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In a recent paper, M. Andersson and C. Vasquez have introduced a new and useful estimate related to Pliss Lemma (see previous blog entry for its statement together with a proof).  One considers some real numbers $a_1,\dots,a_N$ such that for some reals $B<\alpha$ and $0\leq\kappa\leq1$:

• $a_k\geq B$;
• $|\{1\leq k\leq N:a_k\geq \alpha\}|\geq (1-\kappa)N$.

In contrast to Pliss Lemma, one assumes a lower bound (instead of an upper bound) and a high frequency of  large values (instead of just a large average). Andersson and Vasquez then show that the fraction of Pliss times can be made large by taking $\kappa$ is sufficient small.

Lemma (Andersson-Vasquez). Given $B\leq \beta<\alpha$, the fraction of Pliss times, i.e.,  $k$ such that

$\forall 1\leq\ell\leq k\quad \frac1{k-\ell+1}(a_\ell+\dots+a_k) \geq \beta\qquad (*)$

among $1,\dots,N$ is at least $1-\kappa\frac{\alpha-B}{\alpha-\beta}$.

In this situation, we say that $\beta$ is the target average.

Note that this lower bound is close to $1$ when $\kappa$ is small whereas  in Pliss Lemma this only occurs when $\alpha\approx A$.

Andersson and Vasquez provide a direct proof. Here we deduce it as a corollary of Pliss Lemma.

Proof. Obviously we can assume that

(*) $1-\kappa\frac{\alpha-B}{\alpha-\beta}>0$.

Let $a'_k=1$ if $a_k\geq\alpha$, $a'_k=0$ otherwise. We apply Pliss Lemma to this sequence with upper bound $A'=1$, average $\alpha'=1-\kappa$ and target average $\beta'=\frac{\beta-B}{\alpha-B}$. The required inequality $\beta'<\alpha'$ is equivalent to $\kappa(1-\beta')^{-1}<1$ which follows from (*). We obtain a fraction $\frac{\alpha'-\beta'}{A'-\beta'}=1-\kappa\frac{\alpha-B}{\alpha-\beta}$ of Pliss times. To conclude note that if $k$ is such a Pliss time, then the average of $a$ along all intervals $[\ell,k]$ is at least $\beta=\beta'\alpha+(1-\beta')B$. $\square$

A proof of Pliss’ lemma

Pliss’ lemma is a simple combinatorial tool very useful for nonuniformly hyperbolic dynamics. It is about finite sequences of real numbers $a_1,\dots,a_N$ ($N\geq1$). Assume that:

$\tfrac1N(a_1+\dots+a_N) \geq \alpha$ for some $\alpha$ (the average is good).

Fix $\beta<\alpha$. Say that an (integer) interval $I\subset\{1,\dots,N\}$ is bad if $\tfrac1{|I|}\sum_{i\in I} a_i<\beta$. Otherwise the interval is called good. An index $1\leq k\leq n$ is Pliss if all subintervals $[\ell,k]$ for $1\leq \ell\leq k$ are good.

Lemma (Pliss). If $A\geq \max(a_1,\dots,a_N)$ then there are at least $\frac{\alpha-\beta}{A-\beta} N$ Pliss indices among $\{1,\dots,N\}$.

Note: the same bound holds for the trivial estimate:

$|\{1\leq k\leq N: a_k\geq\beta\}| > \frac{\alpha-\beta}{A-\beta} N$

Here is a proof, modeled after Kamae’s proof of the ergodic theorem.

For each non-Pliss index $k$ define $1\leq \ell(k)\leq k$ to be the largest integer such $[\ell,k]$ is bad. Let $B\subset\{1,\dots,N\}$ be the set of non-Pliss indices and let $\hat B:=\bigcup_{k\in B} [\ell(k),k]$. Clearly $B\subset \hat B$. We can assume that $B\ne\emptyset$ since otherwise there is nothing to show.

Claim. The intervals appearing in the above union are disjoint or nested.

Indeed, take two intersecting intervals, i.e., $k,k'\in B$ such that $k and $\ell(k)\leq \ell(k')\leq k$. Note that $[k+1,k']$ is good by maximality of $\ell(k')$. Since $[\ell(k'),k']$ is bad, this implies that $[\ell(k'),k]$ is bad. The maximality of $\ell(k)$ implies that $\ell(k)\geq \ell(k')$, proving the claim.

It follows that $\hat B$ can be written as the disjoint bad intervals. Therefore $\tfrac1{|\hat B|}\sum_{k\in \hat B} a_k<\beta$. This yields:

$\alpha N \leq \sum_{k\in \hat B} a_i+\sum_{k\in[1,N]\setminus \hat B} a_i \leq \beta|\hat B| + (N-|\hat B|)A$

A direct computation then concludes the proof of the lemma. QED

Pliss Lemma generalizes to infinite sequences, say $(\alpha_n)_{n\geq1}$. For backward Pliss indices, i.e., $k\geq1$ such that the intervals $[\ell,k]$ are good for all $1\leq\ell\leq k$,  it is enough to apply the previous, finitary version to intervals $[1,N]$.

Lemma (Pliss, backward, infinite version). Let $\alpha:=\liminf_N \frac1N(a_1+\dots+a_N)$ and let $A:=\sup_{n\geq1} A_n$. Given any $\beta<\alpha$, the lower density of backward Pliss times for parameter $\beta$ is at least $\frac{\alpha-\beta}{A-\beta}$.

