) ëY6u c. Ïd, c l 0 ( u c = π/2)

π ( u c = −π/2) ,2 2π (£ c = π/2) Cz, B(S(c)) ´ëYCz.

âÚn 5.6.1, ·±½ÂN χ : [0, 2π] → S(γ), ¦éz β ∈ [0, 2π], k

χ(β) = S(cβ) Ú B(S(cβ)) = β. ùL² S(γ) ⊂ Σλ ´^ Jordan .

b c ∈ Σλ ¦ ∂∆gc ´²L ±π/2 Ú ±c [. P π/2 Ú c mÝ

B(S(c′)), aquÚn 5.6.1 y², gc Ú gS(c′) *d[/Ý. âÚn 5.6.2

1(Ø, · S(c′) = c. ùL² c ∈ S(γ).

- S(γ)int Ú S(γ)ext ©O Σλ \ S(γ) k.ÚÃ.©|. âíØ 5.5.12, S ´ëY, Ïd S(γint) ¹3 Σλ ¥. âÚn 5.6.3, ùL² S(γint) o¹3 S(γ)int ¥,

o¹3 S(γ)ext ¥.

·K 5.6.4. ÃâN S : Σ1/2 → Σλ ´÷.


y². y²ù·K, ·y² S : γint ∪ γ → S(γ)int ∪ S(γ) Ú S : γext ∪ γ →S(γ)ext ∪ S(γ) Ñ´÷.

âc¡EÚ.:éA'X, S 7ò γint N\ S(γ)int. P γint =

γint ∪ γ ∪ f0 Ú S(γ)int = S(γ)int ∪ S(γ) ∪ g0, Ù¥ f0(z) = z/2 − z3/6 g0(z) =

e2πiθ(z − z3/3). ·äó S : γint → S(γ)int ´÷. ¯¢þ, 3üÓ h1 : γint → DÚ h2 : S(γint)→ D, ¦ h2 S h−1

1 : D→ D ´ëY, h2 S h−11 3ü ± S1

þ´ðN. XJÃâN S Ø´÷, K·òíÑ D kÂ Ø S1. ù

´ØU.

Ï γext dü©||¤, y²E,:. ·é γext ÑXed'

X ∼. é?¿ c1, c2 ∈ γext, c1 ¡ ∼ du c2, XJ (1) c1 = c2 ½ö (2) c1 = c2

|Re(c1)| = π/2. ·^ γ′ext 5L«ûm γext/ ∼. N´wÑ γ′ext ∪ f∞ Óuü . âcaq?Ø, k S : γ′ext → S(γ)′ext ÷. ùp γ′ext = γ′ext ∪ γ ∪ f∞ S(γ)′ext = S(γ)′ext ∪ S(γ) ∪ g∞, Ù¥ f∞(z) = sin(z)/2, g∞(z) = e2πiθ sin(z) S(γ)′ext

S(γ)ext ûm, ùûmþk γ′ext Ód'X ∼. ·Ky..

ã 5.2 ëêm Σλ 9Ù±Ï, Ù¥ λ = e2πiθ θ = (√

5− 1)/2 ^=ê.

ü4 S(γ) ⊂ [−π/2, π/2] × [−3, 3] çÚÚÚÜ©>., §©O

éAX S(γ)ext Ú S(γ)int. ã¡: [−4, 4]× [−3, 3].

üíØ, ·ke¡½n 5.6.5 Ú½n 5.6.6.

½n 5.6.5. - θ k..Ãnê. Ké?¿ c ∈ C \ kπ : k ∈ Z,

gc(z) = e2πiθ

(sin(z)− sin3(z)

3 sin2(c)

)

§5.6. ½n 5.1.1 y² 95

±:% Siegel >.´3Pê¿Âe²L 2 , 4 ½ö 6 fc

.:[.

½n 5.6.6. Jordan 4 S(γ) ⊂ Σλ ò Σλ ©¤üÜ© S(γ)int Ú S(γ)ext, ¦

(1) XJ c ∈ S(γ)int, K ∂∆gc TÐ²L 2 .: c Ú −c.(2) XJ c ∈ S(γ)ext, K ∂∆gc TÐ²L 2 .: π/2 Ú −π/2.

(3) XJ c ∈ S(γ) \ ±π/2, K ∂∆gc TÐ²L 4 .: ±π/2 Ú ±c.(4) XJ c = π/2 ½ −π/2, K ∂∆gc TÐ²L 2 ê 3 .: π/2 Ú −π/2.

½n 5.1.1 Ú½n 5.1.3 y². (Ü½n 5.6.5, ½n 5.6.6 ±9Ún 5.2.3.

18Ù ng tableaux ¢y½n#y²

ùÙ¥, |^ Emerson [30], DeMarco Ú McMullen [18] Ú\äþÄåXÚ,

·Ñ Branner Ú Hubbard ng. tableau ¢y½n#y².

§6.1 Ä tableaux

ngõªÄåXÚ®²é\/ïÄ. Ù¥²;óáu Branner Ú

Hubbard [10, 11]. 3 [11] ¥, ¦Ú\ûïÄngõª Julia 8|Ü(

rkåóä: tableau (L). ¦ÓÄ tableaux ¢y¯K, =äkü

.: tableau oÿ±ngõª¤¢y.

éuäkEXêngõª f , 8Ü

K(f) = z ∈ C | S f n(z)n≥1 k.

¡ f W¿ Julia 8, Ù¥ f n L« f n gEÜ. ÙÖ Ω(f) := C \ K(f) Ã

¡áÚ, K(f) >.½Â f Julia 8 J(f). ®², <ºÝ¼ê

G : C→ [0,+∞) ½Â

G(z) = limn→∞

1

3nlog+ |f n(z)|.

§´ëY, ÷v G(f(z)) = 3G(z) Ú K(f) = G−1(0); [10, 53].

bngõª f Julia 8Ø´ëÏ, K3 f .:á3 Ω(f)

¥. - c0 Ú c1 f ü.:, c0 <º¯@, = G(c0) ≥ G(c1). P

r0 = G(c0) > 0. é?¿ k ≥ 0, 8Ü G−1([0, r0/3k−1)) kõpØmÿ

À¿. zùm¡´Ý k ©ã¡, P Pk. éz¦

G(c0)/3N ≤ G(c1) < G(c0)/3N−1 ê N ≥ 1, .©ã¡ Pk(c1) ½Â¹ c1

G−1([0, r0/3k−1)) ëÏ©|, Ù¥ 0 ≤ k < N + 1. ùp N #NuÃ¡, =du

c1 Ø<º, Ï G(c1) = 0 d3Ã¡õ.©ã¡.

õª f N ng tableau ½IP: (marked grid) ´de^

½Â M(j, k) ∈ 0, 1 | j, k ≥ 0 j + k ≤ N:

M(j, k) = 1 = f k(c1) ∈ Pj(c1).

XJ M(j, k) = 1, K M(j, k) ¡ÍP. IP:±w´1o

Z2 :, Ù¥ j ≥ 0 L«l x ¶ål, k ≥ 0 L«lK y ¶ål. Branner Ú

Hubbard [11]y²ngõg®IP:7L÷vK. N (Ä

96

§6.2. äþÄåXÚ 97

)IP: M = M(j, k) ∈ 0, 1 | j, k ≥ 0 j + k ≤ N ´÷ve^ [11, 19]:

(R0) éz n ≤ N , M(n, 0) = M(0, n) = 1.

(R1) XJ M(j, k) = 1, Ké¤k i ≤ j k M(i, k) = 1.

