Hyperbolic dynamics Non-hyperbolic systems C1-topology Cycles and tangencies Tame/Wild Map of Diff 1(M)

A panorama of dynamical systems using theC1-topology

Christian Bonatti

CNRS & Université de Bourgogne

IMPA-Rio, 10 de Agosto 2009

Hyperbolic dynamics Non-hyperbolic systems C1-topology Cycles and tangencies Tame/Wild Map of Diff 1(M)

Morse Smale dynamics

ϕ : M → R a Morse function. X =−−→gradϕ. f = X1 : M → M.

Morse-Smale: same picture but equilibria = periodic points.

Hyperbolic dynamics Non-hyperbolic systems C1-topology Cycles and tangencies Tame/Wild Map of Diff 1(M)

Morse Smale dynamics

ϕ : M → R a Morse function. X =−−→gradϕ. f = X1 : M → M.

Morse-Smale: same picture but equilibria = periodic points.

Hyperbolic dynamics Non-hyperbolic systems C1-topology Cycles and tangencies Tame/Wild Map of Diff 1(M)

Simple chaotic dynamic

f : t 7→ 10t on the circle S1 = R/Z.

Hyperbolic dynamics Non-hyperbolic systems C1-topology Cycles and tangencies Tame/Wild Map of Diff 1(M)

Simple chaotic dynamic

f : t 7→ 10t is chaotic:density of periodic pointsdensity of point t with f n(t) = 0 for n large.Generic points→ dense orbit.sensitive dependance to initial conditions

but :structurally stableLebesgue is invariant and ergodic,exponential decay of correlations, etc...g : S1 → S1 expanding +C2 =⇒ g 7→ µg << Lebcontinuous in g.

Hyperbolic dynamics Non-hyperbolic systems C1-topology Cycles and tangencies Tame/Wild Map of Diff 1(M)

Simple chaotic dynamic

f : t 7→ 10t is chaotic:density of periodic pointsdensity of point t with f n(t) = 0 for n large.Generic points→ dense orbit.sensitive dependance to initial conditions

but :structurally stableLebesgue is invariant and ergodic,exponential decay of correlations, etc...g : S1 → S1 expanding +C2 =⇒ g 7→ µg << Lebcontinuous in g.

Hyperbolic dynamics Non-hyperbolic systems C1-topology Cycles and tangencies Tame/Wild Map of Diff 1(M)

Simple chaotic dynamic

f : t 7→ 10t is chaotic:density of periodic pointsdensity of point t with f n(t) = 0 for n large.Generic points→ dense orbit.sensitive dependance to initial conditions

but :structurally stableLebesgue is invariant and ergodic,exponential decay of correlations, etc...g : S1 → S1 expanding +C2 =⇒ g 7→ µg << Lebcontinuous in g.

Hyperbolic dynamics Non-hyperbolic systems C1-topology Cycles and tangencies Tame/Wild Map of Diff 1(M)

Hyperbolic sets: Smale solenoid

f : S1 × D2 → S1 × D2 (t , z) 7→ (2t , z10 + e2iπt).

A hyperbolic attractor.

Hyperbolic dynamics Non-hyperbolic systems C1-topology Cycles and tangencies Tame/Wild Map of Diff 1(M)

Hyperbolic sets: Smale horseshoe

An saddle-like hyperbolic set

Hyperbolic dynamics Non-hyperbolic systems C1-topology Cycles and tangencies Tame/Wild Map of Diff 1(M)

Hyperbolic diffeomorphisms

f : M → M

Axiom A + no cycle: same picture as Morse-Smale butEquilibria= hyperbolic sets

Hyperbolic dynamics Non-hyperbolic systems C1-topology Cycles and tangencies Tame/Wild Map of Diff 1(M)

Axiom A + no cycle:⇐⇒ structurally stable+ C2 =⇒ good probabilistic description(SinaiRuelleBowen measures)

Hyperbolic dynamics Non-hyperbolic systems C1-topology Cycles and tangencies Tame/Wild Map of Diff 1(M)

End of Smale’s dream

Diff 1(M) \ {Hyperbolic} 6= ∅ (Abraham Smale 1968, Simon)→ robustly non-hyperbolic systems→ robustly unstable

