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Journal of Dynamical and Control Systems, Vol. 12, No. 4, October 2006, 465–488 ( c©2006)

ERGODIC HYPERBOLIC ATTRACTORSOF ENDOMORPHISMS

DA-QUAN JIANG and MIN QIAN

Abstract. Let μ be an SRB-measure on an Axiom A attractor Δof a C2-endomorphism (M, f). As is known, μ-almost every x ∈ Δis positively regular and the Lyapunov exponents of (f, Tf) at x are

constants λ(i)(f, μ), 1 ≤ i ≤ s. In this paper, we prove that Lebesgue-almost every x in a small neighborhood of Δ is positively regular andthe Lyapunov exponents of (f, Tf) at x are the constants λ(i)(f, μ),1 ≤ i ≤ s. This result is then generalized to nonuniformly com-pletely hyperbolic attractors of endomorphisms. The generic propertyof SRB-measures is also proved.

1. Introduction

Assume that Δ is a hyperbolic attractor of a C2 Axiom A diffeomorphism(M,f) and that v is the Lebesgue measure onM induced by the Riemannianmetric. It is well known that v-almost all x in the basin of attraction W s(Δ)are generic with respect to the SRB-measure μ+ of f on Δ (see [3, Theorem4.12]), i.e.,

limn→+∞

1n

n−1∑

i=0

δfix = μ+. (1)

If μ is an ergodic invariant measure of a C2-diffeomorphism g, the Lyapunovexponents of g not zero are μ-almost everywhere and if the measure μ hasthe SRB-property (i.e., the conditional measures of μ on unstable manifoldsare absolutely continuous with respect to the corresponding Lebesgue mea-sures), then Ledrappier [15] and Pugh and Shub [24] proved that the setof points generic with respect to μ has positive Lebesgue measure. Thisgeneric property of SRB-measures is of a particular interest for physics

2000 Mathematics Subject Classification. 37D20, 37D25, 37C40.Key words and phrases. Hyperbolic attractor, endomorphism, Lyapunov exponent,

SRB-measure, absolute continuity of local stable manifolds.This work was supported by the 973 Funds of China for Nonlinear Science, the NSFC

10271008, and the Doctoral Program Foundation of the Ministry of Education.

465

1079-2724/06/1000-0465/0 c© 2006 Springer Science+Business Media, Inc.

DOI: 10.1007/s10883‐006‐0002‐1

466 DA-QUAN JIANG and MIN QIAN

(see [8, 9, 33]). It makes easy to compute the space averages of various ob-servables approximately via their time averages, even if the SRB-measureis singular, since the initial point can be chosen in the basin of attractionof the attractor uniformly with respect to the Lebesgue measure.

For the hyperbolic attractor ∆ of the Axiom A diffeomorphism (M,f),by the Oseledec multiplicative ergodic theorem, µ+-almost every x ∈ ∆ isLyapunov regular and the Lyapunov exponents of (f, Tf) at x are constantsλ(i)(f, µ+), 1 ≤ i ≤ s. That is, there exists a linear decomposition of TxM ,TxM = U

(1)x ⊕ · · · ⊕ U

(s)x satisfying the condition

limn→±∞

1n

log ‖Txfnu‖ = λ(i)(f, µ+)

for all 0 = u ∈ U(i)x , 1 ≤ i ≤ s. Jiang et al. [12] proved that v-almost every

x ∈W s(∆) is positively regular and the Lyapunov exponents of (f, Tf) at xare the constants λ(i)(f, µ+), 1 ≤ i ≤ s. That is, there exists a sequence oflinear subspaces of TxM , {0} = V

(0)x ⊂ V

(1)x ⊂ · · · ⊂ V

(s)x = TxM satisfying

the condition

limn→+∞

1n

log ‖Txfnu‖ = λ(i)(f, µ+)

for all u ∈ V(i)x \V (i−1)

x , 1 ≤ i ≤ s. Jiang et al. [12] also showed that a similarresult holds for a general nonuniformly completely hyperbolic attractor withan ergodic SRB-measure. Tsujii [36] asserted that an ergodic probabilitymeasure µ of a diffeomorphism f without zero Lyapunov exponents is anSRB-measure if and only if the set of points, which are generic with respectto µ and positively regular with the same constant Lyapunov exponents asthose associated with µ, has a positive Lebesgue measure. However, theproof of the sufficiency is somewhat not easy to access, and the detailedproof of the necessity was given by Jiang et al. [12].

In practical applications, one should choose an initial point when com-puting approximately the Lyapunov exponents. The above “large ergodicproperty” of the Lyapunov exponents associated with an SRB-measure jus-tifies that the initial point can be taken in the basin of attraction of theattractor uniformly with respect to the Lebesgue measure while what onekeeps in mind is the Lyapunov exponents with respect to the SRB-measure.In general, the hyperbolic attractor may have a fractal structure and a sin-gular SRB-measure and, therefore, the Lebesgue measure is a more prefer-able reference measure for sampling the initial point than the SRB-measure,although the Lebesgue measure is, in general, not an invariant measure.

For a random hyperbolic dynamical system generated by small pertur-bations of a deterministic Axiom A diffeomorphism, Liu and Qian [19],and Liu [18] proved that its SRB-measure has a similar generic property asabove; while Jiang et al. [13] showed that the Lyapunov exponents of therandom system have a similar “large ergodic property” as above.

ERGODIC HYPERBOLIC ATTRACTORS OF ENDOMORPHISMS 467

The purpose of this paper is to use the methods developed in the abovereferences to attack the case of endomorphisms.

Let M be a smooth, compact, and connected Riemannian manifold with-out boundary and v be the Lebesgue measure on M induced by the Rie-mannian metric. Let O be an open subset of M and ∆ ⊂ O be an Axiom Aattractor of an endomorphism f ∈ C2(O,M) (see Sec. 2 below for the def-inition of an Axiom A attractor). Qian and Zhang [25] showed that thereexists a unique f -invariant Borel probability measure µ on ∆ satisfying thePesin entropy formula:

hµ(f) =∫

∆

s(x)∑

i=1

λ(i)x +m(i)

x dµ(x),

where hµ(f) is the measure-theoretic entropy of f with respect to µ, and−∞ ≤ λ

(1)x < λ

(2)x < · · · < λ

(s(x))x < +∞ are the Lyapunov exponents of

(f, Tf) at x with the multiplicities m(i)x , 1 ≤ i ≤ s(x). Qian and Zhang [25]

also proved that if ε > 0 is sufficiently small and the set of critical pointsCf = {x ∈ O|det(Txf) = 0} has zero Lebesgue measure, then for v-almostall x ∈ Bε(∆),

limn→+∞

1n

n−1∑

i=0

δfix = µ, (2)

where Bε(∆) = {y ∈ O | d(y,∆) < ε}. Since the SRB-measure µ is f -ergodic, the Lyapunov spectrum of (f, Tf) is µ-almost everywhere equal toa constant

{(λ(i)(f, µ),m(i)(f, µ)) : 1 ≤ i ≤ s}.In Sec. 2, after we review some basic notions and results about Axiom Aendomorphisms, we exploit the absolute continuity of local stable manifoldsand the SRB property of µ to prove the following result.