(R2) XJ M(j, k) = 1, Ké¤k 0 ≤ i ≤ j k M(j − i, k + i) = M(j − i, i).(R3) XJ j + k < N , M(j, k) = 1, M(j + 1, k) = 0, M(j − i, i) = 0 é 0 < i < m ¤

á, M(j −m+ 1,m) = 1, K M(j −m+ 1, k +m) = 0.

(R4) XJ j + k < N , M(j, k) = 1, M(1, j) = 0, M(j + 1, k) = 0, é¤k

0 < i < j k M(j − i, k + i) = 0, K M(1, j + k) = 1.

K (R4) 3 [11] ¥vkÑy, 3¦y²¥(¢^. K (R4) ©

ÄkÑy3©z [34] ¥, ´þ¡ãK (R4) «AÏ/. , Ù

(ãd Kiwi [42], DeMarco Ú McMullen [18] Õá/é.

P M N IP:, XJ f tableau M Ó, K¡ M ù

ngõª f ¤¢y. Branner Ú Hubbard y²e¡½n.

½n 6.1.1 (Tableaux¢y, [11]). ?ÛIP:Ñ±dngõª¤¢y.

5`, ¢y½IP:ngõªØ´. ùÙ·ò^ä

þÄåXÚÑ½n 6.1.1 #y². ÙÌg´: k`²?Û

N IP:±dÓ “Ð” ng children N¢y, ùng

children N±òÿNÝu 3 aõªä. duzngaõªä

Ñ±dngõª¤¢y, l ½n 6.1.1 y.

§6.2 äþÄåXÚ

P¹õª f 3 Ω(f) þ|Ü&E, Emerson [30], DeMarco Ú McMullen [18]

kÚ\äþÄåXÚ.

Äk, ·£Á3 [18] ¥½ÂÄäÄåXÚk'½Â. üXä T

´,ëÏ,ÛÜk,ÃüXE/. ä T º:Ú> (vk!4

) ¤8Ü©OP¤ V (T ) Ú E(T ). ½º: v ∈ V (T ) >/¤k

8 Ev(T ), ÙÄê½Âº: v val(v). äà¤m ∂T ´ØëÏ

m, §±dé T \K ëÏ©|¤8Ü_4, Ù¥ K H T ¤kk

fä.

F : T1 → T2 ´üüXämN, XJ (1) F ´_;, mëY

; (2) F ´üXN (T1 ¥z^>Ñ5/N T2 ¥,^>), K¡ F ´

98 18Ù ng tableaux ¢y½n#y²

©|CX. ©|CX F : T1 → T2 ÛÜNÝ¼ê´÷ve^N

deg : E(T1) ∪ V (T1)→ 1, 2, 3, · · · : ézv ∈ V (T1), k

2 deg(v)− 2 ≥∑

e∈Ev(T1)

(deg(e)− 1), (6.2.1)

±9éz e ∈ Ev(T1), k

deg(v) =∑

e′∈Ev(T1):F (e′)=F (e)

deg(e′). (6.2.2)

ÛNÝ deg(F ) ½Â

deg(F ) =∑

F (e1)=e2

deg(e1) =∑

F (v1)=v2

deg(v1), (6.2.3)

Ù¥ e2 Ú v2 T2 ¥?¿>Úº:. |^ (6.2.2) ±9 T2 ëÏ5, Ný (6.2.3)

½Â´Ün. ùÙ¥, ·=Ä deg(F ) = 3 /.

½Â 6.2.1. - F : T → T üXägC©|CXN, x, y ´ T þü

:. XJé, m,n > 0 k F m(x) = F n(y), K¡ x, y á3Ó;. XJe¡ 4

^¤á, K¡©|CXN F : T → T áõª.

(1) 3áà ∞ ∈ ∂T ;

(2) 3÷v (6.2.1) Ú (6.2.2) ÛÜNÝ¼ê;

(3) ä T vkf ( 1 º:);

(4) ?Ûº:;Ñ¹u½u 3 º:.

d, | (T, F ) ¡áõªä.

©z [18] ¥½n 2.9 L²: XJ©|CXN F : T → T k÷v (6.2.1)

Ú (6.2.2) ÛÜNÝ¼ê, @oùÛÜNÝ¼ê´. aõªk'

[&E±3 [18] ¥é.

- f ńgõª Julia 8ëÏ, ò G Y²8zëÏ©|l¤:

ûm´üXä, P T . u´ f pûäþN F : T → T .

| τ(f) = (T, F ) ¡´ f û. d [18, ½n 3.1], f û τ(f) áõªä. ©z

[18] ¥Ì(Jé¡½n.

½n 6.2.2 (ä¢y, [18]). ?Ûaõªä (T, F ) Ñ´,õª f û.

T káàüXä (5¿ T fØ¹3S). ?ÀÙ¥

áàIP ∞, Kz¿©C ∞ º: 2. XJ T ku½u 3

§6.2. äþÄåXÚ 99

º:, K73C ∞ u½u 3 º:, P v0, ·¡ùº

: T Ä:. XJ T zº:Ñu 3, ?ÀÙ¥u 2 º:, IP

v0.

|ÜpÝ¼ê h : V (T ) → Z ½Â: |h(v)| ë v Ú v0 ¤I>

^ê. d, h ÎÒde¡^(½: XJ v á3ë v0 ∞ á´p, K

h(v) ≤ 0, ÄK h(v) > 0.

XJüº: v, w ÷v h(v) = h(w) + 1 k^>ë v Ú w, Kº: v ¡´

º: w child.

½Â 6.2.3. F : T → T ´üXN, XJézº: v ∈ V (T ), ÷v v

z child Ñ´ F (v) child, @o F ¡´± children .

5¿aõª½´± children . N´wÑé± children N

F : T → T 5`, XJ h(v) = k, K h(F (v)) = k + U(F ), Ù¥ U(F ) ´=6u F

~ê. d, aõªäØÓ´, ± children NØ½´÷.

üXä T ©| (branch)´S (vk)nk=1, Ù¥ h(vk) = k vk+1 ´ vk

child, 1 ≤ k < n, ùpê n ¦´, = n =∞ ½ vn u 1.

½Â 6.2.4. XJ± children N F : T → T ÷ve^, K¡Ù´

N > 0 ng children N.

(a) U(F ) = −1;

(b) 3NÝ¼ê deg : V (T ) ∪ E(T )→ 1, 2, 3 ÷v(b1) é,©| (sk)

Nk=1 k deg(sk) = 2, Ù¥ h(sk) = k 1 ≤ k < N + 1;

(b2) é l ≤ 0 k deg(sl) = 3, Ù¥ s0 = v0 Ä: h(sl) = l ≤ 0.

(b3) ézº: v ∈ T \ ∪k<N+1sk k deg(v) = 1.

(b4) - v1 Ú v2 > e ∈ E(T ) üà:. XJ h(v1) = h(v2) + 1, K deg(e) =

deg(v1).

Ï skk<0 3ä T þÄåXÚ¥¿Ø, S S := (sk)k<N+1 ¡ńg

children N F : T → T .©|.

½Â 6.2.4 ^ (a) L«3 F ê, äXáà ∞ c?Ú.

5¿·=ÄÝ N ¢y¯K, Ï kü<º.: c0 Ú c1 ÷v

G(c0)/3N ≤ G(c1) < G(c0)/3N−1 ngõªU¢y N IP:.

3ù«¹e, dTngõªpaõªäkü<º.: (= U(F ) = −2).

, , ·±ÏLK τ(c1) 3Ýu N ;, Ó½ÂÐaõ

ªä, Ù¥ τ ò G Y²8ëÏ©|l¤ü:@ûN. ¯¢þ, éu?¿


N < ∞ IP:, Ñ3ngõª¦Ù¥.:Ø<º

Ó¢yù½IP:.

