Post on 24-Aug-2020
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!