Theorem 1. Let f ∈ C2(O,M) and ∆ ⊂ O be an Axiom A attractor off , and suppose that Txf is nondegenerate for every x ∈ ∆. Then there existsε > 0 such that Lebesgue-almost every x ∈ Bε(∆) is positively regular andthe Lyapunov spectrum of (f, Tf) at x is a constant {(λ(i)(f, µ),m(i)(f, µ)) :1 ≤ i ≤ s}. That is, there exists a sequence of linear subspaces of TxM ,{0} = V

(0)x ⊂ V

(1)x ⊂ · · · ⊂ V

(s)x = TxM satisfying the condition

limn→+∞

1n

log ‖Txfnu‖ = λ(i)(f, µ)

for all u ∈ V(i)x \ V (i−1)

x , 1 ≤ i ≤ s. In addition, dimV(i)x − dimV

(i−1)x =

m(i)(f, µ), 1 ≤ i ≤ s.

The results of (2) and Theorem 1 can also be generalized to the caseof nonuniformly completely hyperbolic attractors of endomorphisms. More


concretely, suppose that f is a C2-endomorphism of a compact Riemannianmanifold M . Zhu [39] introduced the inverse limit space of (M,f) to over-come the difficulty arising from the noninvertibility and improved the localunstable manifold theorem of Pugh and Shub [24] for (M,f). Qian, Xie,and Zhu [26,27] presented a formulation of the SRB-property for an invari-ant measure µ of the endomorphism (M,f) and proved that this propertyis sufficient and necessary for the Pesin entropy formula:

hµ(f) =∫

M

s(x)∑

i=1

λ(i)x +m(i)

x dµ(x).

Assume that µ is f -ergodic, then for µ-almost every x ∈ M , the Lyapunovspectrum {(λ(i)

x ,m(i)x ) : 1 ≤ i ≤ s(x)} of (f, Tf) at x is equal to a constant:

{(λ(i)(f, µ),m(i)(f, µ)) : 1 ≤ i ≤ s}.In Sec. 3, we employ the absolute continuity of local stable manifolds toprove the following result.

Theorem 2. Suppose that µ is an ergodic invariant measure of the C2-endomorphism (M,f) satisfying the following conditions:

1. log |det(Txf)| ∈ L1(M,µ);2. µ is an SRB-measure of (M,f);3. the Lyapunov exponents of (f, Tf) are µ-almost everywhere not zero,

moreover, the smallest Lyapunov exponent

λ(1)(f, µ) = min{λ(i)(f, µ) : 1 ≤ i ≤ s} < 0.

Then there exists a Borel set ∆ ⊂M such that f∆ = ∆, µ(∆) = 1 and thatfor every x ∈W s(∆) def=

⋃y∈∆

W s(y),

limn→+∞

1n

n−1∑

i=0

δfix = µ,

where W s(y) is the global stable set of f at y and, moreover, v(W s(∆)) > 0.

Every point x ∈W s(∆) \+∞⋃n=0

f−nCf is positively regular and the Lyapunov

spectrum of (f, Tf) at x is the constant {(λ(i)(f, µ),m(i)(f, µ)) : 1 ≤ i ≤ s},where the set of critical points Cf = {y ∈M |det(Tyf) = 0}. If, in addition,

v(Cf ) = 0, then v(W s(∆) \+∞⋃n=0

f−nCf ) > 0.

Conversely, we wonder whether the existence of a positive Lebesgue mea-sure set of points with genericity and positive regularity as above impliesthe SRB-property of µ, as asserted by Tsujii [36] in the situation of diffeo-morphisms.


The above results justify that for uniformly or nonuniformly completelyhyperbolic attractors of endomorphisms, the initial points can be chosenclose to the attractors uniformly with respect to the Lebesgue measures,to compute the space averages of observables approximately via their timeaverages, or to compute approximately the Lyapunov exponents associatedwith the SRB-measures.

In applications, if the differentiable mappings or the equations of motionthat define dynamical systems are completely known, Lyapunov exponentsare computed by a straightforward technique using a phase space plus tan-gent space approach (see [2, 8, 10,34,37]).

2. Lyapunov exponents of Axiom Aattractors of endomorphisms

The purpose of this section is to prove Theorem 1. But first, we needto review some basic notions and results about hyperbolic sets of endomor-phisms.

We begin with the notion of an inverse limit space. Let X be a compactmetric space and T be a continuous mapping on X. The inverse limit spaceXT of the system (X,T ) is defined as the subset of XZ consisting of all fullorbits, i.e.,

XT ={x = {xi}i∈Z

∣∣ xi ∈ X, Txi = xi+1 ∀i ∈ Z}.

Obviously, XT is a closed subset of XZ which is endowed with the producttopology and the metric

d(x, y) =∑

i∈Z

12|i|

d(xi, yi) ∀x, y ∈ XZ.

Let p be the natural projection fromXT to X, i.e., p(x) = x0 for all x ∈ XT ,and τ : XT → XT be a left-shift homeomorphism, then p◦τ = T ◦p. Denotethe set of all T -invariant Borel probability measures on X by MT (X), whilethe notation Mτ (XT ) has a similar meaning. The projection p induces acontinuous bijection between Mτ (XT ) and MT (X) as follows:

pµ(φ) = µ(φ ◦ p) ∀µ ∈ Mτ (XT ), φ ∈ C(X);

(XT , τ, µ) is ergodic if and only if (X,T, pµ) is ergodic.Throughout this paper, we assume that M is a smooth, compact, and

connected Riemannian manifold without boundary and denote by v theLebesgue measure on M induced by the Riemannian metric. Assume thatf is a C1-endomorphism on M , i.e., f ∈ C1(M,M). Write E = p∗TM forthe pullback bundle of the tangent bundle TM by the projection p, and

Ex = p∗xTMp∗�p∗

x

Tx0M


for the natural isomorphisms between fibers Ex and Tx0M :

ξ = (x, v)p∗�p∗

x

v (∀x ∈Mf , v ∈ Tx0M).

A fiber-preserving mapping on E with respect to τ can be defined as

p∗τx ◦ Tf ◦ p∗ : Ex → Eτx

for each x ∈ Mf . For convenience, we will still denote it by Tf . Tf isa linear mapping on each fiber and there is a constant K > 0 such that‖Tf(x)‖ ≤ K for all x ∈Mf .