§6.3 ng children Nòÿ

éu N IP:, ·IEÓng childrenN

¦ÄåXÚ´½ÂÐ. Ó, ·7L`²dùIP:ùng±

children N±òÿNÝu 3 aõªä, l â½n 6.2.2 ½n

6.1.1 ¤á.

- F ′ : T ′ → T ′ Ú F : T → T üng children N. N F ′ : T ′ → T ′ ¡

´ F : T → T òÿ´: T ´ T ′ fä, F ′ 3 T þu F §3

T þkÓNÝ¼ê. - C(v) L« v ¤k children ¤8Ü.

Ún 6.3.1 (òÿÚn). - S = (sk)k<N+1 L« N ≥ 1ng children

N F : T → T .©|. K F : T → T ±òÿNÝu 3 aõª

ä=

(1) é v ∈ V (T ) \ S, XJ v1, v2 ∈ C(v) v1 6= v2, K F (v1) 6= F (v2);

(2) é i 6= 0, XJ u ∈ C(si) u 6= si+1, K F (u) 6= F (si+1);

(3) XJ si k 3 ØÓ children u, v, w F (u) = F (v), K F (w) 6= F (u);

(4) é?¿ v ∈ V (T ), k∑

F (u)=v deg(u) ≤ 3.

y². XJng children N F : T → T ±òÿngaõªä

F ′ : T ′ → T ′, K (6.2.1)-(6.2.3) éu F ′ : T ′ → T ′ ¤á. ±y (6.2.2) íÑ (1),

(2) Ú (3) (6.2.3) íÑ (4). ùy²75.

·ÏL8By²¿©5. Ä T Ä: s0 (= v0), d^ (4) · s0

õkü children. XJ C(s0) = s1,· s0 \þÝu 1 child¦ (6.2.3)

¤á. Pòÿng children N F1 : T1 → T1. Kª (6.2.2) éu T1 ¥@

|ÜpÝu 1 º:´¤á.

b (6.2.1)-(6.2.3) éu Tk ¥@|ÜpÝu k º:´¤á, Ù¥ k ≥ 1.

éz¦ h(v) = k º: v ∈ Tk, u v children. XJª (6.2.2) Ø¤á, v

\þvõÝu 1 children ¦ª (6.2.2) ¤á. y3·ng

children N Fk+1 : Tk+1 → Tk+1. ùN´ Fk : Tk → Tk òÿ, ¦ª (6.2.2)

é Tk+1 ¥@|ÜpÝu k + 1 º:´¤á.

^ù«ª, ·±ò F : T → T òÿ ∞ ng children N

F∞ : T∞ → T∞. d, ªf (6.2.1)-(6.2.3) éu T∞ ¥¤kº:Ú>Ñ¤á. 'aõ

§6.4. lIP:Ðng CHILDREN N 101

ªä¤I 4 ^, ¿5¿ÛNÝ deg(F∞) =∑

F∞(v)=s0deg(v) = 3,

F∞ : T∞ → T∞ ŃÝu 3 aõªä.

5 6.3.2. Ún 6.3.1 4 ^´ [18] ¥Ongõªäè'. éuõ

.: tableaux ¢y¯K, ^ (2), (3) Ú (4)Ø2´7.

½Â 6.3.3. XJng children N F : T → T ÷vÚn 6.3.1 ¥^, K

¡§´Ð (nice).

§6.4 lIP:Ðng children N

ù!¥, ·F"l½½IP:¥JÐng chil-

dren N. ·¡ N IP: M ÓÐng children

N F : T → T ¤¢y, XJ§÷vXe^:

M(j, k) = 1 = deg(F k(sj+k)) ≥ 2.

Ún 6.4.1. z N IP:þÓÐng children

N¤¢y.

y². ·|^8By²ùÚn. éu N = 1, - T1 = (sk)−∞k=1 ¦ sk ´ sk−1

child. ½Â F1 : T1 → T1 F1(sk) = sk−1, Ù¥ k ≤ 1. NÝ¼ê½Â

deg(s1) = 2 deg(si) = 3, Ù¥ i ≤ 0. w,, F1 : T1 → T1 ´Ðng children

N, ¿¢y@ 1 IP:.

XJ N = 2, KküØÓ 2IP:, ¦©OéAX M(1, 1)

IP½vIP. XJ M(1, 1) IP, s1 \þÝ 2 child s2, ·

üXä T2.

ò F1 : T1 → T1 òÿ F2 : T2 → T2 ¦éu¤k k ≤ 2 k F2(sk) = sk−1, K

F2 : T2 → T2 ´Ðng childrenN¿Ó¢y 2M(1, 1) = 1I

P:. aq/,XJ M(1, 1)vkIP, s1 \þÝ 2child s2 s0 \þ

Ý 1 child v1, ·üXä T ′2. y3ò F1 : T1 → T1 òÿ F ′2 : T ′2 → T ′2

¿¦ F ′2(s2) = v1 F ′2(v1) = s0. K F ′2 : T ′2 → T ′2 ´Ðng children N

Ó¢y 2 M(1, 1) = 0 IP:.

b?Û k ≤ N − 1 IP:þÓÐng children

N Fk : Tk → Tk ¤¢y, Ù¥ N < +∞. â8BL§, üXä Tk á3.©|

(si)i≤k c; (forward orbit) ¥. - M ½ N IP:, e

5?Øòl M(N, 0) Ñu, 3IP:¥÷XÀ M(0, N) (å.


Äk, sN−1 \þÝ 2 child sN ,,£ M(N−1, 1). XJM(N−1, 1) IP, KâK (R2) N :ÑIP. ½Â FN(sN) =

sN−1,·Ðng childrenN FN : TN → TN ,TN¢y M(N−1, 1) IP: M .

b M(N − 1, 1) vkIP. FN−1(sN−1) \þÝ 1 child vN−1

½Â FN(sN) = vN−1. ,£ M(N − 2, 2). XJ M(N − 2, 2) IP, ½Â

FN(vN−1) = sN−2. Ný FN : TN → TN ´Ðng children N¢y

M(N − 1, 1) vkIP M(N − 2, 2) IP:.

b M(N − 2, 2) vkIP. éu M ¥?Û M(m,n), P M(m,n) þ

n/ Q(m,n) = (M(j, k)) | j + k ≤ m + n, 0 ≤ j ≤ m n ≤ k ≤ m + n.XJ Q(N − 2, 2) 6= Q(N − 2, 1), F 2N−1(sN−1) \þÝ 1 child vN−2, ½Â

FN(vN−1) = vN−2. ,£ M(N − 3, 3). XJ Q(N − 2, 2) = Q(N − 2, 1), K

M(N − 2, 1) vkIP. ½Â FN(vN−1) = FN−1(sN−1), Ún 6.3.1 ¥ 4 ^ÑN

ý÷v.

bXéuN FN ·®²3; sN 7→ FN(sN) 7→ · · · 7→ F kN (sN) þ½Â, Ù

¥ k ≥ 1 (5¿ FN 3 F kN (sN) þvk½Â), vN−i = F iN (sN) ´ F iN−1(sN−1) #

\Ý 1 child, ùp 1 ≤ i ≤ k. d, éu 1 ≤ i ≤ k k M(N − i, i) = 0, ù´Ï

XJ·-IP @o8BEÒ(å.

y3£ M(N − k − 1, k + 1) þ, ·^Úc¡aq?Ø. XJ M(N − k −1, k+ 1) IP, ½Â FN(vN−k) = sN−k−1. 3ù«¹e, ·IyÚn 6.3.1 ¥

^ (4). ¯¢þ, y3, Ø vN−k, âE, sN−k−1 vkÙ§Ý 1 _

. ùL²∑

F (u)=sN−k−1deg(u) ≤ 3.