Hyperbolic dynamics Non-hyperbolic systems C1-topology Cycles and tangencies Tame/Wild Map of Diff 1(M)

Local phenomena/global structure

Recent results on Diff 1(M) for the C1-topology −→New dream of a global view of Diff 1(M):

Spliting Diff 1(M) using dichotomies:U+

i and U−i open subsets of Diff 1(M);U+

i ∩ U−i = ∅

U+i ∪ U

−i = Diff 1(M)

f ∈ U+i ⇐⇒ f presents a robust local phenomenon Pi

f ∈ U−i ⇐⇒ f admits a robust global structure forbidding Pi

Description of the dynamics in the open regions⋂

i Uεii .

Hyperbolic dynamics Non-hyperbolic systems C1-topology Cycles and tangencies Tame/Wild Map of Diff 1(M)

Local phenomena/global structure

Recent results on Diff 1(M) for the C1-topology −→New dream of a global view of Diff 1(M):

Spliting Diff 1(M) using dichotomies:U+

i and U−i open subsets of Diff 1(M);U+

i ∩ U−i = ∅

U+i ∪ U

−i = Diff 1(M)

f ∈ U+i ⇐⇒ f presents a robust local phenomenon Pi

f ∈ U−i ⇐⇒ f admits a robust global structure forbidding Pi

Description of the dynamics in the open regions⋂

i Uεii .

Hyperbolic dynamics Non-hyperbolic systems C1-topology Cycles and tangencies Tame/Wild Map of Diff 1(M)

Characterization of chaotic dynamics

Theorem [Pujals-Sambarino 2000, B-Gan-Wen 2006,Crovisier]

{Morse-Smale} ∪ { Horseshoes} = Diff 1(M)

Hyperbolic dynamics Non-hyperbolic systems C1-topology Cycles and tangencies Tame/Wild Map of Diff 1(M)

C1-Perturbations lemmas

Closing and connecting lemmas (Pugh 68,Mañé,Hayashi98,B-Crovisier 04) :

Hyperbolic dynamics Non-hyperbolic systems C1-topology Cycles and tangencies Tame/Wild Map of Diff 1(M)

Global dynamics/periodic orbits

For C1-generic diffeomorphisms:

The chain recurrent set is the closure of the periodic points[Pugh 1968][B-Crovisier 2004]Each chain transitive set is the Hausdorff limit of periodicorbits [Crovisier 2006]A chain recurrence class containing a periodic point p isthe homoclinic class H(p).[B-Crovisier]Every ergodic measure is the weak and Hausdorff limit ofperiodic measure [Mañé].

Hyperbolic dynamics Non-hyperbolic systems C1-topology Cycles and tangencies Tame/Wild Map of Diff 1(M)

Global dynamics/periodic orbits

For C1-generic diffeomorphisms:

The chain recurrent set is the closure of the periodic points[Pugh 1968][B-Crovisier 2004]Each chain transitive set is the Hausdorff limit of periodicorbits [Crovisier 2006]A chain recurrence class containing a periodic point p isthe homoclinic class H(p).[B-Crovisier]Every ergodic measure is the weak and Hausdorff limit ofperiodic measure [Mañé].

Hyperbolic dynamics Non-hyperbolic systems C1-topology Cycles and tangencies Tame/Wild Map of Diff 1(M)

Global dynamics/periodic orbits

For C1-generic diffeomorphisms:

The chain recurrent set is the closure of the periodic points[Pugh 1968][B-Crovisier 2004]Each chain transitive set is the Hausdorff limit of periodicorbits [Crovisier 2006]A chain recurrence class containing a periodic point p isthe homoclinic class H(p).[B-Crovisier]Every ergodic measure is the weak and Hausdorff limit ofperiodic measure [Mañé].

Hyperbolic dynamics Non-hyperbolic systems C1-topology Cycles and tangencies Tame/Wild Map of Diff 1(M)

Global dynamics/periodic orbits

For C1-generic diffeomorphisms:

The chain recurrent set is the closure of the periodic points[Pugh 1968][B-Crovisier 2004]Each chain transitive set is the Hausdorff limit of periodicorbits [Crovisier 2006]A chain recurrence class containing a periodic point p isthe homoclinic class H(p).[B-Crovisier]Every ergodic measure is the weak and Hausdorff limit ofperiodic measure [Mañé].