Let O be an open subset of M , f ∈ C1(O,M), and ∆ = f(∆) ⊂ O be acompact invariant set of f . Since ∆ is f -invariant, the inverse limit space∆f of the system (∆, f), the natural projection p from ∆f to ∆, and theleft-shift homeomorphism τ on ∆f can be defined as above. From later onin this section, we also denote

Mf = {x = {xi}i∈Z : xi ∈ O, f(xi) = xi+1 ∀i ∈ Z}.A compact f -invariant set ∆ is said to be hyperbolic if there exists a con-tinuous splitting p∗T∆M = E|∆f = Es ⊕ Eu such that

1. Tf(Es) ⊂ Es, Tf(Eu) ⊂ Eu;2. ‖Tfn(ξs)‖ ≤ Aλn

0‖ξs‖ ∀ξs ∈ Es, n ∈ Z+,‖Tfn(ξu)‖ ≥ A−1λ−n

0 ‖ξu‖ ∀ξu ∈ Eu, n ∈ Z+,where λ0 ∈ (0, 1) and A ≥ 1 are constants. For each x ∈ ∆f , the decompo-sition of the fiber Ex = Es

x ⊕ Eux may depend on the past. For x = y with

x0 = y0, it may happen that p∗Eux = p∗Eu

y . However, p∗Esx depends only

on x0.A compact f -invariant set ∆ is called an Axiom A basic set of f if1. ∆ is hyperbolic;2. ∆ ⊂ P (f), where P (f) is the set of periodic points of f ;3. f is topologically positively transitive on ∆, i.e., there exists x ∈ ∆

such that Orb+(x) = {f ix}+∞i=0 is dense in ∆;

4. there exists an open set V ⊃ ∆ satisfying⋂i∈Z

f i(V ) = ∆.

An Axiom A basic set ∆ of f is called an Axiom A attractor if there existarbitrarily small open neighborhoods U of ∆ such that fU ⊂ U .

Example 1. Let A be a (d × d)-matrix with elements in Z and absolutedeterminant greater than 1, all of whose eigenvalues have absolute valuedifferent from 1. Then the linear mapping A on Rd induces a multi-to-one endomorphism fA on the d-dimensional torus Td = Rd/Zd, as in thesituation of linear Anosov diffeomorphisms on torus. For each x ∈ Td, defineEs

x, Eux as TfA-invariant subspaces of TxTd associated with eigenvalues of

absolute value less than 1 and greater than 1, respectively. Then one cansee that the whole torus Td is a hyperbolic set of fA. Applying the local


product structure and shadowing property presented in [25, 32] to fA, onecan prove that the periodic points of fA are dense in Td, like the Anosovclosing lemma [3] in the diffeomorphism situation. Hence Td is an Axiom Aattractor of fA. Obviously, fA preserves the Lebesgue measure on Td. Forexample,

A =(n 11 1

), n ≥ 3,

with eigenvalues

0 < λ1 =n+ 1 −

√n2 − 2n+ 52

< 1 < λ2 =n+ 1 +

√n2 − 2n+ 52

,

induces a hyperbolic endomorphism on T2.

Let ∆ be a hyperbolic invariant set of f ∈ Cr(O,M) (r ≥ 1) andE|∆f = Es ⊕ Eu be the hyperbolic splitting. For each x ∈ ∆f and δ > 0,we write

Esx(δ) = {es ∈ Es

x : ‖es‖ < δ},Eu

x (δ) = {eu ∈ Eux : ‖eu‖ < δ},

Ex(δ) = Esx(δ) ⊕ Eu

x (δ).

For small ε > 0, x ∈ ∆, and x ∈ ∆f , the local stable and unstable manifoldsare defined, respectively, as follows:

W sε (x) def=

{y ∈M

∣∣ d(fnx, fny) < ε,∀n ∈ Z+},

Wuε (x) def=

{y0 ∈M

∣∣ ∃y ∈Mf such that

y0 = p(y), d(x−n, y−n) < ε ∀n ∈ Z+}.

For x ∈ ∆f , the global unstable set Wu(x) of f at x in Mf is defined by

Wu(x) def={y ∈Mf

∣∣∣∣ lim supn→+∞

1n

log d(y−n, x−n) < 0},

and the global unstable set of f at x in M is defined by Wu(x) def= pWu(x).Let

Wuε (x) =

{y ∈Mf

∣∣ d(y−n, x−n) < ε ∀n ∈ Z+},

which is called a local unstable set of f at x in Mf . Then p : Wuε (x) →

Wuε (x) is bijective. For x ∈ ∆, the global stable set W s(x) of f at x in M

is defined by

W s(x) def={y ∈M


1n

log d(fny, fnx) < 0}.


One has

Wu(x) =+∞⋃

n=0

fn(Wuε (τ−nx)) ∀x ∈ ∆f ,

W s(x) =+∞⋃

n=0

f−n(W sε (fnx)) ∀x ∈ ∆.

(3)

If ∆ is an Axiom A attractor of f , then there exists a sufficiently small ε > 0such that Wu

ε (x) ⊂ ∆ ∀x ∈ ∆f [25, Proposition 2.1], and hence

Wu(x) = ∪+∞n=0f

n(Wuε (τ−nx)) ⊂ ∆ ∀x ∈ ∆f .

Denote by Bx(δ) the open ball on M of radius δ centered at x ∈ M ,and by Bx(δ) the open ball on Mf of radius δ centered at x ∈ Mf . Bythe property of the continuous splitting E|∆f = Es ⊕ Eu, there exists aconstant a > 0 such that for any e = es ⊕eu with es ∈ Es, eu ∈ Eu, one has

max{‖es‖, ‖eu‖

}≤ a

2‖e‖.

The following proposition is a result of changing coordinates from the the-orem about stable and unstable manifolds for hyperbolic invariant sets ofendomorphisms (see [25, Theorems 2.1 and 2.2] and [38]).

Proposition 1. Let f ∈ Cr(O,M) (r ≥ 1) and ∆ ⊂ O be a hyperbolicinvariant set of f . Then there exists a number δ0 > 0 such that for eachx ∈ ∆f , if y ∈ Bx(δ0/2) ∩ ∆f , then there are Cr-mappings

φsx,y : Es

x(aδ0) → Eux , φu

x,y : Eux (aδ0) → Es

x

satisfying

W sδ0

(y0) = By0(δ0) ∩ expx0◦p∗ ◦ (id, φs

x,y)Esx(aδ0),

Wuδ0

(y) = By0(δ0) ∩ expx0◦p∗ ◦ (φu

x,y, id)Eux (aδ0),

sup{‖Tesφs

x,y‖ : y ∈ Bx(δ02

) ∩ ∆f , es ∈ Esx(aδ0)

}≤ 1

2,

sup{‖Teuφu

x,y‖ : y ∈ Bx(δ02

) ∩ ∆f , eu ∈ Eux (aδ0)

}≤ 1

2.

Proposition 2. Let f ∈ C2(O,M) and ∆ ⊂ O be a hyperbolic invariantset of f . For each x ∈ ∆, if there exist some positively regular point y0 ∈W s(x) and the smallest Lyapunov exponent of (f, Tf) at y0, λ

(1)y0 > −∞,

then every y ∈ W s(x) with det(Tfkyf) = 0 for all k ≥ 0, is positivelyregular and the Lyapunov spectrum of (f, Tf) at y is the same as that atthe point y0.

Proof. First, we consider the case y0 ∈ W sε (x). The assumption λ

(1)y0 >

−∞ implies that det(Tfky0f) = 0 for all k ≥ 0. Then, provided that ε issufficiently small for each y ∈W s

ε (x), we have det(Tfkyf) = 0 for all k ≥ 0.Therefore, we can exploit the Ruelle perturbation theorem for the spectrum


of matrix products (see [31, Theorem 4.1]) to prove that y is positivelyregular and the Lyapunov spectrum of (f, Tf) at y is the same as that atthe point y0. The argument is almost the same as the proof of Theorem 3.1in [12], therefore, we omit details here.

