Statistical stability of mostly expanding diffeomorphisms

Andersson, Martin; Vásquez, Carlos H.

doi:10.1016/j.anihpc.2020.04.007

Andersson, Martin ¹ ; Vásquez, Carlos H.²

¹ Universidade Federal Fluminense, Departamento de Matemática Aplicada, Rua Professor Marcos Waldemar de Freitas Reis, s/n, 24210-201, Niterói, Brazil
² Instituto de Matemática, Pontificia Universidad Católica de Valparaíso, Blanco Viel 596, Cerro Barón, Valparaíso, Chile

Annales de l'I.H.P. Analyse non linéaire, Tome 37 (2020) no. 6, pp. 1245-1270.

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Résumé
Abstract

Nous étudions comment les mesures physiques varient avec la dynamique sous-jacente, dans la classe ouverte des difféomorphismes $C^{r}$ , $r > 1$ , fortement partiellement hyperboliques pour lesquelles les exposants de Lyapunov centraux de tout u-état de Gibbs sont positifs. Lorsque transitifs, de tels difféomorphismes possédent une unique mesure physique qui persiste et varie continûment avec la dynamique.

Un des ingrédients principaux de la preuve est un nouveau lemme de type Pliss qui, appliqué dans le contexte adéquat, implique que la fréquence des temps hyperboliques est proche de un. Une autre nouveauté est l'introduction d'une nouvelle caractérisation des cu-états de Gibbs. Chacun de ses deux aspects ayant leur propre intérêt.

Le cas non transitif est aussi traité : dans ce contexte, le nombre de mesures physiques est une fonction semi-continue supérieure du difféomorphisme, et les mesures physiques varient continûment sous des hypothèses naturelles.

We study how physical measures vary with the underlying dynamics in the open class of $C^{r}$ , $r > 1$ , strong partially hyperbolic diffeomorphisms for which the central Lyapunov exponents of every Gibbs u-state is positive. If transitive, such a diffeomorphism has a unique physical measure that persists and varies continuously with the dynamics.

A main ingredient in the proof is a new Pliss-like Lemma which, under the right circumstances, yields frequency of hyperbolic times close to one. Another novelty is the introduction of a new characterization of Gibbs cu-states. Both of these may be of independent interest.

The non-transitive case is also treated: here the number of physical measures varies upper semi-continuously with the diffeomorphism, and physical measures vary continuously whenever possible.

MR Zbl

DOI : 10.1016/j.anihpc.2020.04.007

Classification : 37D30, 37C40, 37D25
Mots-clés : Partial hyperbolicity, Lyapunov exponents, SRB measures, Stable ergodicity, Statistical stability

Affiliations des auteurs :

Andersson, Martin ¹ ; Vásquez, Carlos H. ²

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     title = {Statistical stability of mostly expanding diffeomorphisms},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1245--1270},
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Andersson, Martin; Vásquez, Carlos H. Statistical stability of mostly expanding diffeomorphisms. Annales de l'I.H.P. Analyse non linéaire, Tome 37 (2020) no. 6, pp. 1245-1270. doi : 10.1016/j.anihpc.2020.04.007. http://archive.numdam.org/articles/10.1016/j.anihpc.2020.04.007/

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