Hyperbolic dynamics Non-hyperbolic systems C1-topology Cycles and tangencies Tame/Wild Map of Diff 1(M)

Global dynamics/periodic orbits

For C1-generic diffeomorphisms:

The chain recurrent set is the closure of the periodic points[Pugh 1968][B-Crovisier 2004]Each chain transitive set is the Hausdorff limit of periodicorbits [Crovisier 2006]A chain recurrence class containing a periodic point p isthe homoclinic class H(p).[B-Crovisier]Every ergodic measure is the weak and Hausdorff limit ofperiodic measure [Mañé].

Hyperbolic dynamics Non-hyperbolic systems C1-topology Cycles and tangencies Tame/Wild Map of Diff 1(M)

Behind our philosophy: a conjecture

Definitionp periodic, hyperbolic . The chain recurrence class C(p) is ro-bustly non hyperbolic if C(pg) is not hyperbolic ∀g close to f .

Conjecture 1

Hyp ∪ {Robustly nonhyp classC(Pf )} = Diff 1(M)

If wrong:C1-open setWhyp. For C1-generic f ∈ Whyp:

every homoclinic class H(p) is a hyperbolic basic setbut ∃ H(pn)→ C aperiodic class

Hyperbolic dynamics Non-hyperbolic systems C1-topology Cycles and tangencies Tame/Wild Map of Diff 1(M)

Behind our philosophy: a conjecture

Definitionp periodic, hyperbolic . The chain recurrence class C(p) is ro-bustly non hyperbolic if C(pg) is not hyperbolic ∀g close to f .

Conjecture 1

Hyp ∪ {Robustly nonhyp classC(Pf )} = Diff 1(M)

If wrong:C1-open setWhyp. For C1-generic f ∈ Whyp:

every homoclinic class H(p) is a hyperbolic basic setbut ∃ H(pn)→ C aperiodic class

Hyperbolic dynamics Non-hyperbolic systems C1-topology Cycles and tangencies Tame/Wild Map of Diff 1(M)

Behind our philosophy: a conjecture

Definitionp periodic, hyperbolic . The chain recurrence class C(p) is ro-bustly non hyperbolic if C(pg) is not hyperbolic ∀g close to f .

Conjecture 1

Hyp ∪ {Robustly nonhyp classC(Pf )} = Diff 1(M)

If wrong:C1-open setWhyp. For C1-generic f ∈ Whyp:

every homoclinic class H(p) is a hyperbolic basic setbut ∃ H(pn)→ C aperiodic class

Hyperbolic dynamics Non-hyperbolic systems C1-topology Cycles and tangencies Tame/Wild Map of Diff 1(M)

Heterodimensional cycles and homoclinic tangencies

Fragil (non-robust) phenomena

Hyperbolic dynamics Non-hyperbolic systems C1-topology Cycles and tangencies Tame/Wild Map of Diff 1(M)

f ∈ Diff 1(M) \ Hyp(M)C1 − perturbation

−−−−−−−−−−−−−−−−→weak hyperbolic periodic orbits:

1 small Lyapunov exponents2 weak domination Es(p)⊕< Eu(p)

[PS][W][G]−−−−−−−−−−−→

homoclinic tangency

weak points:C1 − perturbation

−−−−−−−−−−−−−−−−→close points with different indices

C1 − perturbation ???−−−−−−−−−−−−−−−−−−−−→

heterodimensional cycles.

Hyperbolic dynamics Non-hyperbolic systems C1-topology Cycles and tangencies Tame/Wild Map of Diff 1(M)

f ∈ Diff 1(M) \ Hyp(M)C1 − perturbation

−−−−−−−−−−−−−−−−→weak hyperbolic periodic orbits:

1 small Lyapunov exponents2 weak domination Es(p)⊕< Eu(p)

[PS][W][G]−−−−−−−−−−−→

homoclinic tangency

weak points:C1 − perturbation

−−−−−−−−−−−−−−−−→close points with different indices

C1 − perturbation ???−−−−−−−−−−−−−−−−−−−−→

heterodimensional cycles.