By (3), the general case y0 ∈ W s(x) can be reduced to the previousone. However, we need to note the fact that for a positively regular pointy ∈ W s(x), if det(Tfk0yf) = 0 for some k0 ≥ 0, then the smallest Lya-punov exponent of (f, Tf) at y, λ(1)

y = −∞; otherwise, for each n ≥ 0, theLyapunov spectrum of (f, Tf) at fny is the same as that at y.

Now we prove the absolute continuity of the local stable manifolds of anAxiom A attractor of an endomorphism. Assume that ∆ is an Axiom Aattractor of f ∈ C2(O,M). For small ε > 0 and x ∈ ∆f , we denoteby F(x, ε) the collection of local stable manifolds W s

ε (y) passing throughy ∈Wu

ε (x) ⊂ ∆. We set

U(x, ε) =⋃

y∈W uε (x)

W sε (y);

U(x, ε) is an open neighborhood of x0 = px in M . A submanifold W ofM is said to be transversal to the family F(x, ε) if the following conditionshold:

(i) W ⊂ U(x, ε) and p∗x ◦ exp−1x0W is the graph of a C1-mapping ψ :

Eux (ε) → Es

x;(ii) W intersects any W s

ε (y), y ∈ Wuε (x), at exactly one point and this

intersection is transversal, i.e., TzW ⊕ TzWsε (y) = TzM , where z =

W ∩W sε (y).

We denote by vW the Lebesgue measure on W induced by the Riemannianmetric on W inherited from M . Now consider two submanifolds W1 andW2 transversal to F(x, ε). Since {W s

ε (y)}y∈∆ is a continuous family of C2

embedded discs, there exist two submanifolds W1 and W2, respectively, ofW1 and W2 such that we can well define the so-called Poincare mapping

PW1,W2: W1 ∩ U(x, ε) → W2 ∩ U(x, ε)

by settingPW1,W2

: z �→ W2 ∩W sε (y)

for z = W1 ∩ W sε (y), y ∈ Wu

ε (x), and, moreover, PW1,W2is a homeo-

morphism. A mapping T : X → Y between two σ-finite measure spaces(X,A, µ) and (Y,B, ν) is said to be absolutely continuous if the followingthree conditions hold:

1. T is injective;2. if A ∈ A, then TA ∈ B;3. A ∈ A and µ(A) = 0 imply ν(TA) = 0.


Proposition 3. There exists a number ε0 > 0 such that for each x ∈ ∆f

and ε ∈ (0, ε0), the family of C2-embedded disks F(x, ε) = {W sε (y)}y∈W u

ε (x)

is absolutely continuous in the following sense: For every two submanifoldsW1 and W2 contained in U(x, ε) and transversal to the family F(x, ε), thePoincare mapping PW1,W2

constructed as above is absolutely continuous withrespect to the Lebesgue measures vW1 and vW2 .

A detailed proof of this proposition can be carried out by a completelysimilar argument to that of Part II of [14]. We omit the details here. Theproof of the absolute continuity of local stable manifolds for Anosov dif-feomorphisms was given in [1] (see also [20, Chap. III, Theorem 3.1]). Foruniformly partially hyperbolic systems it was formulated in [5]. The case ofnonuniformly partially hyperbolic systems was considered in [14,22,24].

Suppose that ∆ is an Axiom A basic set of f ∈ C2(O,M). Qian andZhang [25] exploited the local product structure of the Axiom A basic set∆ [25, Theorem 2.2] to show that (∆f , d1, τ) is a Smale space, where d1 is asuitable metric on ∆f . Therefore, ∆f has Markov partitions of arbitrarilysmall diameter. (For the definition of Smale spaces and their Markov par-titions, see [30, Chap. 7].) One can obtain the symbolic representations of(∆f , d1, τ) via its Markov partitions. Applying the thermodynamic formal-ism of symbolic spaces, one can prove that every Holder continuous functionon ∆f has a unique equilibrium state with respect to τ .

Define φu : ∆f → R as φu(x) = − log |det(Tf |Eux)| for all x ∈ ∆f . φu is

Holder continuous and has a unique equilibrium state µφu with respect toτ [25], i.e.,

hµφu (τ) +∫φudµφu = sup

µ∈Mτ (∆f )

(hµ(τ) +

∫φudµ

)= Pτ (φu),

where hµ(τ) is the measure-theoretic entropy of τ with respect to µ, andPτ (φu) is the topological pressure of φu with respect to τ . In the case where∆ is an Axiom A attractor, Pτ (φu) = 0 and the measure µ def= pµφu is theunique Borel probability measure on ∆ satisfying the Pesin entropy formula:

hµ(f) =∫

∆

s(x)∑

i=1

λ(i)x +m(i)

x dµ(x).

Now we review the SRB-property of the measure µ constructed above.For any fixed ν ∈ Mf (∆), let ν be the unique τ -invariant Borel probabilitymeasure on ∆f so that pν = ν. Let du = dimEu. A measurable partitionξ of ∆f is said to be subordinate to Wu-manifolds of f with respect to ν iffor ν-a.e. x ∈ ∆f , the member of ξ which contains x, denoted by ξ(x), hasthe following properties:

1. p|ξ(x) : ξ(x) → p(ξ(x)) is bijective;


2. there exists a du-dimensional C1-embedded submanifold Wx of Msuch that Wx ⊂ Wu(x), p(ξ(x)) ⊂ Wx, and p(ξ(x)) contains an openneighborhood of x0 = px in the submanifold topology of Wx.

We say that ν has the SRB-property if for every measurable partition ξof ∆f subordinate to Wu-manifolds of f with respect to ν, p(νξ(x)) � vu

x

for ν-a.e. x ∈ ∆f , where {νξ(x)}x∈∆f is a canonical system of conditionalmeasures of ν associated with ξ, and vu

x is the Lebesgue measure on Wx

induced by its inherited Riemannian metric as a submanifold of M . (Werefer to [19, 28] for the details of conditional measures given a measurablepartition which is closely connected to regular conditional probability dis-tributions in probability theory.) The proposition below coincides with [26,Corollary 1.1.2].

Proposition 4. Let f ∈ C2(O,M) and ∆ ⊂ O be an Axiom A attractorof f , and assume that Txf is nondegenerate for every x ∈ ∆. Then themeasure µ = pµφu is the unique f-invariant Borel probability measure on ∆which is characterized by each of the following properties:

1. if ε > 0 is sufficiently small, then

limn→+∞

1n

n−1∑

k=0

δfkx = µ

for Lebesgue-almost every x ∈ Bε(∆) def= {y ∈M |d(y,∆) < ε};2. the Pesin entropy formula holds for the system (∆, f, µ);3. µ has the SRB-property.