P43E¥,·o\\ childrenÝÑu 1oâK (R2),E

(å. XJ M(N −k−1, k+ 1)vkIP,u Q(N −k−1, k+ 1)Ú Q(N −k−1, j),

Ù¥ 0 < j ≤ k. XJé?¿ 0 < j ≤ k, k Q(N − k − 1, k + 1) 6= Q(N − k − 1, j),

F(k+1)N−1 (sN−1) \þ child vN−k−1 ¿½Â FN(vN−k) = vN−k−1. ,·£

M(N − k − 2, k + 2). XJ3, j ¦ Q(N − k − 1, k + 1) = Q(N − k − 1, j), @o

·`²¯. XJ M(N − k − 1, k), M(N − k − 1, j − 1) M(N − k, j − 1)

þIP, F(k+1)N−1 (sN−1) \þ child vN−k−1 ¿½Â FN(vN−k) = vN−k−1. ,

·£ M(N − k− 2, k+ 2). ÄK, ·½ÂFN(vN−k) = F jN−k−j−1(sN−k−j−1). a

q/, â tableaux KÚ·EÚn 6.3.1 ¥ 4 ^þ¤á.

^ù, ·o´±Ðng children N¿¢y½

N IP:. d8B, ½ny².

5 6.4.2. ·éÚn 6.4.1y²¥ M(N − 2, 2) = 0Ú Q(N − 2, 2) = Q(N − 2, 1)

§6.4. lIP:Ðng CHILDREN N 103

/Ñ5º. ¯¢þ, ÷vÚn 6.3.1 ¥^ (2), ·Iu M(N −2, 1), M(N − 2, 0) Ú M(N − 1, 1). Ï M(N − 2, 1) = 0, ·½Â FN(vN−1) =

FN−1(sN−1). , , XJ´/ M(N − k − 1, k + 1) = 0 é, 0 < j ≤ k k

Q(N − k − 1, k + 1) = Q(N − k − 1, j), K·IaquÚn 6.4.1 ¥y².

½IP:±ü½õÐng children N¢y (ã 6.1 ±

9 §6.5). Nýéu N ≤ 4, N IP:UÓÐ

ng children N¢y.

s−1

s0

s1

s2

s3

s4

s5

s−1

s0

s1

s2

s3

s4

s5

0

1

2

3

4

5

0 1 2 3 4 5

ã 6.1 N = 5 IP:, §±düÐng children N

¢y. ¥mä´dÚn 6.4.1 ¥E. ¢%:L.

%:KL. .

½n 6.1.1 y². (Ü½n 6.2.2 ±9Ún 6.3.1 Ú 6.4.1 (J.

N 1 2 3 4 5 6 7

MGN 1 2 4 8 16 33 69

NCN 1 2 4 8 18 42 103

N 8 9 10 11 12 13 14

MGN 144 303 641 1361 2895 6174 13188

NCN 260 670 1753 4644 12433 33581 91399

N 15 16 17 18 19 20 21

MGN 28229 60515 129940 279415 601742 1297671 2802318

NCN 250452 690429 1913501 5328648 14902959 41841737 117887513

L 6.1 1 ≤ N ≤ 21 , IP:ÚÐng children Nê8. ÎÒ

MGN Ú NCN ©OL«IP:ÚÐng children Nê8.


§6.5 IP:ÚÐng children Nê8

âc¡ãK (R0)-(R4), ·±O½ØÓIP:ê8. a

q/, dÚn 6.3.1 ¥ã 4 ^, ·±O½ØÓÐng

children Nê8 (L 6.1).

5¿3©z [18] ¥, ØÓÐng children Nê8 ([18] ¥¡´ä

ê8) ®²O N ≤ 17. ùp·ò(Jí2 N ≤ 21.

1ÔÙ McMullen NÃ¡áÚ>.

Hausdorff ê

§7.1 ½nã

éuê Q ≥ 3, McMullen N

fp(z) = zQ + p/zQ

ÄåXÚ®²þïÄ ([20, 22, 24, 69, 61]). p é, ùAÏkn¼

ê±w¤´ f0(z) = zQ ü6Ä. 5¿ùp·ò (4.1.1) ¥ëê λ ¤ p Ì

´ [75] ¥±.

l [20, 49] ®²: éu p, fp Julia 8´ Cantor ±. 3ù«¹

e, ¤k Fatou ©|Ñ ∞ ¤3 Fatou ©|¤áÚ. ·^ Bp L« ∞ áÚ,

K>. ∂Bp ´^ Jordan (âÚn 7.2.3 ¯¢þ´[). ¯¢þ, [61]

¥y² ∂Bp XJ J(fp) Ø´ Cantor 8, K>. ∂Bp o´^ Jordan . 3ù

Ù¥, ·y²e¡½n.

½n 7.1.1. - Q ≥ 3, Kéu¦ J(fp)´ Cantor± p, ∂Bp Hausdorff

ê´

dimH(∂Bp) = 1 +|p|2

logQ+O(|p|3). (7.1.1)

AO/, XJ Q 6= 4, Kp O(|p|3) ±d O(|p|4) O.

íØ, p é, þã½nÑ J(fp) Hausdorff ê

e..

éuõª Pc(z) = zd + c, Ù¥ d ≥ 2 c é¦ Pc ´V, Pc Julia 8

Hausdorff ê®²3 [65], [75] Ú [15] ¥©OO, Ùêúª©OÐm'

u c 1, 1nÚ1o. nØþ, pÑ±ëY/¦Ñ5. , , X,

p, O¬5E,.

§7.2 úªí

½n 7.1.1 y²aqu [54] Ú [75]. ¤k¡Oò33e!. e5,

·o´b½ p v (p = 0 ´#N).

105

106 1ÔÙ McMullen NÃ¡áÚ>. Hausdorff ê

ã 7.1 fp(z) = zQ + p/zQ Julia 8, Ù¥ p = 0.005 Q = 3, 4. üÑ´

Cantor ±. ã¡: [−1.25, 1.25]× [−1.25, 1.25].

Øõª Pc(z) = zd + c, McMullenxëêmkÛ: p = 0. pªu

0 , Julia 8 J(fp) 3 Hausdorff ÿÀeØÂñ J(f0) (ü ± S1), [22].

, , Ã¡áÚ>. ∂Bp %´Âñ. ¯¢þ, ·±`² (Ún 7.2.3)

∂Bp ´ü ± S1 X$Ä. d, ·Äk£X$Ä½Â.

½Â 7.2.1 (X$Ä, [48]). - E E¥¡ C f8, N h : D×E → C ¡´ E ± D ëêm, 0 Ä:X$Ä, XJTN÷v

(1) éz z ∈ E, β 7→ h(β, z) 'u β ∈ D ´X;

(2) éz β ∈ D, z 7→ h(β, z) 3 E þ´ü;

(3) h(0, z) = z é¤k z ∈ E ¤á.

½Â 7.2.1 ¥ü D ±dÙ§ÿÀO.

½n 7.2.2 (λ Ún, [48]). E X$Ä h : D×E → C ±/òÿ E

X$Ä h : D × E → C. éz β ∈ D, N h(β, ·) : E → C ±òÿE¥¡g[/N.

l [69] ®²: McMullen «M := p ∈ C− 0; J(fp) Cantor ± ´:%. ù`² V =M∪ 0 ¹ 0 ÿÀ.

