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## Entropie des difféomorphismes sans décomposition dominée

Lors d’une journée autour de la soutenance de la thèse de Jordan EMME, j’ai présenté les résultats obtenus avec Sylvain CROVISIER et Todd FISHER sur l’entropie des difféomorphismes sans domination en régularité C^1.

J’ai expliqué différentes questions sur l’entropie topologique et notamment le problème de (non)densité des difféomorphismes “stables pour l’entropie” (ie, dont l’entropie topologique est localement constante) et les réponses apportées par nos résultats basés sur un renforcement de résultats classiques de Newhouse et plus récemment de Bonatti, Catalan, Tahzibi et Gourmelon et d’autres.

Voici mes transparents et la prépublication  sur arxiv.

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## Dichotomie: exposants tous nuls / hyperbolicité et ergodicité

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’ensemble $E$ 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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## Rafael Potrie: Locally generic Diffeomorphisms With No Attractor

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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## Ergodicity of partially hyperbolic symplectomorphisms

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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## Sylvain CROVISIER’s HDR

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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## Stable ergodicity and partial hyperbolicity

Ya. Pesin gave a talk (Chevaleret, June 19, 2009) summarizing the work on Push-Shub Stable Ergodicity Conjecture and presenting some new results of his in the non-conservative case.

This conjecture states that there is a $C^1$-dense and open set of ergodic diffeomorphisms among those which are partially hyperbolic and volume preserving. Note the tension between perturbative techniques which are mainly known in the $C^1$ setting and smooth ergodic theory in the $C^2$ setting.

One tries to establish accessibility, i.e., that it is possible to link any two points following alternatively stable and unstable manifolds. This is now known to hold:

• $C^1$-densely according to Dolgopyat and Wilkinson;
• if the central dimension is 1, $C^r$-densely according to Didier ( $r=2$) and Rodriguez-Hertz, Roderiguez-Hertz and Ures.

Ergodicity then follows under $C^2$ smoothness and center-bunching (i.e., near conformality of the central bundle) according to Burns and Wilkinson and Rodriguez-Hertz, Roderiguez-Hertz and Ures.

The talk then turned to the non-conservative case, first recalling the result of Burns, Dolgopyat and Pesin establishing stable ergodicity provided: (i) all the exponents are strictly negative on a set of positive volume; (ii) $C^2$ smoothness; (iii) essential accessibility.

More precisely, $f$ is a partially hyperbolic, $C^2$ diffeomorphism of a compact manifold with a topological attractor $\Lambda$. The role of the volume (no longer preserved) is played by u-measures.

They have been built “from the outside” by pushing the volume by Pesin-Sinai and Bonatti-Diaz-Viana. They have been characterized “from the inside”, by pushing the leaf volume by Pesin-Sinai. According to Bonatti-Diaz-Viana, any physical measure is a u-measure. According to Dolgopyat, a unique u-measure is a physical measure.

It is not difficult to see that uniqueness of the u-measure may fail even for topologically transitive attractor.

Pesin stated three theorems:

Theorem A.

1. There exists $\mu$, a u-measure as soon as the Lyapunov exponents along the central bundle are all strictly negative and all unstable leaves are dense in the attractor.
2. if $\mu$ is unique as a u-measure, it is unique as an S.R.B. measure.
3. $\mu$ is Bernoulli.

This is related to results of Alvès-Bonatti-Viana. It applies to some examples with central dimension 1 (otherwise the sum of the central exponents is negative and this is a problem).

Theorem B.

In the setting of Theorem A, any $C^2$-diffeomorphism $g$ which is $C^1$-close to $f$:

1. automatically satisfies the negativity of its central exponents;
2. has a unique u-measure $\mu$ (which is SRB);
3. this $\mu$ has full basin and is Bernoulli.

It is interesting to note that this theorem fails if one instead assumes that the central exponents are strictly positive. The authors do not claim that all unstable leaves are dense.

Theorem C.

Let $f$ be a $C^2$-diffeomorphism with a partially hyperbolic attractor $\Lambda_f$. Assume (i) there is a unique u-measure with strictly positive central exponents $\mu$-almost everywhere; (ii) all unstable leaves are dense in $\Lambda_f$. Then $f$ is stably ergodic.

This uses the result by Dolgopyat that the limit of a sequence of u-measures for a sequence of partially hyperbolic diffeomorphisms is itself a u-measure for the limit diffeomorphism (when they both exist). It also uses Pliss Lemma to get mesoscopic local central disk.

It is noted that they are few results in the presence of critical points (see Viana’s multidimensional attractors, however).

It is also observed that integrability is known to be breakable by a $C^1$-perturbation (recall the result of Dolgopyat-Wilkinson). This is unknown in the $C^2$-topology, except for central dimension 1 where a result of JP Marco exists.

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