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Posts Tagged ‘generic dynamics’

Dans un travail récemment diffusé sur arxiv, Jana RODRIGUEZ HERTZ montre le théorème suivant:

Théorème. Soit un difféomorphisme f:M\to M d’une variété compacte tridimensionnelle, de classe C^1 et préservant la mesure volume m. Génériquement, f vérifie l’une des deux assertions suivantes:

  1. m-p.p.les trois exposants de Lyapunov sont nuls;
  2. m-p.p. aucun des trois exposants n’est nul. De plus (a) (f,m) est ergodique; (b) f est partiellement hyperbolique, i.e., admet une décomposition dominée TM=E\oplus^{<} F volume hyperbolique séparant les exposants strictement positifs et strictement négatifs.

 

Ce résultat fait partie d’un ensemble initié par le théorème de Bochi (ETDS 2002, annoncé par Mané en 1983) généralisé par Bochi et Viana (Ann. Math. 2005, en version preprint sur arxiv) sous la forme: pour un difféomorphisme générique d’une variété compacte, de classe C^1 et préservant le volume, la décomposition d’Oseledets (définie m-p.p. par les valeurs des exposants) s’étend en une décomposition dominée. En 2009, Avila et Bochi (Trans AMS 2012, en version preprint sur arxiv) avaient montré qu’en toute dimension, on a génériquement soit on a des exposants nuls presque partout, soit il existe un ensemble dense et de mesure non-nulle sans exposants de Lyapunov et sur lequel la dynamique est ergodique.

Avila, Crovisier and Wilkinson ont annoncé la généralisation en toute dimension du théorème de J. Rodriguez Hertz.

Ingrédients de la preuve. Le point principal (par rapport à ce qui est connu) réside en l’affirmation: si l’ensembleE des points ayant trois exposants \lambda_1(x)<\lambda_2(x)=0<\lambda_3(x) n’est pas de mesure nulle alors f est volume hyperbolique, non seulement au-dessus de E (ceci découle des techniques de Bochi et Viana)  mais globalement. On pourra alors conclure en combinant le théorème d’ergodicité de Hertz-Hertz-Urès et la technique perturbative de Bonatti-Barraviera (qui permet de moyenniser l’exposant central).

La preuve de l’affirmation se fait en considérant K, l’ensemble où f est partiellement hyperbolique privé de l’union des classes d’accessibilité ouvertes. Selon la proposition 5.3, les classes d’accessibilité définissent sur K une lamination compacte.

L’ensemble des feuilles compactes de K est encore une lamination d’après Haefliger. Si celle-ci n’est pas vide,  on peut trouver une feuille de bord qui donne un tore périodique sur lequel f est Anosov. Il existe donc deux points homocliniquement reliés, ce qui amène la contradiction dans ce cas. Supposons donc qu’aucune des feuilles de K n’est compacte.

Les composantes connexes de M\setminus K sont périodiques par préservation du volume. On les complète en ajoutant leurs feuilles de bord. On peut les écrire comme une union d’une partie compacte G et d’un fibré en droites F au-dessus de surfaces non-compactes (arbitrairement petites), l’intersection des deux étant une union d’anneaux.

Soit l’ensemble des points qui reviennent une infinité de fois dans G, soit l’ensemble des points restant après N itérations dans I est d’intérieur non-vide. On peut ensuite utiliser le lemme de fermeture d’Anosov pour trouver les deux points homocliniquement reliés et conclure.

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Dans le cadre du groupe de travail sur les cocycles au-dessus des dynamiques hyperboliques, Jiagang YANG a présenté ce 3 février 2012 ses récents travaux exploitant les propriétés de continuité de l’entropie topologique ou mesurée en fonction de la mesure et/ou de la transformation, notamment pour l’étude de la dynamique générique et de faible régularité.

1. Entropie topologique

Notons \mathcal T(M) l’ensemble des difféomorphismes admettant une tangence homocline. Notons B_f(x,\epsilon,n):=\{y:\forall 0\leq k<n\; d(f^ky,f^kx)<\epsilon\} la boule dynamique. Notons h_{top}(f) l’entropie topologique de f.

