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NON-HYPERBOLIC ERGODIC MEASURES AND HORSESHOES IN PARTIALLY HYPERBOLIC HOMOCLINIC CLASSES

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

Published online by Cambridge University Press:  07 January 2019

Dawei Yang  and
Jinhua Zhang
Dawei Yang
Affiliation:
School of Mathematical Sciences, Soochow University, Suzhou, 215006, P.R. China ([email protected]; [email protected])
Jinhua Zhang
Affiliation:
Laboratoire de Mathématiques d’Orsay, CNRS - Université Paris-Sud, Orsay 91405, France ([email protected])
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Abstract

We study a rich family of robustly non-hyperbolic transitive diffeomorphisms and we show that each ergodic measure is approached by hyperbolic sets in weak$\ast$-topology and in entropy. For hyperbolic ergodic measures, it is a classical result of A. Katok. The novelty here is to deal with non-hyperbolic ergodic measures. As a consequence, we obtain the continuity of topological entropy.

Keywords

blender horseshoepartial hyperbolicityentropyperiodic measurenon-hyperbolic ergodic measure

MSC classification

Primary: 37D25: Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.) 37A35: Entropy and other invariants, isomorphism, classification 37C40: Smooth ergodic theory, invariant measures
Secondary: 37C50: Approximate trajectories (pseudotrajectories, shadowing, etc.) 37D30: Partially hyperbolic systems and dominated splittings
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 19 , Issue 5 , September 2020 , pp. 1765 - 1792
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Copyright
© Cambridge University Press 2019

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Footnotes

This work was done when J. Zhang visited Soochow University in July 2017. J. Zhang would like to thank Soochow University for hospitality. D. Yang was partially supported by NSFC 11671288 and NSFC 11790274. J. Zhang was partially supported by the ERC project 692925 NUHGD. J. Zhang is the corresponding author.

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