Ún 7.2.3. 3X$Ä H : V × S1 → C ± V ëêm± 0 Ä:¦é

u¤k p ∈ V k H(p,S1) = ∂Bp.

y². ·Äky² f0(z) = zQ z½5±Ï:3 V ¥´X/$Ä. -

z0 ∈ S1 ù±Ï k ±Ï:. éu p, N fp ´ f0 6Ä.

§7.2. úªí 107

âÛ¼ê½n, 3 0 U0 ¦éu¤k p ∈ U0, z0 ¤ fp ½5±

Ï: zp, äkÓ±Ï k. ,¡, éu¤k p ∈ M, Ï fp vk½5±Ï

;, fp z½5±Ï;3M ¥Ñ´X/$Ä ( [50] ¥½n 4.2).

Ï V ´üëÏ, éu p ∈ U0, 3XN Z : V → C ¦ Z(p) = zp. -

Fix(f0) f0 ¤k½5±Ï:. K½Â h(p, z0) = Z(p) N h : V × Fix(f0)→ C´X$Ä. 5¿ S1 = Fix(f0), â½n 7.2.2, 3 h òÿ, P H :

V × S1 → C. w, H(p,S1) ´ J(fp) ëÏ©|.

e5, ·`²éu¤k p ∈ V , H(p, S1) = ∂Bp. âV Julia 8X$Ä

5, Iy²éuëê p ∈ (0, ε) k H(p,S1) = ∂Bp ¤á, Ù¥ ε > 0.

Ä f0 fp 6Ä, Ù¥ëê p ∈ (0, ε) é, f0 ØÄ: z0 = 1 ¤ fp

½5ØÄ: zp, ù½5ØÄ:´¢C 1. N fp kü¢ØÄ:. Ù¥

´ zp ,Ć 0 z∗p . w, zp fp Ý 0 Xº:. Ïd

H(p, 1) = zp ∈ ∂Bp. ùL²éu¤k p ∈ (0, ε) k H(p,S1) = ∂Bp.

½Â 7.2.4 ([65]). - M ê N ¢)Û6/, J ´ M ;f8,

V ´ J 3 M ¥m. ¡ J ´¢)ÛN f : V →M ½f (repeller) X

Je¡n^¤á:

(1) 3~ê C > 0, α > 1 ¦ ‖(Txf n)u‖ ≥ Cα‖u‖ é?¿ x ∈ J, u ∈ TxM Ún ≥ 1 ¤á;

(2) J = x ∈ V :é¤k n > 0 k f nx ∈ V ;(3) é?¿ J m8 O, 3, n > 0 ¦ J ⊂ f n(O).

â½Â 7.2.4, >. ∂Bp N fp ½f.

½n 7.2.5 ([65,íØ 5]). XJx¢)ÛN fλ ½f Jλ )Û6uëê λ (=

(λ, x)→ fλ(x) ´¢)ÛN), @o Jλ Hausdorff ê¢)Û6u λ.

·½Â¼ê H : V → R+ H(p) = dimH(∂Bp). e5·ÄkéÑ H

Ä5. du ∂B0 = S1, K H(0) = 1. â Ruelle ½n, · H ´¢

)Û¼ê. Ïd p 3 0 NC, ·k

H(p) =∑s,t≥0

astpspt, a00 = 1. (7.2.1)

â fp(z) = fp(z) Ú f 2e2πi/(Q−1)p

(eπi/(Q−1)z) = eπi/(Q−1)f 2p (z) k

H(p) = H(p) = H(p), H(e2πi/(Q−1)p) = H(p).


ÏdXê÷v

ast = ast = ats, ast = aste2πi(s−t)/(Q−1).

AO/, XJ s− t 6= 0 mod (Q− 1), K ast = 0. Ïd·k

H(p) =

1 + a20(p2 + p2) + a11|p|2 +O(|p|4), if Q = 3,

1 + a11|p|2 +O(|p|3), if Q = 4,

1 + a11|p|2 +O(|p|4), if Q ≥ 5.

O ∂Bp Hausdorff ê, ·Ie¡(J ( [31], ½n 9.1, ·K 9.6 Ú

9.7).

½n 7.2.6 (Falconer, [31]). - S1, · · · , Sm Rn 4f8 D þØ N¦

|Si(x)− Si(y)| ≤ ci|x− y|, Ù¥ ci < 1. K

(1) 3;8 J ¦ J =⋃m

i=1 Si(J).

(2) J Hausdorff ê H(J) ÷v H(J) ≤ s, Ù¥∑m

i=1 csi = 1.

(3) XJ?Ú¦ |Si(x)− Si(y)| ≥ bi|x− y|, Ù¥ i = 1, · · · ,m, K H(J) ≥ s, Ù

¥∑m

i=1 bsi = 1.

y3, ·k

Ún 7.2.7. é?Û p ∈ V, ∂Bp Hausdorff ê D = H(p) de¡ªû½:∑z∈Fix(fnp )∩∂Bp

|(f np )′(z)|−D = O(1). (7.2.2)

y². - wp ∈ ∂Bp fp Ý 0 Xº:. ·±ò wp w¤ü:

w+p Ú w−p ò ∂Bp w¤´äkà: w+

p Ú w−p 4ã. N f np : ∂Bp → ∂Bp

k Qn _©|, P S1, · · · , SQn , zÑò ∂Bp N4ã, ¦§U_

ü. d, ∂Bp =⋃Si(∂Bp). AO/, S1(∂Bp) Ú SQn(∂Bp) ¹à: wp. éu

1 < j < Qn, Sj(∂Bp) =¹ f np ØÄ:. â Koebe ½n¿5¿ ∂Bp ´

[, K3ü n Ã'~ê C1, C2, ¦

C1

|(f np )′(ζ)|≤ |Si(x)− Si(y)|

|x− y|≤ C2

|(f np )′(ζ)|, ∀ 1 ≤ i ≤ Qn, x, y ∈ Si(∂Bp),

Ù¥ ζ f np á3 Si(∂Bp) ¥ØÄ:.

â½n 7.2.6, ·k s1 ≤ D ≤ s2, Ù¥∑

ζ Csjj |(f np )′(ζ)|−sj = 1, j = 1, 2. ùL

² n éÿ, Ú∑

ζ |(f np )′(ζ)|−D ´0u C−D1 Ú C−D2 mê∑z∈Fix(fnp )∩∂Bp

|(f np )′(z)|−D =∑ζ

|(f np )′(ζ)|−D − |(f np )′(wp)|−D = O(1).

§7.2. úªí 109

Úny..

½n 7.1.1 y². 5¿ p = 0 , Julia 8 J(fp) ü ±±ëêz

z(t) = e2πit ¿¦

fp(z(t)) = z(Qt). (7.2.3)

éu p 6= 0, fp : ∂Bp → ∂Bp ŃÝ Q CXN. Ï ∂Bp Óuü

±, l ∂Bp ±ëêz¦ (7.2.3)¤á. âÚn 7.2.3, · ∂Bp þ: z(t)

´X/$Ä. ùL², 3 0 S, ·±ò z(t) Ðm

z(t) = e2πit(1 + pU1(t) + p2 U2(t) +O(p3)), (7.2.4)

Ù¥ Um(t) éu m ≥ 1 ÷v Um(t + 1) = Um(t). ò (7.2.4) \ (7.2.3), ,' p

k'ÐmXê, ·ke¡ª

U1(Qt)−QU1(t) = e−2πi(2Q)t, (7.2.5)

U2(Qt)−QU2(t) =Q(Q− 1)

2U2

1 (t)− e−2πi(2Q)tQU1(t). (7.2.6)

Ný5¼ê§ φ(Qt)−Qφ(t) = e−2πit k)

φ(t) = − 1

Q

∞∑l=0

Q−le−2πiQlt. (7.2.7)

Ï ·±)Ñ§ (7.2.5) Ú (7.2.6)

U1(t) = φ(2Qt), (7.2.8)

U2(t) =Q(Q− 1)

2

∞∑l1, l2=1

Q−(l1+l2)φ(2(Ql1 +Ql2)t) +Q

∞∑l=1

Q−lφ(2(Ql +Q)t).(7.2.9)

¯¢þ, p Um(t) ±ÏL8BOÑ5, Ù¥m ≥ 3. ùò¬~E,.