Forward Pliss times are a bit more delicate since an index $1\leq k\leq N$ can be forward Pliss for $(a_1,\dots,a_N)$ but not for $(a_1,\dots,a_M)$ for some $M>N$. Given $k\geq1$ which is not forward Pliss, let $\ell(k)$ be the smallest integer $\ell\geq k$ such that $[k,\ell(k)]$ is bad. As before, the intervals $[k,\ell(k)]$ $(k\geq1)$ are disjoint or nested.  Setting $\hat N:=\sup_{k\leq N} \ell(k)$, note that for each $k\leq\hat N$, $\ell(k)\leq\hat N$. Thus, we can estimate the number of forward Pliss times in $[1,\hat N-1]$ by the finitary Pliss Lemma and obtain:

Lemma (Pliss, forward, infinite version). Let $\alpha:=\liminf_N \tfrac1N(a_1+\dots+a_N)$ and let $A:=\sup_{n\geq1} A_n$. Given any $\beta<\alpha$, the upper density of backward Pliss times for parameter $\beta$ is at least $\frac{\alpha-\beta}{A-\beta}$.

One can combine the two to obtain:

Lemma (Pliss, backward, bi-infinite version). Let $\alpha=\min(\liminf_{N} \tfrac1{N}(a_{-N}+\dots+a_{-1}), \limsup_N\tfrac1N(a_1+\dots+a_N)$ and let $A:=\sup_{n\in\mathbf Z} A_n$. Given any $\beta<\alpha$, the upper density of backward Pliss times for parameter $\beta$ is at least $\frac{\alpha-\beta}{A-\beta}$.

Note: the upper density of $S\subset \mathbf Z$ is $\limsup_{N,M\to\infty} \tfrac1{M+N+1}(a_{-M}+\dots+a_N)$.

Séminaire: Etats d’équilibre et classes homoclines

Le 18 janvier, j’ai présenté au séminaire de dynamique de l’Institut Mathématique de Jussieu un nouvel aspect du travail accompli avec Sylvain CROVISIER et Omri SARIG. Nous obtenons l’unicité de l’état d’équilibre pour un potentiel régulier dès qu’on se restreint aux mesures ergodiques hyperboliques et à une classe homocline mesurée. Ceci s’applique aux difféomorphismes $C^{1+}$ de variétés compactes avec un potentiel Hölder-continu ou bien géométrique.

Measures of maximal entropy for surface diffeomorphisms

With Sylvain CROVISIER and Omri SARIG, we show that $C^\infty$ surface diffeomorphisms with positive topological entropy have at most finitely many ergodic measures of maximal entropy in general, and at most one in the topologically transitive case. This answers a question of Newhouse, who proved that such measures always exist.

To do this we generalize Smale’s spectral decomposition theorem to non-uniformly hyperbolic surface diffeomorphisms, we introduce homoclinic classes of measures, and we study their properties using codings by irreducible countable state Markov
shifts.

The preprint is on arxiv (you can also download the preprint here).

The Spectral Decomposition for Smooth Surface Diffeomorphisms

At the third conference on Dynamics of Differential Equations, I explained the following Spectral Decomposition Theorem for arbitrary smooth surface diffeomorphisms. It is mostly part of the joint work with Sylvain Crovisier and Omri Sarig.

Given some dynamics, say that $X\equiv Y$ if the symmetric difference of the two subsets does not carry measures with positive entropy.

Theorem (B-Crovisier-Sarig). For a $C^\infty$ diffeomorphism of a closed surface, let $\{H(O_i)\}_{i\in I}$ be its infinite homoclinic classes. Then the non-wandering set satisfies: $\Omega(f) \equiv \bigcup_{i\in I} H(O_i)$ with $H(O_i)\cap H(O_j)\equiv\emptyset$ and $f:H(O_i)\to H(O_i)$ topologically transitive. More precisely, for each $i\in I$, there is a compact set $A_i$ and an integer $_i\geq1$ such that $f^{p_i}:A_i\to A_i$ is topologically mixing, $H(O_i)=\bigcup_{j=0}^{p_i-1} f^j(A_i)$, $A_i=f^{p_i}(A_i)$ and $f^{-j}(A_i)\cap f^{-k}A_i\equiv\emptyset$ if $0\leq j.

If $f|\Omega(f)$ is topologically transitive and $h_{{\rm top}}(f)>0$, then there is a unique infinite homoclinic class. If, additionally, $f|\Omega(f)$ is topologically mixing, then this homoclinic class is itself topologically mixing.

I also explained that we have a good understanding of the dynamics inside the pieces (measures of maximum entropy, periodic orbits, Borel classification and the periods involved in the three descriptions are the above period $p_i$). See these slides for details.

“Facteurs de Bowen” à Quimper

A l’occasion de la journée Systèmes dynamiques, probabilité, statistiques à Quimper, j’ai présenté la notion de facteur de Bowen et son utilité pour l’étude des difféomorphismes de surfaces. On peut ainsi rendre injective les extensions finies construites par Omri Sarig, en ne perdant qu’une partie faiblement errante (union dénombrable de parties errantes).