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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.

Une courte preuve du théorème d’Ornstein par Downarowicz et Serafin

Le théorème d’Ornstein (1970) est un des sommets de la théorie ergodique. C’est l’aboutissement des recherches initiées par Kolmogorov sur la classification des schémas de Bernoulli et le point de départ de résultats très généraux. Downarowicz et Serafin ont publié une élégante preuve de ce théorème difficile.

Rappelons l’énoncé du théorème (dans sa version la plus simple). Un décalage de Bernoulli $\Sigma$ est un système dynamique probabiliste constitué de l’espace $\{0,1,\dots,N\}^{\mathbb Z}$ muni de sa tribu borélienne et d’une mesure produit $\mu_{p_0,p_1,\dots} \quad (p_n\geq0,\;\sum p_n=1)$ et du décalage $\sigma:(s_n)_{n\in\mathbb Z}\mapsto (s_{n+1})_{n\in\mathbb Z}$. Un isomorphisme entre systèmes dynamiques probabilistes $(X_i,\mathcal B_i,\mu_i,\sigma_i),\; i=1,2$ est une bijection $\psi:X_1\to X_2$ vérifiant $\psi^{-1}(\mathcal B_2)=\mathcal B_1$ et $\psi\circ\sigma_1=\sigma_2\circ\psi$.

Théorème (Ornstein 1970).  Deux décalages de Bernoulli $\Sigma^{(i)},\; i=1,2$ sont isomorphes si et seulement s’ils ont la même entropie: $H(p^{(1)}):=-\sum_n p_n^{(1)}\log p_n^{(1)} = H(p^{(2)})$ ($p^{(1)},p^{(2)}$ sont les vecteurs de probabilité définissant les deux décalages; $0\log 0 = 0$ par convention).

La preuve de Downarowicz et Serafin utilise (comme d’autres approches avant eux) un argument de Baire sur les couplages.  L’ensemble des couplages est une partie fermée du compact des mesures boréliennes invariantes et de probabilité pour le produit (dans la topologie * faible – $N<\infty$). L’ensemble $\mathcal E$ des couplages ergodiques est un $G_\delta$, donc encore un espace de Baire. L’énoncé principal de Downarowicz et Serafin est:

Théorème.  Soit $\Sigma^{(i)},\; i=1,2$ deux décalages de Bernoulli. Si $H(p^{(1)}) = H(p^{(2)})$ alors l’ensemble des facteurs $\Sigma^{(1)}\to\Sigma^{(2)}$ est une partie générique de $\mathcal E$.

Le théoreme d’Ornstein s’en déduit immédiatement, l’intersection de deux parties génériques d’un espace de Baire étant générique et donc non-vide.

La partie générique du théorème est obtenue comme l’intersection des ouverts $\mathcal F_\epsilon$, l’ensemble des $\epsilon$-facteurs $\Sigma^{(1)}\to\Sigma^{(2)}$:  les couplages $\xi$ tels que pour tout $b\in\{0,1,\dots,N\}$, il existe un borélien $B$ tel que la différence symétrique entre $\Sigma^{(1)}\times\{b\}$ et $B\times \Sigma^{(2)}$ soit de $\xi$-mesure strictement inférieure à $\epsilon$.

Les $\mathcal F_\epsilon,\;\epsilon>0$ étant manifestement ouverts et emboîtés, il suffit de montrer que tout couplage ergodique $\xi$ peut être approché par un élément de $\mathcal F_\epsilon$. Cette approximation se fait en deux étapes.

Etape 1. Construction d’un facteur $\mu\to\nu'$ pour $h(\nu')\sim h(\nu)$ ($\mu:=\mu^{(1)},\nu:=\mu^{(2)}$)

Etape 2. Modification du facteur en un élément de $\mathcal F_\epsilon$.

K. McGoff: Random Subshifts of Finite Type

It is often interesting to study the properties of objects picked at random in some class. This may shed some light on whether some observed system has is typical in some probabilistic model or may even turn out to be an efficient way of  obtaining an interesting behavior. This idea was put to spectacular use by Erdös in combinatorics (see Random graphs, Bollobas?), and closer to us, by Gromov (and later Yann Ollivier) for finitely presented groups.

In a very recent preprint, K. McGoff has considered random subshift of finite type (see An introduction to symbolic dynamics by Marcus and Lind). A subshift of finite type is a compact set of the form

$X:=\{x\in\{1,\dots,N\}^{\mathbb N}: \forall p\in\mathbb Z\; x_px_{p+1}\dots x_{p+L-1}\in\mathcal A\}$

where $N\in\mathbb N^*$ and $\mathcal A\subset\{1,\dots,N\}^L$ for some $L$, the finite sequences being considered up to translation of the indices. The self-map $\sigma: (x_p)_{p\in\mathbb Z}\mapsto (x_{p+1})_{p\in\mathbb Z}$ (called the shift) turns $X$ into a dynamical system.

Note that the following inclusion is usually strict: $\mathcal B(X,L):=\{x_p\dots x_{p+L-1}:x\in X,\; p\in\mathbb Z\} \subset \mathcal F$.

Given $\mathcal F$, some of the most natural questions are:

1. Is $X$ empty?
2. What is the entropy of $X$? This quantifies how big $X$ is. It is the positive number $h(X):=\lim_{n\to\infty}\frac1n\log \#X(n)$ where $X(n)$ is the set of all the finite sequences of length $n$ which occurs as a subsequence of any $x\in X$ and $\#X(n)$ is the cardinality of that set.
3. How many irreducible components does $X$ have? An irreducible component  is the closure of an orbit, i.e., $\{\sigma^n(x):n\in\mathbb Z\}$, which is not included in a larger one.