Théorème (Liao, Viana, Yang). Tout difféomorphisme de {Diff}^1(M)\setminus\overline{\mathcal T(M)} est robustement entropie-expansif: il existe \epsilon>0 et un voisinage \mathcal U\ni f tels que pour tous g\in\mathcal U, on a: h_{top}(g,B_g(x,\epsilon,\infty))=0.

Conséquences.

  1. f\in{Diff}^1(M)\setminus\overline{\mathcal T(M)}\mapsto h_{top}(f) est semi-continu supérieurement (scs).
  2. La conjecture de l’entropie de Shub est vérifiée pour tout f\in{Diff}^1(M)\setminus\overline{\mathcal T(M)}.

2. Entropie mesurée dans le cas conservatif

Théorème (Yang). L’ensemble des points de continuité de f\in{Diff}^1(M)\mapsto h_{vol}(f) est générique.

Corollaire. La formule de Pesin h_{vol}(f)=\int \sum_i \lambda_i(f,x)^+\, dvol s’étend au cas C^1.

Corollaire. L’ensemble des points de continuité de f\mapsto (x\mapsto \lambda_i(f,x)\in L^1(vol) est générique.

3. Anosov topologiquement transitifs

Si f_n\in{Diff}^2(M) $C^1$-convergent vers f\in{Diff}^2(M) alors les mesures SRB des f_n convergent vaguement vers celle de f.

Un difféomorphisme f\in{Diff}^1(M) générique, topologiquement transitif, possède exactement une mesure physique.

4. Transformation plutôt contractantes

M. Andersson a montré que la condition être “plutôt contractant” (mostly contracting) est ouverte dans la topologie C^2. Est-ce encore le cas pour la topologie C^1?

Proposition (Yang). L’ensemble des difféomorphismes de classe C^1 partiellement hyperboliques avec: dimension centrale égale à 1; plutôt contractants; d’exposant centre-instable strictement négatif forment un ouvert C^1.

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We consider a diffeomorphism f:M\to M where M is a compact manifold.

A topological attractor is a compact subset \Lambda\subset M which is (i) invariant: f(\Lambda)=\Lambda; (ii) chain recurrent: for any x,y\in\Lambda, any \epsilon>0, there exists a finite sequence x_0=x,x_1,\dots,x_N=y\in M such that d(f(x_i),x_{i+1})<\epsilon; and (iii) whose basin, \{x\in M:\lim_{n\to\infty} d(f^nx,\Lambda)=0\},  is a neighborhood of \Lambda. This last property can be stated as: it admits a neighborhood U such that \Lambda=\bigcap_{n\geq0} f^n(U).

One might be tempted to think that any (or almost every) diffeomorphism admits at least one attractor. This is the case on surfaces (according to Pujals-Sambarino) and in higher dimension far from homoclinic tangencies (according to Crovisier-Pujals).  However this fails in full generality.

More precisely, in a 2009 preprint, Bonatti, Li and Yang have built, on any compact manifold of dimension at least 3,  a locally C^1-generic diffeomorphism with no topological attractor (i.e., this property holds for all diffeomorphism in a subset containing a dense intersection of countably many open subsets of some fixed open subset of Diff^1(M)).

Rafael Potrie has explained to our work group (on June 21)  his construction of such an example (see the preprint here). His example is derived from Anosov. This  allows a precise control of its dynamics:

  • there is a unique minimal Milnor attractor. More precisely, it is a compact invariant set which is chain recurrent, whose basin has full Lebesgue measure and such that any invariant proper compact subset has a basin of zero Lebesgue measure.
  • each chain recurrent class besides the Milnor attractor is contained in a cycle of center-stable-leaves.

He carefully explained his construction. He starts with a hyperbolic automorphism of the 3-torus with a simple expanding eigenvalue and a pair of conjugate, non-real, contracting eigenvalues. As in Bonatti-Viana’s paper, one modifies it in a ball of some small radius \delta>0 around a fixed point q to get:

  • q becomes a saddle point with one wealky contracting eigenvalues and two expanding ones such that the area of any 2-plane is expanding. Also, its local stable curve has length larger than \delta;
  • there are thin unstable, resp., center-stable cones, invariant under f, resp., f^{-1}, and close to the unstable, resp. stable, spaces of the linear map.