Hyperbolic dynamics Non-hyperbolic systems C1-topology Cycles and tangencies Tame/Wild Map of Diff 1(M)

f ∈ Diff 1(M) \ Hyp(M)C1 − perturbation

−−−−−−−−−−−−−−−−→weak hyperbolic periodic orbits:

1 small Lyapunov exponents2 weak domination Es(p)⊕< Eu(p)

[PS][W][G]−−−−−−−−−−−→

homoclinic tangency

weak points:C1 − perturbation

−−−−−−−−−−−−−−−−→close points with different indices

C1 − perturbation ???−−−−−−−−−−−−−−−−−−−−→

heterodimensional cycles.

Hyperbolic dynamics Non-hyperbolic systems C1-topology Cycles and tangencies Tame/Wild Map of Diff 1(M)

f ∈ Diff 1(M) \ Hyp(M)C1 − perturbation

−−−−−−−−−−−−−−−−→weak hyperbolic periodic orbits:

1 small Lyapunov exponents2 weak domination Es(p)⊕< Eu(p)

[PS][W][G]−−−−−−−−−−−→

homoclinic tangency

weak points:C1 − perturbation

−−−−−−−−−−−−−−−−→close points with different indices

C1 − perturbation ???−−−−−−−−−−−−−−−−−−−−→

heterodimensional cycles.

Hyperbolic dynamics Non-hyperbolic systems C1-topology Cycles and tangencies Tame/Wild Map of Diff 1(M)

Robust Tangency

Hyperbolic dynamics Non-hyperbolic systems C1-topology Cycles and tangencies Tame/Wild Map of Diff 1(M)

Robust cycle

Hyperbolic dynamics Non-hyperbolic systems C1-topology Cycles and tangencies Tame/Wild Map of Diff 1(M)

Conjecture 2

Local C1-density of hetero. cycles =⇒ Robust cycle

This conjecture is essentially proved [B-Diaz]

Conjecture 3

Local C1-density of tangencies =⇒ Robust tangency

Proved by [B-Diaz] when the tangency appears on a periodicpoint in a heterodimensional cycle.

Hyperbolic dynamics Non-hyperbolic systems C1-topology Cycles and tangencies Tame/Wild Map of Diff 1(M)

Conjecture 2

Local C1-density of hetero. cycles =⇒ Robust cycle

This conjecture is essentially proved [B-Diaz]

Conjecture 3

Local C1-density of tangencies =⇒ Robust tangency

Proved by [B-Diaz] when the tangency appears on a periodicpoint in a heterodimensional cycle.

Hyperbolic dynamics Non-hyperbolic systems C1-topology Cycles and tangencies Tame/Wild Map of Diff 1(M)

Two first dichotomies

Hyperbolic dynamics Non-hyperbolic systems C1-topology Cycles and tangencies Tame/Wild Map of Diff 1(M)

Third dichotomy

Theorem [Abdenur,B-Crovisier]

For C1-generic f ,C isolated class C ⇐⇒ C is robustly isolated.

Definitionf tame if every class C is robustly isolated⇔ the number of class of f is finite an locally constant.

T (M) = {tame f} is an open set containing Hyp(M).

W(M) = Diff 1(M) \ T (M).C1-generic f ∈ W(M) have infinitely many classes.

Hyperbolic dynamics Non-hyperbolic systems C1-topology Cycles and tangencies Tame/Wild Map of Diff 1(M)

Third dichotomy

Theorem [Abdenur,B-Crovisier]

For C1-generic f ,C isolated class C ⇐⇒ C is robustly isolated.

Definitionf tame if every class C is robustly isolated⇔ the number of class of f is finite an locally constant.

T (M) = {tame f} is an open set containing Hyp(M).

W(M) = Diff 1(M) \ T (M).C1-generic f ∈ W(M) have infinitely many classes.

Hyperbolic dynamics Non-hyperbolic systems C1-topology Cycles and tangencies Tame/Wild Map of Diff 1(M)

Third dichotomy

Theorem [Abdenur,B-Crovisier]

For C1-generic f ,C isolated class C ⇐⇒ C is robustly isolated.