Let us stress that these questions are well-known to have an algorithmic answer. If the maximal length of sequences appearing in $\mathcal F$ is $L$, one defines a $N^{L-1}\times N^{L-1}$-matrix with 0,1-entries. The largest positive eigenvalue is the exponential of the entropy (or zero if and only if $X=\emptyset$) and its multiplicity is the number of irreducible components. But of course $N^{L-1}$ is very large if $L$ is large.

K. McGoff considers the following random model with parameter $0\leq \alpha\leq 1$ and a subshift of finite type $X$. Given $L$ a very large integer, $\mathcal F\subset\mathcal B(X,L)$ is defined by declaring that each sequence of length belongs to $\mathcal F$ independently and with probability $1-\alpha$. He then considers the limit of the probabilities when $L\to\infty$. Let us quote his main results in a somewhat weakened form and specialized to $X=\{1,\dots,N\}^{\mathbb Z}$ for simplicity.

There is a dichotomy between small $\alpha$ (empty shift with positive limit probability,  zero entropy with full limit probability, non-trivial distribution of the number of components) and large $\alpha$ (non-empty, entropy close to $\log(\alpha\lambda)$ and a single, aperiodic irreducible component with full limit probability).

The above results for small $\alpha$ are shown for all $\alpha<1/N$. For large $\alpha$, the non-emptyness holds for $\alpha\geq 1/N$, the convergence of the entropy for $\alpha>1/N$ and the irreducibility and aperiodicity for $\alpha$ close enough to 1.

C^2 surface diffeomorphisms always have a symbolic extension

Most of topological dynamics studies systems of the form $T:X\to X$ where $T$ is a continuous self-map and $X$ is a compact metric space. One approach is to “reduce” such systems to symbolic dynamical system, i.e., $\sigma:S\to X$ where $S$ is a closed subset of $\{1,\dots,d\}^{\mathbb Z}$ and $\sigma((x_n)_{n\in\mathbb Z})=(x_{n+1})_{n\in\mathbb Z}$ such that $\sigma(S)=S$.

J. Auslander asked about the obstructions for a topological system $T:X\to X$ to have a symbolic extension, i.e., a symbolic system $\sigma:S\to S$ and a continuous surjection $\pi:S\to X$ commuting with the dynamics: $\pi\circ\sigma =T\circ\pi$. There is an obvious one: a symbolic system (and therefore its topological factors) has finite topological entropy. Is there any other?

M. Boyle showed that this was indeed the case. With D. Fiebig and U. Fiebig, he showed that asymptotically h-expansive systems (including $C^\infty$ self-maps of compact manifolds by a result of mine based on Yomdin’s theory) always have a “nice” symbolic extension. T. Downarowicz and S. Newhouse showed that generic $C^1$ map have no symbolic extension whatever, leaving open the question of diffeomorphisms with finite smoothness.

T. Downarowicz and A. Maas showed that $C^r$ interval maps also always have symbolic extensions for $1.

David BURGUET has finally proved the same for arbitrary $C^2$ surface diffeomorphisms, see his preprint here.

Behind these works there is a rich and beautiful topological/ergodic/functional-analytical theory of entropy (called the entropy structure by T. Downarowicz) which does yet have the audience it deserves, in my opinion.

Discontinuity of the topological entropy for Lozi maps

I have shown that, like diffeomorphisms, piecewise affine surface homeomorphisms are approximated in entropy by horseshoes, away from their singularities. It follows in particular that their topological entropy is lower-semicontinuous: a small perturbation cannot cause a macroscopic drop in entropy.

The continuity of the entropy for such maps had been an open problem for some time. Rigorous numerical estimates by Duncan SANDS and Yutaka ISHII seemed to suggest some discontinuous drops, but investigation at a small scale suggested these drops to be steep yet continuous variations.

Izzet B. YILDIZ has solved this question by finding for Lozi maps $f_{a,b}(x,y)=(1-a|x|+by,x)$ on $\mathbb R^2$, small numbers $\epsilon_1,\epsilon_2>0$ such that, setting $(a,b)=(1.4+\epsilon_1,0.4+\epsilon_1)$, for all $0<\epsilon<\epsilon_2$:

• $h_{top}(f_{a,b})=0$;
• $h_{top}(f_{a+\epsilon,b})>\frac14\log\frac12(\sqrt{5}+1)$.

The verification turns out to be quite simple (once you know where to look!). The non-wandering set of $f_{a,b}$ is shown to be reduced to be reduced to the fixed points of its fourth iterates, yielding the zero entropy immediately. $f_{a+\epsilon,b}$ on the other hand is shown to admit 2 disjoint closed quadrilaterals $U,V$ such that $f^4(U)$ hyperbolically crosses both $U$ and $V$ and $f^4(V)$ hyperbolically crosses $U$. This means that the sides of $U$ and $V$ can be branded alternatively s and u with the following property. The image of a u side crosses each of $U,V$ it meets, intersecting both their s sides and none of their u sides. This again yields the entropy estimate.

Ergodicity of smooth systems with product measures

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?

A classical theorem (Marstrand 1954) asserts that, given any Borel subset $X\subset\mathbb R^d$, the obvious inequality of the Hausdorff dimensions: $\dim(\pi(X))\leq \min(k,\dim(X))$ is in fact an equality for almost all orthogonal projections $\pi:\mathbb R^d\to\mathbb R^k$. As is often the case it is usually very dificult to prove equality for a given projection.