According to Bonatti-Viana, there is a constant L<\infty such that, any center-stable disk of radius bigger than 2\delta and any unstable curve of length at least L intersect.

Recall that a quasi-attractor is a compact subset \Lambda satisfying (i), (ii) and the following weakening version of (iii): it admits a decreasing sequence of neighborhoods U_n such  that \Lambda = \bigcap_{n\geq0} U_n and \overline{f(U_n)} \subset  U_n. An application of Zorn’s Lemma shows that all homeomorphisms of compact sets have quasi-attractors.

Potrie shows that his diffeomorphism has a unique quasi-attractor by observing that any small piece of unstable curve contained in any U_n is eventually mapped into a long curve that must meet the large stable disk of a fixed point r away from p,q. It follows that any quasi-attractor contains r. Hence the uniqueness.

To prove that the quasi-attractor is not an attractor, Potrie observes first that it must contains q and that, generically by a result of Bonatti-Crovisier, it must coincide with the homoclinic class of q. The choice of the eigenvalues of the automorphism implies that the center-stable space of this class has no finer dominated splitting. T_xf|E^{cs}  contracts areas. By Bonatti-Diaz-Pujals, this generically implies that the class is accumulated by sources.

The proof that the Milnor attractor is minimal is done in two steps. For C^2 systems, it follows from the existence of a Sinai-Ruelle-Bowen measure.  For C^1 systems, it follows from the fact that the existence of a minimal Milnor attractor supported by the closure of the unstable set of r is a G_\delta, which is dense by the C^2 result.

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Federico RODRIGUEZ-HERTZ presented at the IHP conference for Katok’s 65th birthday several results pertaining to the ergodicity of systems with some product structure.

Theorem (RHRHTU 2008). Let f be a C^2 diffeomorphism of a compact manifold which preserves volume. Let p be a hyperbolic periodic point. Define \Lambda^u(p):=\{x\text{ backward-Lyapunov regular}: W^u(x)\cap W^s(p) has a transverse point \}. Define \Lambda^s(p) similarly.

If vol(\Lambda^s(p))>0 and vol(\lambda^u(p))>0 then these two sets are equal (up to a zero volume set) and the restriction of f there, is ergodic with no zero exponent.

Corollary. Ergodicity is C^1-open and dense within C^2 partially hyperbolic diffeomorphisms with central dimension at most 2.


Theorem (M.A. Rodriguez-Hertz). A generic 3-dimensional volume-preserving either has all its Lyapunov exponents vanishing Lebesgue-almost everywhere, or has a dominated splitting separating positive and negative Lyapunov exponents and is ergodic with respect to the volume.

Remark. The proof is expected to generalize to symplectomorphisms in arbitrary dimensions.

Remark. This is a stronger but less general result that that of AVILA and BOCHI according to which, in arbitrary dimension, the C^1-generic volume-preserving diffeomorphism either has a dominated splitting and a set of positive volume of non-uniformly hyperbolic points defining an ergodic component of the volume or there is, almost everywhere, at least one zero exponent.

Consider a bundle with fiber a smooth compact manifold and a bundle map acting smoothly on the fibers and preserving on the base a measure.

Theorem. Assume that there are well-defined holonomy maps on the fibers: h^s_{xy}:F_x\to F_y for y\in W^s(x). If the exponents along the fiber vanish, then these holonomy preserve the conditional measures. If the base map is hyperbolic and the base measure has a product structure then the conditional measures depend continuously on the base point.

This extends a classical result of Ledrappier for matrix cocycles. The above has the following application:

Theorem (RHRHTU). Let f:S^1\times\mathbb T^2\to S^1\times\mathbb T^2  be partially hyperbolic with a central foliation into circles. If f has the accessibility property then either f is topologically conjugate to an isometric extension of an Anosov diffeomorphism (and f has a unique maximal entropy measure which has a zero central exponent), or f has at least 2 maximal entropy measures and finitely many of them and their Lyapunov exponents are bounded away from zero.

Question. Is the set of diffeomorphism with finitely many ergodic maximal entropy measures C^r-dense?