Definitionf tame if every class C is robustly isolated⇔ the number of class of f is finite an locally constant.

T (M) = {tame f} is an open set containing Hyp(M).

W(M) = Diff 1(M) \ T (M).C1-generic f ∈ W(M) have infinitely many classes.

Hyperbolic dynamics Non-hyperbolic systems C1-topology Cycles and tangencies Tame/Wild Map of Diff 1(M)

Third dichotomy

Theorem [Abdenur,B-Crovisier]

For C1-generic f ,C isolated class C ⇐⇒ C is robustly isolated.

Definitionf tame if every class C is robustly isolated⇔ the number of class of f is finite an locally constant.

T (M) = {tame f} is an open set containing Hyp(M).

W(M) = Diff 1(M) \ T (M).C1-generic f ∈ W(M) have infinitely many classes.

Hyperbolic dynamics Non-hyperbolic systems C1-topology Cycles and tangencies Tame/Wild Map of Diff 1(M)

Third dichotomy

Theorem [Abdenur,B-Crovisier]

For C1-generic f ,C isolated class C ⇐⇒ C is robustly isolated.

Definitionf tame if every class C is robustly isolated⇔ the number of class of f is finite an locally constant.

T (M) = {tame f} is an open set containing Hyp(M).

W(M) = Diff 1(M) \ T (M).C1-generic f ∈ W(M) have infinitely many classes.

Hyperbolic dynamics Non-hyperbolic systems C1-topology Cycles and tangencies Tame/Wild Map of Diff 1(M)

Map of Diff 1(M)

Hyperbolic dynamics Non-hyperbolic systems C1-topology Cycles and tangencies Tame/Wild Map of Diff 1(M)

Map of Diff 1(M)

Hyperbolic dynamics Non-hyperbolic systems C1-topology Cycles and tangencies Tame/Wild Map of Diff 1(M)

Conjecture (Palis)

Hyp ∪ {Tangencies} ∪ {Cycles} = Diff 1(M)

Proved in dimension 2 [PujalsSambarino(2000)];proved for attrator/repellers in any dimension [Crovisier Pujals].Conjecture 4Robust tangency =⇒ Robust cycles

Announced in dimension 2 (Moreira), progresses in anydimensions.Conjecture 5Far from tangencies =⇒ Tame

Progresses (J. Yang, B-Gan-Li-D. Yang, Crovisier D. Yang).

Hyperbolic dynamics Non-hyperbolic systems C1-topology Cycles and tangencies Tame/Wild Map of Diff 1(M)

Conjecture (Palis)

Hyp ∪ {Tangencies} ∪ {Cycles} = Diff 1(M)

Proved in dimension 2 [PujalsSambarino(2000)];proved for attrator/repellers in any dimension [Crovisier Pujals].Conjecture 4Robust tangency =⇒ Robust cycles

Announced in dimension 2 (Moreira), progresses in anydimensions.Conjecture 5Far from tangencies =⇒ Tame

Progresses (J. Yang, B-Gan-Li-D. Yang, Crovisier D. Yang).

Hyperbolic dynamics Non-hyperbolic systems C1-topology Cycles and tangencies Tame/Wild Map of Diff 1(M)

Conjecture (Palis)

Hyp ∪ {Tangencies} ∪ {Cycles} = Diff 1(M)

Proved in dimension 2 [PujalsSambarino(2000)];proved for attrator/repellers in any dimension [Crovisier Pujals].Conjecture 4Robust tangency =⇒ Robust cycles

Announced in dimension 2 (Moreira), progresses in anydimensions.Conjecture 5Far from tangencies =⇒ Tame

Progresses (J. Yang, B-Gan-Li-D. Yang, Crovisier D. Yang).

Hyperbolic dynamics Non-hyperbolic systems C1-topology Cycles and tangencies Tame/Wild Map of Diff 1(M)

Conjecture (Palis)

Hyp ∪ {Tangencies} ∪ {Cycles} = Diff 1(M)

Proved in dimension 2 [PujalsSambarino(2000)];proved for attrator/repellers in any dimension [Crovisier Pujals].Conjecture 4Robust tangency =⇒ Robust cycles

Announced in dimension 2 (Moreira), progresses in anydimensions.Conjecture 5Far from tangencies =⇒ Tame

Progresses (J. Yang, B-Gan-Li-D. Yang, Crovisier D. Yang).