In fact, if ξ is a measurable partition of ∆f subordinate to Wu-manifoldsof f with respect to µ, and let ρx be the density of p(µφu,ξ(x)) with respectto vu

x , then for µφu-almost all x ∈ ∆f , there exists a countable number ofdisjoint open subsets Un(x), n ∈ N, of Wx such that

⋃

n∈N

Un(x) ⊂ ξ(x), vux(ξ(x) \

⋃

n∈N

Un(x)) = 0

and on each Un(x), ρx is a strictly positive function satisfying

ρx(y)ρx(z)

=+∞∏

i=1

exp(φu(p|−1ξ(x)y))

exp(φu(p|−1ξ(x)z))

∀y, z ∈ Un(x).

In particular, log ρx restricted to each Un(x) is Lipschitz along Wx.

By the assumption det(Txf) = 0 for all x ∈ ∆, the smallest Lyapunovexponent λ(1)

x of (f, Tf) is µ-almost everywhere not −∞. Since µ is f -ergodic, the Lyapunov spectrum of (f, Tf) is µ-almost everywhere equal toa constant {(λ(i)(f, µ),m(i)(f, µ)) : 1 ≤ i ≤ s}. Denote by Γµ the set ofpositively regular points x ∈ O such that the Lyapunov spectrum of (f, Tf)at x is the constant {(λ(i)(f, µ),m(i)(f, µ)) : 1 ≤ i ≤ s}. We write Γc

µ =


O \ Γµ. Let Γµ be the set of positively regular points x ∈ ∆f such that theLyapunov spectrum of (τ, Tf) at x is {(λ(i)(f, µ),m(i)(f, µ)) : 1 ≤ i ≤ s},then pΓµ ⊂ Γµ. Denote by vB the normalized Lebesgue measure

v

v(B)on

a Borel subset B of M with v(B) > 0.

Proof of Theorem 1. Let R = {R1, R2, · · · , Rk0} be a Markov partition of∆f with diameter smaller than min(δ0, ε0)/2, where δ0 is the constant speci-fied in Proposition 1 and ε0 is the constant from Proposition 3. The elementsof R are closed proper rectangles, and some of its elements intersect withone another on the boundary. We can modify the elements of R appropri-ately on the boundary to make them not intersect with one another. ThenR becomes a measurable partition of ∆f .

Denote by ξ the measurable partition of ∆f into sets having the formRi∩Wu

δ0(y) for Ri ∈ R and y ∈ Ri. Let ∂R = ∂+R∪∂−R be the boundary

of R as defined in [30, Chap. 7]. Since µφu(∂R) = 0 (see [30, Chap. 7]),the measurable partition ξ of ∆f is subordinate to Wu-manifolds of f withrespect to µ. Let {µφu,ξ(y)}y∈∆f be a canonical system of conditional mea-sures of µφu associated with ξ. For each y ∈ ∆f , denote by vξ

y the normalizedLebesgue measure on p(ξ(y)) induced by the inherited Riemannian metric.By the Oseledec multiplicative ergodic theorem,

µφu(∆f ∩ Γµ) =∫

∆f

µφu,ξ(y)

((∆f ∩ Γµ) ∩ ξ(y)

)dµφu(y) = 1, (4)

then for µφu -almost all y ∈ ∆f ,

µφu,ξ(y)

(Γµ ∩ ξ(y)

)= 1. (5)

By the SRB-property of µ (Proposition 4), for µφu -almost all y ∈ ∆f ,p(µφu,ξ(y)) is equivalent to vξ

y. For each Ri ∈ R, since µφu(Ri) > 0, there

exists xi ∈ Ri such that µφu,ξ(xi)

(Γµ ∩ ξ(xi)

)= 1 and p(µφu,ξ(xi)) is equiv-

alent to vξxi

, and hence

vξxi

(Γµ ∩ p(ξ(xi))) = 1. (6)

For the above point xi, let η be the measurable partition{

exp(xi)0 ◦p∗({es} × Eu

xi(aδ0))

}es∈Es

xi(aδ0)

of (exp(xi)0 ◦p∗)(Exi(aδ0)), and let η be the restriction of η to

Uidef=

⋃

y∈W uδ0

(xi)∩p(Ri)

W sδ0∧ε0(y),

where δ0 ∧ ε0 = min(δ0, ε0). For each y ∈ Ui, denote by vηy the normalized

Lebesgue measure on η(y) induced by the inherited Riemannian metric. Let


{vUi

η(y)}y∈Uibe a canonical system of conditional measures of vUi associated

with the partition η. Then, by the Fubini theorem, applied to Exi∩ p∗xi

◦exp−1

(xi)0Ui, and a simple argument, one can prove that for vUi-almost all

y ∈ Ui, the measure vUi

η(y) is equivalent to vηy and there exists a number

C > 1 such that

C−1 ≤dvUi

η(y)

dvηy

≤ C (7)

holds vηy -almost everywhere on η(y).

By Proposition 1, for each y ∈ Ui, η(y) is transversal to the family

F(xi, δ0 ∧ ε0) def= {W sδ0∧ε0(z)}z∈W u

δ0(xi)∩p(Ri).

p(ξ(xi)) = Wuδ0

(xi) ∩ p(Ri) is also transversal to the family F(xi, δ0 ∧ ε0).We can assume that δ0 ∧ ε0 is sufficiently small, so that det(Txf) = 0 forany x ∈ Bδ0∧ε0(∆). Then by Proposition 2, we have

Pp(ξ(xi)),η(y)(Γµ ∩ p(ξ(xi))) = Γµ ∩ η(y),Pp(ξ(xi)),η(y)(Γc

µ ∩ p(ξ(xi))) = Γcµ ∩ η(y). (8)

Then from (6), (8), and the absolute continuity of F(xi, δ0 ∧ ε0) (Proposi-tion 3), we have

vηy (Γc

µ ∩ η(y)) = vηy

(Pp(ξ(xi)),η(y)(Γc

µ ∩ p(ξ(xi))))

= 0. (9)

By (7), for vUi-almost all y ∈ Ui, the measure vUi

η(y) is equivalent to vηy and,

therefore,vUi

η(y)(Γcµ ∩ η(y)) = 0

and vUi

η(y)(Γµ ∩ η(y)) = 1. Then we obtain

vUi(Γµ ∩ Ui) =∫

Ui

vUi

η(y)(Γµ ∩ η(y))dvUi(y) = 1. (10)

Let

G =k0⋃

i=1

Ui, ε′ =δ0 ∧ ε0

2.

Then G is an open neighborhood of ∆ in M satisfying the condition

W sε′(∆) def=

⋃

x∈∆

W sε′(x) ⊂ G

and v(Γcµ ∩W s

ε′(∆)) = 0.

Remark 1. Following the proof of Theorem 1 and applying the Birkhoffergodic theorem instead of the Oseledec multiplicative ergodic theorem,one can prove (2), the generic property of µ. Qian and Zhang [25] gave


a different proof of this property following the main line of Sinai [35] andBowen [3, Theorem 4.12].

3. Nonuniformly completely hyperbolic attractorsof endomorphisms

This section is devoted to the proof of Theorem 2. First, we reviewsome necessary concepts and results from the smooth ergodic theory forendomorphisms [26,27,39].