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Amie WILKINSON presented new results towards the Pugh-Shub Stable ergodicity conjecture. In particular, with A. AVILA and J. BOCHI, she proved that ergodicity is generic in C^1 partially hyperbolic symplectomorphisms. She noted that, by a result of SAGHIN and Z. XIA, a stably ergodic symplectomorphism is automatically partially hyperbolic (which fails for conservative diffeomorphisms by an example of A. TAHZIBI).

The ingredients of the proof are:

  • Bochi’s alternative: a symplectomorphism which is not Anosov has only zero exponents Lebesgue almost everywhere.
  • A new criterion for ergodicity and saturation for C^2 partially hyperbolic diffeomorphisms  involving a new, non-uniform variant of the center bunching property (used to prove that a set is simultaneously saturated wrt to the stable and unstable foliations)
  • A perturbation technique introduced by M.-C. ARNAUD in her proof of Mañé’s ergodic closing lemma
  • The ergodic  diffeomorphisms of the disk arbitrarily close to the identity, built by ANOSOV and KATOK
  • A Baire argument

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On November 25th, 2009, Sylvain CROVISIER defended his habilitation à diriger des recherches titled Perturbation de la dynamique de difféomorphismes en petite régularité. He first explained basic perturbation techniques:

  • the Anosov-Katok procedure: you use more and more distorted conjugacies such that the limiting dynamics has new properties;
  • the closing lemma of Pugh and the subsequent connecting lemmas of Hayashi and Bonatti-Crovisier: you try to glue pieces of orbits while avoiding intermediate visits to the support of the pertubation. This may force you to drop pieces of the orbits. Crovisier nevertheless managed to prove a generalized shadowing lemma constructing orbits guaranteed to visit some neighborhoods.

He then explained how the above was put to work. Together with François Béguin and Frédéric Le Roux, he used the first tool to realize almost arbitrary measurable dynamics as minimal, uniquely ergodic homeomorphisms on arbitray compact manifolds.

With Christian Bonatti, Enrique Pujals and other co-workers, he used the second set of tools (together with constructions of dominated splittings by Wen, Liao-Wen and his own central models) to get deep new results on the dynamics of C^1 generic diffeomorphisms.

Some of the most important ones state that, after a small C^1 perturbation, any diffeomorphism either has a strong global structure (i.e., a phenomenon in the sense of Pujals) or presents a simple obstruction (a mechanism, which one would like to be robust). Namely, up to C^1 perturbations, any diffeomorphism of a compact manifold is:

  • Morse-Smale unless it has a  transverse homoclinic intersection (this provides a description of an open and dense subset of the C^1 diffeomorphisms called the weak Palis conjecture);
  • partially hyperbolic with a central bundle which is one-dimensional or a sum of two one-dimensional sub-bundles, unless it has a heterodimensional cycle or a homoclinic tangency;
  • essentially hyperbolic (hyperbolic from the point of view of its attractors and repellers), unless it has a heterodimensional cycle or a homoclinic tangency.

In answer to the jury’s questions, Sylvain CROVISIER made several additional and more subjective comments:

  • It is true that Mañé already characterized the non-hyperbolic diffeomorphisms as those having periodic points that can be made to bifurcate but the goal here is to get robust obstructions;
  • It is not clear that dynamics with infinitely many chain recurrent classes can exist robustly: there is no known mechanism for that;
  • It seems very difficult to get beyond C^1 with anything like the current techniques – even C^1+1/log seems out of reach;
  • It is a reasonable to question to try and develop more precise description of the dynamics especially for the situations that hold on C^1 open sets and therefore occur on C^2 open sets, a usual requisite of ergodic theory techniques;
  • The techniques have yielded results very analoguous to Zeeman Tolerance Stability conjecture. However such kinds of stability seem to have more philosophical appeal than a true rôle in the mathematical theory, as opposed to structural stability.
  • It is not clear that the above decomposition of the space of C^1 diffeomorphisms extend to higher smoothness (even disregarding the enormous technical difficulties pertaining to the closing lemmas). For instance Bonatti and Diaz hope to show that any diffeomorphism can be C^1 approximated by a hyperbolic one or by one with a homoclinic tangency but this is not the case in the C^2 topology, because of Newhouse phenomenon.
  • The techniques seem insufficient to study prevalent or Kolmogorov-typical dynamics.

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