5¿ f np 3 ∂Bp þØÄ:

Fix(f np ) ∩ ∂Bp = z(tj) : tj = j/(Qn − 1), j = 0, 1, · · · , Qn − 2. (7.2.10)

d [75], ·Ú\²þÎÒ

〈G(t)〉n =1

Qn − 1

Qn−2∑j=0

G(tj). (7.2.11)

ù²þÎÒk'k^ª´

〈e2πimt〉n =

1 XJ m ≡ 0 mod Qn − 1,

0 Ù§.(7.2.12)


du

(f np )′(z(tj)) =n−1∏m=0

f ′p(z(Qmtj)) = Qn

n−1∏m=0

(zQ−1(Qmtj)−

p

zQ+1(Qmtj)

), (7.2.13)

·±ò (7.2.2)

O(1) = Q−nD(Qn − 1)

⟨n−1∏m=0

∣∣∣∣zQ−1(Qmt)− p

zQ+1(Qmt)

∣∣∣∣−D⟩n

. (7.2.14)

N¹¥OL²éu¤k¿© n, ·k

O(1) = Q−nD(Qn − 1)(1 +D2n|p|2 +O(np3)). (7.2.15)

½, n, p vÿ, ·k

O(1) = exp(n(−(D − 1) logQ+D2|p|2)

). (7.2.16)

ùL²

D = 1 +|p|2

logQ+O(|p|3), (7.2.17)

ù´½n 7.1.1 ¥Iúª.

§7.3 ON¹

ù!òÌy²ªf (7.2.15). Äk,·éÎÒz. ·©O^ zm, U1,m

Ú U2,m 5L« z(Qmt), U1(Qmt) Ú U2(Qmt). - σ = e2πit, â (7.2.4), ·k∣∣zQ−1m − p/zQ+1

m

∣∣ =∣∣∣(1 + Vm)Q−1 − σ−2Qm+1

p/(1 + Vm)Q+1∣∣∣

=∣∣∣1 + (Q− 1)Vm + (Q− 1)(Q− 2)V 2

m/2− σ−2Qm+1

p (1− (Q+ 1)Vm) +O(p3)∣∣∣ ,(7.3.1)

Ù¥ Vm = U1,m p+ U2,m p2 +O(p3). Ïd∣∣zQ−1

m − p/zQ+1m

∣∣−D2 =∣∣∣1 +

[(Q− 1)U1,m − σ−2Qm+1

]p

+[(Q− 1)(Q− 2)U2

1,m/2 + (Q+ 1)σ−2Qm+1

U1,m + (Q− 1)U2,m

]p2 +O(p3)

∣∣∣−D2=

∣∣∣∣1− D

2Amp+

D

8Bmp

2 +O(p3)

∣∣∣∣ ,(7.3.2)

§7.3. ON¹ 111

Ù¥

Am = (Q− 1)U1,m − σ−2Qm+1

, (7.3.3)

Bm = (Q− 1) (D(Q− 1) + 2)U21,m − 2(D(Q− 1) + 4Q)σ−2Qm+1

U1,m (7.3.4)

−4(Q− 1)U2,m + (D + 2)σ−4Qm+1

.

,·k∣∣zQ−1m − p/zQ+1

m

∣∣−D =

(1− D

2Amp+

D

8Bmp

2

)(1− D

2Amp+

D

8Bmp

2

)+O(p3)

= 1− D

2(Amp+ Amp) +

D2

4|p|2AmAm +

D

8(Bmp

2 +Bmp2) +O(p3).

(7.3.5)

Ún 7.3.1. - u, v ∈ N, é?Û n, K (1) 2Qv/(Qn − 1) 6≡ 0 mod 1; (2)

2Qv(Qu + 1)/(Qn − 1) 6≡ 0 mod 1.

y². Ï (Q,Qn − 1) = 1, ùL² (Qv, Qn − 1) = 1. qÏ n é 0 < 2 <

Qn − 1, l 2Qv/(Qn − 1) ØU´ê.

éu1(Ø, b u = ns+ r, Ù¥ 0 ≤ r < n, K

2Qv(Qu + 1)

Qn − 1≡ 2Qv Q

ns(Qr − 1) + 2

Qn − 1≡ 2Qv(Qr + 1)

Qn − 1mod 1.

Ï (Qv, Qn−1) = 1 né 0 < 2(Qr+1) < Qn−1,ùL² 2Qv(Qu+1)/(Qn−1)

ØU´ê.

âÚn 7.3.1, (Ü²þ5 (7.2.12), Ný

íØ 7.3.2. 〈Am〉n = 0, 〈AmAk〉n = 0 〈Bm〉n = 0, Ù¥ 0 ≤ m, k ≤ n− 1.

â (7.3.5), ·k⟨n−1∏m=0

∣∣zQ−1m − p/zQ+1

m

∣∣−D⟩n

= 1 +D2

4|p|2

n−1∑m, k=0

〈AmAk〉n +O(|p|3). (7.3.6)

ò (7.3.3) \ AmAk, ·k

〈AmAk〉n = (Q− 1)2〈U1,mU1,k〉n + 〈σ2Q(Qk−Qm)〉n− (Q− 1) (〈σ−2Qm+1

U1,k〉n + 〈σ2Qk+1

U1,m〉n).(7.3.7)

Ún 7.3.3. - u ∈ N, K (Qu − 1)/(Qn − 1) ´ê= u = ns é,

s ∈ N ¤á.


y². ¿©5´w,, ·=y75. b u = ns + r, Ù¥ 0 ≤ r < n, âb

·k

Qu − 1

Qn − 1≡ Qns(Qr − 1)

Qn − 1mod 1.

Ï (Qns, Qn−1) = 1,·Ñ (Qu−1)/(Qn−1)´ê= (Qr−1)/(Qn−1)

´ê, = r = 0.

âÚn 7.3.3 Ú²þÎÒ5 (7.2.12), k

n−1∑m, k=0

〈U1,mU1,k〉n =1

Q2

n−1∑m, k=0

∞∑l1, l2=0

1

Q l1+l2

⟨σ−2Q(Ql1+m−Ql2+k)

⟩n

=1

Q2

n−1∑m, k=0

( ∑l1+m=l2+k

1

Q l1+l2+∑v 6=0

∑l1+m=l2+k+nv

1

Q l1+l2

)

=1

Q2

(n−1∑m=0

m∑k=0

∞∑l1=0

1

Q 2l1+m−k +n−1∑k=0

k−1∑m=0

∞∑l2=0

1

Q 2l2+k−m

)

+1

Q2

(+∞∑v=1

n−1∑m, k=0

∞∑l2=0

1

Q 2l2+k−m+nv+−∞∑v=−1

n−1∑m, k=0

∞∑l1=0

1

Q 2l1+m−k−nv

)

=1

Q2 − 1

(Q+ 1

Q− 1n+O(1)

)+O(1) =

n

(Q− 1)2+O(1).