Throughout this section, we assume that M is a smooth, compact, andconnected Riemannian manifold without boundary and f is a C2-endo-morphism on M , i.e., f ∈ C2(M,M). The inverse limit space Mf of thesystem (M,f), the natural projection p from Mf to M , and the left-shifthomeomorphism τ on Mf can be defined as in Sec. 2.

By the Oseledec multiplicative ergodic theorem, there exists a Borel sub-set Γ ⊂M with fΓ ⊂ Γ and µ(Γ) = 1 for any µ ∈ Mf (M), such that eachpoint x ∈ Γ is positively regular and the Lyapunov spectrum of (f, Tf) at x,{(λ(i)

x ,m(i)x ) : 1 ≤ i ≤ s(x)} (with λ(1)

x < λ(2)x < · · · < λ

(s(x))x ) is well defined.

From now on, we fix µ ∈ Mf (M) satisfying the integrability condition

log |det(Txf)| ∈ L1(M,µ). (11)

DefineΓ∞ =

{x ∈ Γ

∣∣Txf is degenerate or λ(1)x = −∞

}.

The integrability condition (11) implies that µ(Γ∞) = 0. Let

Γ′ = Γ \+∞⋃

n=0

f−nΓ∞.

It is easy to see that fΓ′ ⊂ Γ′, µ(Γ′) = 1, and for any x ∈ Γ′, Txf is anisomorphism and λ(1)

x > −∞. Let

∆ = Mf \+∞⋃

n=−∞τn(p−1Γ∞).

Obviously, τ∆ = ∆ and for any x = {xn}n∈Z ∈ ∆ we have xn ∈M \Γ∞ forall n ∈ Z. As a consequence of the integrability condition (11), µ(∆) = 1,where µ ∈ Mτ (Mf ) is such that pµ = µ. By the Oseledec multiplicativeergodic theorem, there exists a Borel set ∆ ⊂ ∆ satisfying the conditionsτ∆ = ∆ and µ(∆) = 1. Furthermore, for every x = {xn}n∈Z ∈ ∆, thereexists an integer s(x), a splitting of the tangent space Tx0M = E

(1)x ⊕ · · · ⊕

E(s(x))x , numbers −∞ < λ

(1)x < λ

(2)x < · · · < λ

(s(x))x < +∞, and integers

m(i)x , 1 ≤ i ≤ s(x), such that

1. s(x), λ(i)x ,m

(i)x are τ -invariant, dimE

(i)x = m

(i)x ;


2. the splitting is invariant under Tf , i.e.,

Tx0f(E(i)x ) = E

(i)τx , 1 ≤ i ≤ s(x);

3. for any m ∈ Z, let

Tm(x) =

Tx0fm if m > 0,

id if m = 0,(Txm

f−m)−1 if m < 0,

then

limm→±∞

1m

log ‖Tm(x)ξ‖ = λ(i)x

for all 0 = ξ ∈ E(i)x , 1 ≤ i ≤ s(x);

4. x0 ∈ Γ′ and s(x) = s(x0), λ(i)x = λ

(i)x0 and m

(i)x = m

(i)x0 for all i =

1, · · · , s(x), where {(λ(i)x0 ,m

(i)x0 ) : 1 ≤ i ≤ s(x0)} is the Lyapunov

spectrum of (f, Tf) at x0.

The numbers {λ(i)x : 1 ≤ i ≤ s(x)} are called the Lyapunov exponents of

(Mf , τ, µ) at x, and m(i)x is called the multiplicity of λ(i)

x .For simplicity, except for additional statements, we assume that µ is f -

ergodic. Then the Lyapunov spectrum of (f, Tf) is µ-almost everywhereequal to a constant {(λ(i)(f, µ),m(i)(f, µ)) : 1 ≤ i ≤ s}, and so is µ-a.e. theLyapunov spectrum of (Mf , τ, µ). As is stated in Theorem 2, we assumethat λ(i)(f, µ) = 0 for all i. If λ(s)(f, µ) = max{λ(i)(f, µ) : 1 ≤ i ≤ s} < 0,then the support of µ contains only one attracting periodic orbit (see [31,Corollary 6.2]). The local stable manifold of f at each point x on this orbitis a neighborhood of x. The desired result is obvious in this trivial case, sowe assume that λ(s)(f, µ) > 0. As is stated in Theorem 2, we also assumethat λ(1)(f, µ) = min{λ(i)(f, µ) : 1 ≤ i ≤ s} < 0, otherwise, the assumptionλ(i)(f, µ) = 0 for all i, implies that λ(i)(f, µ) > 0 for all i and hence (M,f)is an expanding mapping considered in [11,27].

Zhu [39] and Qian, Xie, and Zhu [26, 27] developed a rigorous theoryof the theorem about stable and unstable manifolds for endomorphisms,borrowing the techniques given in Liu and Qian [19]. Given f and a pointx ∈Mf , the global unstable set Wu(x) of f at x in Mf is defined by

Wu(x) def={y ∈Mf


1n

log d(y−n, x−n) < 0},

while the global unstable set of f at x in M is defined by Wu(x) def= pWu(x)if x ∈ ∆ and λ(s(x))

x > 0 and, otherwise, by Wu(x) = {x0}. For x ∈M , theglobal stable set W s(x) of f at x in M is defined by

W s(x) def={y ∈M


1n

log d(fny, fnx) < 0}.


We can assume that the Lyapunov spectrum of (Mf , τ, µ) at each x ∈ ∆ isthe constant {(λ(i)(f, µ),m(i)(f, µ)) : 1 ≤ i ≤ s}. For each x ∈ ∆, we set

Eux =

⊕

λ(i)x >0

E(i)x , Es

x =⊕

λ(i)x <0

E(i)x , ds = dimEs

x, du = dimEux .

As is shown in [26, 27, 39], there exists a countable number of compactsubsets ∆i, i ∈ N, of Mf with

⋃

i∈N

∆i ⊂ ∆, µ(∆ \⋃

i∈N

∆i) = 0

such that the following conditions hold.1. Eu

x and Esx depend continuously on x ∈ ∆i.

2. For each ∆i, the local unstable manifolds Wuloc(x) of f in M at x, x ∈

∆i, constitute a continuous family of C1,1 embedded du-dimensionaldisks. There are positive numbers λu

i , εui < λui /200, ru

i < 1, γui , αu

i ,and βu

i such that the following properties hold for each x ∈ ∆i:(i) there exists a C1,1-mapping hx : Ux → Es

x, where Ux is an opensubset of Eu

x which contains

Eux (αu

i ) def={ξ ∈ Eu

x : ‖ξ‖ < αui

},

satisfying(a) hx(0) = 0, T0hx = 0;(b) Lip(hx) ≤ βu

i , Lip(T·hx) ≤ βui , where T·hx : ξ → Tξhx;

(c) Wuloc(x) = exppx(Graph(hx));

(ii) for any y0 ∈ Wuloc(x) there exists unique y ∈ Mf such that py =

y0, d(y−n, x−n) ≤ rui e

−εui n for all n ∈ N, and d(y−n, x−n) ≤

γui e

−λui n for all n ∈ N;

(iii) let

Wuloc(x) =

{y ∈Mf

∣∣ y−n ∈Wuloc(τ

−nx) ∀n ∈ N},

which is called a local unstable set of f at x in Mf , then p :Wu

loc(x) →Wuloc(x) is bijective.