Hyperbolic dynamics Non-hyperbolic systems C1-topology Cycles and tangencies Tame/Wild Map of Diff 1(M)

Conjecture (Palis)

Hyp ∪ {Tangencies} ∪ {Cycles} = Diff 1(M)

Proved in dimension 2 [PujalsSambarino(2000)];proved for attrator/repellers in any dimension [Crovisier Pujals].Conjecture 4Robust tangency =⇒ Robust cycles

Announced in dimension 2 (Moreira), progresses in anydimensions.Conjecture 5Far from tangencies =⇒ Tame

Progresses (J. Yang, B-Gan-Li-D. Yang, Crovisier D. Yang).

Hyperbolic dynamics Non-hyperbolic systems C1-topology Cycles and tangencies Tame/Wild Map of Diff 1(M)

Conjectural map of Diff 1(M)

Hyperbolic dynamics Non-hyperbolic systems C1-topology Cycles and tangencies Tame/Wild Map of Diff 1(M)

Tame: globale structures

Theoremf tame far from tangencies =⇒:each class has a partially hyperbolic splitting

Ess ⊕< Ec1 ⊕< · · · ⊕< Ec

k ⊕< Euu

with dim Eci = 1.

Conjecture 6A geometric criterium charaterizing the isolated classes usingblenders (big hyperbolic sets)

Hyperbolic dynamics Non-hyperbolic systems C1-topology Cycles and tangencies Tame/Wild Map of Diff 1(M)

Tame: globale structures

Theoremf tame far from tangencies =⇒:each class has a partially hyperbolic splitting

Ess ⊕< Ec1 ⊕< · · · ⊕< Ec

k ⊕< Euu

with dim Eci = 1.

Conjecture 6A geometric criterium charaterizing the isolated classes usingblenders (big hyperbolic sets)

Hyperbolic dynamics Non-hyperbolic systems C1-topology Cycles and tangencies Tame/Wild Map of Diff 1(M)

Tame: globale structures

Theorem [BDP]f tame far from tangencies =⇒:each class has a dominated splitting

Ecs ⊕< Ecu

No idea of characterization.

Hyperbolic dynamics Non-hyperbolic systems C1-topology Cycles and tangencies Tame/Wild Map of Diff 1(M)

Wild: local phenomenon

DefinitionA property P of a homoclinic class H(pf ) is self replicating (orviral) if

the property P is robust andC1-small perturbation creates new homoclinic orbits H(qg),separated from H(pg) by a filtration, and satisfying P.

Conjecture 7

f inW(M)⇐⇒ f has as viral class

Conjecture 7 =⇒ uncountably many classes for C1-genericf ∈ W(M)

Hyperbolic dynamics Non-hyperbolic systems C1-topology Cycles and tangencies Tame/Wild Map of Diff 1(M)

Wild: local phenomenon

DefinitionA property P of a homoclinic class H(pf ) is self replicating (orviral) if

the property P is robust andC1-small perturbation creates new homoclinic orbits H(qg),separated from H(pg) by a filtration, and satisfying P.

Conjecture 7

f inW(M)⇐⇒ f has as viral class

Conjecture 7 =⇒ uncountably many classes for C1-genericf ∈ W(M)

Hyperbolic dynamics Non-hyperbolic systems C1-topology Cycles and tangencies Tame/Wild Map of Diff 1(M)

Conjectural map of Diff 1(M)

Hyperbolic dynamics Non-hyperbolic systems C1-topology Cycles and tangencies Tame/Wild Map of Diff 1(M)

Conjectural map of Diff 1(M)

Hyperbolic dynamics Non-hyperbolic systems C1-topology Cycles and tangencies Tame/Wild Map of Diff 1(M)

Muito obrigado a todos,

e viva a cooperação franco brasileira!

Hyperbolic dynamics Non-hyperbolic systems C1-topology Cycles and tangencies Tame/Wild Map of Diff 1(M)

Muito obrigado a todos,

e viva a cooperação franco brasileira!