(7.3.8)

þ¡ªê¡üª

n−1∑m=0

m∑k=0

1

Qm−k =nQ

Q− 1− Q−Q−(n−1)

(Q− 1)2=

nQ

Q− 1+O(1), (7.3.9)

n−1∑m=0

m−1∑k=0

1

Qm−k =n

Q− 1− Q−Q−(n−1)

(Q− 1)2=

n

Q− 1+O(1). (7.3.10)

(7.3.8) OL², n ªu ∞ , ¹ l1 +m 6= l2 + k éAÚk=6

u Q þ., ·^ O(1) L«. ù*:3e¡aqO¥´. =, (J¥

ÌÜ©Ñ´5g l1 +m = l2 + k ù«¹.

aqu (7.3.8) ¥O, ·k

n−1∑m, k=0

⟨σ2Q(Qk−Qm)

⟩n

= n+O(1) (7.3.11)

§7.3. ON¹ 113

n−1∑m, k=0

(⟨σ−2Qm+1

U1,k

⟩n

+⟨σ2Qk+1

U1,m

⟩n

)= − 2

Q

n−1∑m, k=0

∞∑l=0

1

Q l

⟨σ−2Q(Qm−Qk+l)

⟩n

= − 2

Q

n−1∑m=0

m∑k=0

1

Qm−k +O(1) = − 2n

Q− 1+O(1).

(7.3.12)

(Ü (7.3.7), (7.3.8), (7.3.11) Ú (7.3.12), ·k

n−1∑m, k=0

〈AmAk〉n = n+ n+ 2n+O(1) = 4n+O(1). (7.3.13)

â (7.3.6), ùL²⟨n−1∏m=0

∣∣zQ−1m − p/zQ+1

m

∣∣−D⟩n

= 1 +D2n|p|2 +O(np3). (7.3.14)

ª (7.2.15) y².

ë©z

[1] L. V. Ahlfors, Complex Analysis: Third Edition, McGraw-Hill, New York, 1979.

[2] L. V. Ahlfors, Lectures on Quasiconformal Mappings, University Lecture Series, 38. Amer.Math. Soc., Providence, RI, 2006.

[3] L. V. Ahlfors and L. Bers, Riemann’s mapping theorem for variable metrics. Ann. ofMath., 72 (1960), 385-404.

[4] A. Avila, X. Buff and A. Cheritat, Siegel disks with smooth boundaries. Acta Math., 193(2004), 1-30.

[5] I. N. Baker, An entire function which has wandering domains. J. Austral. Math. Soc., 22(1976), 173-176.

[6] A. Beardon, Iteration of rational functions. Grad. Texts Math., 132, Springer-Verlag,1991.

[7] A. Beurling and L. V. Ahlfors, The boundary correspondence under quasiconformal map-pings. Acta Math., 96 (1956), 125-142.

[8] P. Blanchard, Complex analytic dynamics on the Riemann sphere. Bull. Amer. Math.Soc., 11, (1984), 85-141.

[9] M. Bonk, Uniformization of Sierpinski carpets in the plane. Invent. Math., 185 (2011),559-665.

[10] B. Branner and J. Hubbard, The iteration of cubic polynomials, part I: the global topologyof parameter space. Acta Math., 160 (1988), 143-206.

[11] B. Branner and J. Hubbard, The iteration of cubic polynomials, part II: patterns andparapatterns. Acta Math., 169 (1992), 229-325.

[12] M. Braverman and M. Yampolsky, Computability of Julia Sets, Algorithms and Compu-tation in Mathematics, 23. Springer Verlag, Berlin, Heidelberg, 2009.

[13] L. Carleson and T. W. Gamlin, Complex Dynamics, Universitext: Tracts in Mathematics.Springer Verlag, New York, 1994.

[14] A. Cheritat, Ghys-like models for Lavaurs and simple entire maps. Conformal Geometryand Dynamics. Amer. Math. Soc. 10 (2006), no. 6, 227-256.

[15] P. Collet, R. Dobbertin and P. Moussa, Multifractal analysis of nearly circular Julia setand thermodynamical formalism. Ann. Inst. H Poincare, 56 (1992), 91-122.

[16] G. Cui, Dynamics of rational maps, topology, deformation and bifurcation, preprint, May2002 (Early version: Geometrically finite rational maps with given combinatorics, 1997).

[17] G. David and S. Semmes, Fractured fractals and broken dreams, volume 7 of Oxford Lec-ture Series in Mathematics and its Applications. The Clarendon Press Oxford UniversityPress, New York, 1997.

114

ë©z 115

[18] L. DeMarco and C. McMullen, Trees and dynamics of polynomials. Ann. Sci. Ecole Norm.Sup. 41 (2008), 337-383.

[19] L. DeMarco and A. Schiff, Enumerating the basins of infinity of cubic polynomials. J.Difference Equ. Appl. 16 (2010), 451-461.

[20] R. Devaney, Baby Mandelbrot sets adorned with halos in families of rational maps, Com-plex dynamics, 37-50, Contemp. Math., 396, Amer. Math. Soc., Providence, RI, 2006.

[21] R. Devaney, Intertwined Internal Rays in Julia Sets of Rational Maps, Fund. Math. 206(2009), 139-159.

[22] R. Devaney and A. Garijo, Julia Sets Converging to the Unit Disk. Proc. AMS, 136(2008), 981-988.

[23] R. Devaney and D. Look, Buried Sierpinski curve Julia sets. Discrete Contin. Dyn. Sys.13 (4) (2005), 1035-1046.

[24] R. Devaney, D. Look and D. Uminsky, The Escape Trichotomy for Singularly PerturbedRational Maps. Indiana University Mathematics Journal 54 (2005), 1621-1634.

[25] R. Devaney and E. D. Russell, Connectivity of Julia Sets for Singularly Perturbed RationalMaps. To appear.

[26] A. Douady, Disques de Siegel et anneaux de Herman. Sem. Bourbaki, 39 (677), 1986/1987.

[27] A. Douady and C. J. Earle, Conformally natural extensions of homeomorphisms of thecircle. Acta Math., 157 (1986), 23-48.

[28] A. Douady and J. H. Hubbard, Etude Dynamique des Polynomes Complexes: I and II.Publ. Math. Orsay, 1984-1985.

[29] A. Douady and J. H. Hubbard, On the dynamics of polynomial-like mappings. Ann. Sci.Ecole Norm. Sup. (4) 18 (1985), no. 2, 287-343.

[30] N. Emerson, Dynamics of polynomials with disconected Julia sets. Discrete Contin. Dyn.Syst. 9 (2003), 801-834.

[31] K. J. Falconer, Fractal geometry: mathematical foundations and applications. John Wiley& Sons, 1990.

[32] L. Geyer, Siegel discs, Herman rings and the Arnold family. Trans. Amer. Math. Soc., 353(2001) no. 9, 3661-3683.

[33] L. R. Goldberg and L. Keen, A finiteness theorem for a dynamical class of entire functions.Ergod. Th. & Dynam. Sys. 6 (1986), 183-192.

[34] D. Harris, Turning curves for critically recurrent cubic polynomials. Nonlinearity, 12(1999), 411-418.

[35] P. Haıssinsky and K. Pilgrim, Quasisymmetrically inequivalent hyperbolic Julia sets. Re-vista Math. Iberoamericana, 28 (2012), 1025-1034.

116 ë©z

[36] J. Heinonen, Lectures on analysis on metric spaces. Universitext. Springer-Verlag, NewYork, 2001.

[37] M. Herman, Conjugaison quasi symmetrique des homeomorphismes analytiques du cerclea des rotations. Preliminary manuscript.