3. For each ∆i, the local stable manifolds W sloc(x) of f in M at x, x ∈

p∆i, constitute a continuous family of C1,1 embedded ds-dimensionaldisks. There are positive numbers λs

i , εsi < λs

i/200, rsi < 1, γs

i , αsi ,

and βsi such that the following properties hold for each x ∈ ∆i:

(i) there exists a C1,1-mapping lx : Ox → Eux , where Ox is an open

subset of Esx which contains

Esx(αs

i )def={ξ ∈ Es

x : ‖ξ‖ < αsi

},

satisfying(a) lx(0) = 0, T0lx = 0;(b) Lip(lx) ≤ βs

i , Lip(T·lx) ≤ βsi , where T·lx : ξ → Tξlx;


(c) W sloc(px) = exppx(Graph(lx));

(ii) for any y ∈W sloc(px), d(f

ny, fn(px)) ≤ rsi e

−εsi n for all n ∈ N, and

d(fny, fn(px)) ≤ γsi e

−λsi n for all n ∈ N.

One has

Wu(x) =+∞⋃

n=0

fn(Wuloc(τ

−nx)), W s(px) =+∞⋃

n=0

f−n(W sloc(f

n(px))).

For each x ∈ ∆ and sufficiently small q > 0, let

U(x, q) = exppxEx(q),

where

Ex(q) ={(es, eu) ∈ p∗TpxM : es ∈ Es

x, eu ∈ Eu

x , ‖es‖ < q, ‖eu‖ < q}.

The following proposition is a consequence of changing coordinates from theabove theorem about stable and unstable manifolds.

Proposition 5. For each ∆i given above, there exists a number δi > 0such that for each x ∈ ∆i, y ∈ Bx(δi/2) ∩ ∆i, y ∈ U(x, δi/2) ∩ p∆i, thereexist C1-mappings

φux,y : Eu

x (δi) → Esx, φs

x,y : Esx(δi) → Eu

x

satisfying the following conditions:

Wuloc(y) ∩ U(x, δi) =

(exppx p∗ Graphφu

x,y

)∩ U(x, δi),

W sloc(y) ∩ U(x, δi) =

(exppx p∗ Graphφs

x,y

)∩ U(x, δi),

sup{‖φu

x,y(w)‖ + ‖Twφux,y‖ : w ∈ Eu

x (δi), y ∈ Bx

(δi2

)∩ ∆i

}≤ 1

2,

sup{‖φs

x,y(w)‖ + ‖Twφsx,y‖ : w ∈ Es

x(δi), y ∈ U

(x,δi2

)∩ p∆i

}≤ 1

2.

Now we consider the absolute continuity of the local stable manifolds.For x ∈ ∆i and q ∈ (0, δi], we denote by F∆i

(x, q) the collection of localstable manifolds W s

loc(y) passing through y ∈ p∆i ∩ U(x, q2 ). We set

∆i(x, q) =⋃

y∈p∆i∩U(x,q/2)

W sloc(y) ∩ U(x, q).

A submanifold W of M is said to be transversal to the family F∆i(x, q) if

the following conditions hold:(i) W ⊂ U(x, q) and p∗x exp−1

px W is the graph of a C1-mapping ψ :Eu

x (q) → Esx;

(ii) W intersects any W sloc(y), y ∈ p∆i ∩ U(x, q/2), at exactly one point

and this intersection is transversal, i.e., TzW ⊕ TzWsloc(y) = TzM ,

where z = W ∩W sloc(y).


For a submanifold W transversal to F∆i(x, q), we define

|W | = supw∈Eu

x (q)

‖ψ(w)‖ + supw∈Eu

x (q)

‖Twψ‖,

where ψ is defined as above. We denote by vW the Lebesgue measureon W induced by the Riemannian metric on W inherited from M . Nowwe consider two submanifolds W1 and W2 transversal to F∆i

(x, q). Since{W s

loc(y)}y∈p∆iis a continuous family of C1-embedded disks, there exist

two open submanifolds W1 and W2, respectively, of W1 and W2 such thatwe can well define the so-called Poincare mapping

PW1,W2: W1 ∩ ∆i(x, q) → W2 ∩ ∆i(x, q)

by settingPW1,W2

: z �→ W2 ∩W sloc(y)

for z = W1 ∩ W sloc(y), y ∈ p∆i ∩ U(x, q/2), and, moreover, PW1,W2

is ahomeomorphism.

The proposition below is a counterpart of the Pesin absolute continu-ity theorem [22, Theorem 3.2.1] for diffeomorphisms (see also [14, Part II,Theorem 4.1]).

Proposition 6. There exists a number qi ∈ (0, δi] such that for everyx ∈ ∆i and every two submanifolds W1 and W2 contained in U(x, qi),transversal to the family F∆i

(x, qi) and satisfying the condition |Wi| ≤ 1/2,i = 1, 2, the Poincare mapping PW1,W2

constructed as above is absolutelycontinuous with respect to the Lebesgue measures vW1 and vW2 .

Now we recapitulate the formulation of the SRB-property following Qian,Xie, and Zhu [26, 27]. A measurable partition ξ of Mf is said to be subor-dinate to Wu-manifolds of (f, µ) if for µ-a.e. x ∈Mf , ξ(x), the member ofξ to which x belongs, has the following properties:

1. p|ξ(x) : ξ(x) → p(ξ(x)) is bijective;2. there exists a dimEu

x -dimensional embedded C1-submanifold Wx ofM such that Wx ⊂ Wu(x), p(ξ(x)) ⊂ Wx and p(ξ(x)) contains anopen neighborhood of x0 = px in the submanifold (or subset) topologyof Wx.

We say that µ has the SRB property if for every measurable partition ξof Mf subordinate to Wu-manifolds of (f, µ), p(µξ(x)) � vu

x for µ-a.e.x ∈ Mf , where {µξ(x)}x∈Mf is a canonical system of conditional measuresof µ associated with ξ, and vu

x is the Lebesgue measure on Wx inducedby its inherited Riemannian metric as a submanifold of M (vu

x = δx0 ifWu(x) = {x0}).

Now we assume that µ is an ergodic invariant measure of the C2-endo-morphism (M,f) satisfying the conditions of Theorem 2:

1. log |det(Txf)| ∈ L1(M,f);


2. µ has the SRB-property;3. the Lyapunov exponents of (f, Tf) are µ-almost everywhere not zero;

the smallest Lyapunov exponent λ(1)(f, µ) < 0.

Proof of Theorem 2. (a) We can assume that for every x ∈ ∆, px is genericwith respect to µ, i.e.,

limn→+∞

1n

n−1∑

i=0

δfipx = µ, (12)

and the Lyapunov spectrum of (Mf , τ, µ) at x is

{(λ(i)(f, µ),m(i)(f, µ)) : 1 ≤ i ≤ s}.

For every fixed x ∈ ∆ and any y ∈W s(px),

lim supn→+∞

1n

log d(fny, fnpx) < 0,

and hencelim

n→+∞ d(fny, fnpx) = 0.

