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GEOMETRICAL AND MEASURE-THEORETIC STRUCTURES OF MAPS WITH A MOSTLY EXPANDING CENTER

Part of: Smooth dynamical systems: general theory Dynamical systems with hyperbolic behavior Ergodic theory

Published online by Cambridge University Press:  12 July 2021

Jiagang Yang Open the ORCID record for Jiagang Yang [Opens in a new window]
Jiagang Yang*
Affiliation:
Departamento de Geometria, Instituto de Matemática e Estatística, Universidade Federal Fluminense, Niterói, Brazil (yangjg@impa.br)
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Abstract

In this article we study physical measures for $\operatorname {C}^{1+\alpha }$ partially hyperbolic diffeomorphisms with a mostly expanding center. We show that every diffeomorphism with a mostly expanding center direction exhibits a geometrical-combinatorial structure, which we call skeleton, that determines the number, basins and supports of the physical measures. Furthermore, the skeleton allows us to describe how physical measures bifurcate as the diffeomorphism changes under $C^1$ topology.

Moreover, for each diffeomorphism with a mostly expanding center, there exists a $C^1$ neighbourhood, such that diffeomorphism among a $C^1$ residual subset of this neighbourhood admits finitely many physical measures, whose basins have full volume.

We also show that the physical measures for diffeomorphisms with a mostly expanding center satisfy exponential decay of correlation for any Hölder observes. In particular, we prove that every $C^2$ , partially hyperbolic, accessible diffeomorphism with 1-dimensional center and nonvanishing center exponent has exponential decay of correlations for Hölder functions.

Keywords

partial hyperbolicitydiffeomorphisms with a mostly expanding centerphysical measuredecay of correlations

MSC classification

Primary: 37C40: Smooth ergodic theory, invariant measures
Secondary: 37D30: Partially hyperbolic systems and dominated splittings 37A25: Ergodicity, mixing, rates of mixing 37A35: Entropy and other invariants, isomorphism, classification
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 22 , Issue 2 , March 2023 , pp. 919 - 959
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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