[38] J. Hubbard, Local connectivity of Julia sets and bifurcation loci: three theorems of J.-C.Yoccoz, in “Topological Methods in Modern Mathematics”, Ed. by Goldberg and Phillips,Publish or Perish, (1993), 467-511.

[39] A. Katok and B. Hasselblatt, “Introduction to the Modern Theory of Dynamical System-s”, Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press,Cambridge, 1995.

[40] L. Keen, Julia sets of rational maps, in “Complex Dynamical Systems: The MathematicsBehind the Mandelbrot and Julia Sets”, Ed. by R. Devaney, Proc. Symp. Appl. Math. 49,Amer. Math. Soc., (1994), 71-89.

[41] L. Keen and G. Zhang, Bounded-type Siegel disks of a one-dimensional family of entirefunctions. Ergod. Th. & Dynam. Sys. 29 (2009), 137-164.

[42] J. Kiwi, Puiseux series polynomial dynamics and iteration of complex cubic polynomials.Ann. Inst. Fourier 56 (2006), 1337-1404.

[43] O. Lehto and K. I. Virtanen, Quasiconformal Mappings in the Plane, Springer Verlag,Berlin, Heidelberg, New York, 1973.

[44] o§. E©ÛÚ. ®: ®ÆÑ, 2004.

[45] M. Lyubich, Dynamics of rational transforms: The topological picture. Russ. Math. Surv.41 (1986), 35-95.

[46] M. Lyubich, Dynamics of quadratic polynomials. I, II. Acta Math., 178 (1997), 185-247,247-297.

[47] J. Mackay and J. Tyson, Conformal dimension: Theory and applications, University Lec-ture Series, 54. American Mathematical Society, Providence, RI, 2010.

[48] R. Mane, P. Sad and D. Sullivan, On the dynamics of rational maps. Ann. Sci. EcoleNorm. Sup. (4) 16 (1983), 193-217.

[49] C. McMullen, Automorphisms of rational maps. in Holomorphic Functions and Moduli I,Math. Sci. Res. Inst. Publ. 10, Springer, 1988.

[50] C. McMullen, Complex Dynamics and Renormalization, Annals of Mathematics Studies,135, Princeton Univ. Press, Princeton, NJ, 1994.

[51] C. McMullen and D. Sullivan, Quasiconformal homeomorphisms and dynamics. III. TheTeichmuller space of a holomorphic dynamical system. Adv. Math. 135 (1998), no. 2,351-395.

ë©z 117

[52] J. Milnor, Periodic orbits, external rays and the Mandelbrot set: An expository account,Asterisque 261 (2000), xiii, 277-333.

[53] J. Milnor, Dynamics in One Complex Variable: Third Edition, Annals of MathematicsStudies, 160, Princeton Univ. Press, Princeton, NJ, 2006.

[54] A. Osbaldestin, 1/s–expansion for generalized dimensions in a hierarchical s–state Pottsmodel. J. Phys. A: Math. Gen. 28 (1995), 5951-5962.

[55] H. Peitgen, D. Saupe editors, The Science of Fractal Images, Springer Verlag, New York,1988.

[56] W. Peng, Y. Yin and Y. Zhai, Density of hyperbolicity for rational maps with CantorJulia sets. Ergod. Th. & Dynam. Sys. 32 (2012), 1711-1726.

[57] C. Petersen, Local connnectivity of some Julia sets containing a circle with an irrationalrotation. Acta Math., 177 (1996), 163-224.

[58] C. Petersen and S. Zakeri, On the Julia set of a typical quadratic polynomial with a Siegeldisk. Ann. Math., 159 (2004), 1-52.

[59] K. Pilgrim and L. Tan, Rational maps with disconnected Julia sets. Asterisque, 261(2000), 349–383.

[60] C. Pommerenke, Boundary Behaviour of Conformal Maps, Springer Verlag, New York,1992.

[61] W. Qiu, X. Wang and Y. Yin, Dynamics of McMullen maps. Advances in Mathematics,229 (2012), 2525-2577.

[62] W. Qiu, L. Xie and Y. Yin, Fatou Components and Julia Sets of Singularly PerturbedRational Maps with Positive Parameter, Acta Math. Sin. (Engl. Ser.), 28 (10) (2012),1937-1954.

[63] W. Qiu and Y. Yin, Proof of the Branner-Hubbard conjecture on Cantor Julia sets. Sciencein China, Series A, 52 (2009), No. 1, 45–65.

[64] L. Rempe, Rigidity of escaping dynamics for transcendental entire functions. Acta Math.,203 (2009), 235-267.

[65] D. Ruelle, Repellers for real analytic maps. Ergodic Theory Dynamical Systems, 2 (1982),99-107.

[66] M. Shishikura, Herman’s theorem on quasisymmetric linearization of analytic circle home-orphisms. Manuscript, 1990.

[67] C. L. Siegel, Iteration of analytic functions. Ann. of Math., 43 (1942), 607–612.

[68] N. Steinmatz, Rational Iteration: Complex Analytic Dynamical Systems. De Gruyter,Berlin, 1993.

118 ë©z

[69] N. Steinmatz, On the dynamics of McMullen family. Conformal Geometry and Dynamics,10 (2006), 159–183.

[70] D. Sullivan, Conformal dynamical systems. In Geometric Dynamics. Leture Notes No.1007, Springer Verlag, 1983.

[71] D. Sullivan, Quasiconformal homeomorphisms and dynamics. I. Solution of the Fatou-Juliaproblem on wandering domains. Ann. of Math. (2) 122 (1985), no. 3, 401-418.

[72] G. Swiatek, On critical circle homeomorphisms. Bol. Soc. Brasil. Mat. (N.S.) 29 (2) (1998),329-351.

[73] L. Tan and Y. Yin, Local connectivity of the Julia sets for geometrically finite rationalmaps. Science in China, Series A (1) 39 (1996), 39–47.

[74] G. T. Whyburn, Topological characterization of the Sierpinski curve, Fund. Math 45(1958), 320-324.

[75] M. Widom, D. Bensimon, L. P. Kadanoff and S. J. Shenker, Strange objects in the complexplane. Journal of Statistical Physics, 32 (1983), 443-454.

[76] L. Xie, The dynamics of McMullen maps with real positive parameter, Ph.D. Thesis (inChinese), Fudan University, 2011.

[77] S. Zakeri, Dynamics of cubic Siegel polynomials. Commun. Math. Phys, 206 (1999), 185-233.

[78] S. Zakeri, On Siegel disks of a class of entire maps. Duke Math. J., 152 (2010), no. 3,481-532.

[79] G. Zhang, On the dynamics of e2πiθ sin(z). Illinois Journal of Mathematics, 49 (2005),1171-1179.

[80] G. Zhang, All bounded type Siegel disks of rational maps are quasi-disks. Invent. Math.,185 (2011), 421-466.

®uLÚ®¤Ø©

1©Yang Fei and Yin Yongcheng, A new proof of the realization of cubic tableaux,Bull. Aust. Math. Soc., 87 (2013), 207-215.

2©Yang Fei, On the dynamics of e2πiθ sin(z) + α sin3(z). Accepted by Acta Math.Sinica, English Series.

3©Qiu Weiyuan, Yang Fei and Yin Yongcheng, Rational maps whose Julia sets areCantor circles, Arxiv 1301.2692v1. Accepted by Ergodic Theory and Dynamical Systems.

4©Yang Fei and Wang Xiaoguang, The Hausdorff dimension of the boundary of theimmediate basin of infinity of McMullen maps, Arxiv 1204.1282v1. Submitted.

5©Qiu Weiyuan, Yang Fei and Yin Yongcheng, A geometric characterization of theJulia sets of McMullen maps. Preprint, 2013.

119

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