Then from (12) one can easily obtain

limn→+∞

1n

n−1∑

i=0

δfiy = µ.

For each x ∈ p∆i and any y ∈ W sloc(x), d(f

ny, fnx) ≤ γsi e

−λsi n for

all n ∈ N. Exploiting this fact, as in the uniformly hyperbolic case, we

can prove that each y ∈ W s(x) \+∞⋃n=0

f−nCf is positively regular with the

Lyapunov spectrum {(λ(i)(f, µ),m(i)(f, µ)) : 1 ≤ i ≤ s}.(b) Denote by vB the normalized Lebesgue measure

v

v(B)on a Borel

subsetB ofM with v(B) > 0. Fix a compact set ∆i such that µ(∆i) > 0 andlet qi be the constant specified in Proposition 6. Then we can find a pointx ∈ ∆i and a number q ∈ (0, qi] such that µ(Bx( q

2 )∩p−1U(x, q/2)∩∆i) > 0and µ(p−1∂U(x, q/2)) = 0. Let U = U(x, q) and η be the measurablepartition {

exppx({ws} × Eux (q))

}ws∈Es

x(q)

of U(x, q). Denote by vηy the normalized Lebesgue measure on η(y), y ∈

U(x, q), induced by the inherited Riemannian metric, and let {vUη(y)}y∈U

be the canonical system of conditional measures of vU associated with thepartition η. Then, clearly, the Fubini theorem implies that for v-almost


all y ∈ U , the measure vUη(y) is equivalent to vη

y and there exists a numberC > 1 such that

C−1 ≤dvU

η(y)

dvηy

≤ C (13)

holds vηy -almost everywhere on η(y).

By [26, Proposition 3.2], we can construct a measurable partition ξ ofMf subordinate to Wu-manifolds of (f, µ). Let {µξ(y)}y∈Mf be a canonicalsystem of conditional measures of µ associated with the partition ξ. Then,by the assumption, for µ-almost all y ∈Mf , p(µξ(y)) is actually equivalentto vξ

y, the normalized Lebesgue measure on p(ξ(y)) induced by the inheritedRiemannian metric. Since

µ(Bx

(q2

)∩ p−1U

(x,q

2

)∩ ∆i

)

=∫

Mf

µξ(y)

(Bx

(q2

)∩ p−1U

(x,q

2

)∩ ∆i ∩ ξ(y)

)dµ(y) > 0, (14)

there exists some point y ∈ Bx(q/2) ∩ p−1U(x, q/2) ∩ ∆i satisfying thecondition

µξ(y)

(Bx

(q2

)∩ p−1U

(x,q

2

)∩ ∆i ∩ ξ(y)

)> 0;

p(ξ(y))∩U(x, q/2) contains an open neighborhood of py in the submanifoldtopology of Wy, and p(µξ(y)) is equivalent to vξ

y, and hence

vξy

(∆i(x, q) ∩ p(ξ(y)) ∩ U

(x,q

2

))

≥ vξy

(p(Bx

(q2

)∩ ∆i ∩ ξ(y)

)∩ U(x,

q

2))> 0. (15)

Let

W sloc(p∆i) =

⋃

z∈p∆i

W sloc(z), W s

loc(p∆) =⋃

z∈p∆

W sloc(z).

By Proposition 5, p(ξ(y)) ∩U(x, q/2) and each η(y), y ∈ U , are transversalto the family F∆i

(x, qi). Then by the absolute continuity of F∆i(x, qi)

(Proposition 6), for each z ∈ U ,

vηz (W s

loc(p∆i) ∩ η(z))

≥ vηz

(Pp(ξ(y))∩U(x,q/2),η(z)

(∆i(x, q) ∩ p(ξ(y)) ∩ U(x, q/2)

))> 0. (16)

Relations (13) and (16) imply that for v-almost all z ∈ U ,

vUη(z)(W

sloc(p∆i) ∩ η(z)) > 0.


Therefore,

vU(W s

loc(p∆i) ∩ U)

=∫

U

vUη(z) (W s

loc(p∆i) ∩ η(z)) dvU (z) > 0, (17)

v(W sloc(p∆)) ≥ v(W s

loc(p∆i)) ≥ v(W sloc(p∆i) ∩ U) > 0. (18)

We have proved that

v(W s(p∆)) ≥ v(W sloc(p∆)) > 0

and every y ∈ W s(p∆) is generic with respect to µ. Moreover, we have

shown that every z ∈ W s(p∆) \+∞⋃n=0

f−nCf is positively regular with the

Lyapunov spectrum {(λ(i)(f, µ),m(i)(f, µ)) : 1 ≤ i ≤ s}. If, in addition,v(Cf ) = 0, then one can inductively prove that v(f−nCf ) = 0 for all n ∈ N.Hence

v

(+∞⋃

n=0

f−nCf

)= 0

and

v

(W s(p∆) \

+∞⋃

n=0

f−nCf

)≥ v

(W s

loc(p∆) \+∞⋃

n=0

f−nCf

)> 0.

It has been proved that the desired result holds for ∆ = p∆.

We can extend the result of Theorem 2 to general f -invariant measures.

Proposition 7. Assume that µ is an invariant Borel probability measureof a C2-endomorphism (M,f) satisfying the following conditions:

1. log |det(Txf)| ∈ L1(M,µ);2. µ is an SRB-measure of (M,f);3. the Lyapunov exponents of (f, Tf) are µ-almost everywhere not zero;

moreover, for µ-almost every x ∈M , the smallest Lyapunov exponentλ

(1)x of (f, Tf) at x is less than zero.

Then the following assertions hold :1. up to a set of zero µ-measure, the support of µ is decomposed into a

countable number of f-invariant measurable sets Ak, k ∈ N;2. the normalization of µ on each Ak, µk = µ/µ(Ak) is f-ergodic;3. the basin of attraction of each Ak, W s(Ak) def=

⋃x∈Ak

W s(x), has a

positive Lebesgue measure; for all y ∈W s(Ak), limn→+∞ d(fny,Ak) = 0;

4. for each k and any y ∈W s(Ak),

limn→+∞

1n

n−1∑

i=0

δfiy = µk;


5. for each k, the points of W s(Ak)\+∞⋃n=0

f−nCf are positively regular with

constant Lyapunov exponents, where the set of critical points Cf ={y ∈M | det(Tyf) = 0}. If, in addition, v(Cf ) = 0, then

v(W s(Ak) \

+∞⋃

n=0

f−nCf

)> 0.

Since the ergodic components of an SRB-measure also have the SRB-property [15], by Theorem 2, their supports have disjoint attracting basinswith positive Lebesgue measures. Hence up to a set of zero µ-measure,there are at most countably many ergodic components and the results ofProposition 7 follow immediately.

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(Received April 30 2005, received in revised form February 26 2006)

Authors’ addresses:Da-Quan JiangLMAM, School of Mathematical Sciences,Peking University, Beijing 100871, P.R. ChinaE-mail: [email protected]

Min QianLMAM, School of Mathematical Sciences,Peking University, Beijing 100871, P.R. China