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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Periodic measures and partially hyperbolic homoclinic classes


Authors: Christian Bonatti and Jinhua Zhang
Journal: Trans. Amer. Math. Soc. 372 (2019), 755-802
MSC (2010): Primary 37D30, 37C40, 37C50, 37A25, 37D25
DOI: https://doi.org/10.1090/tran/7252
Published electronically: April 18, 2019
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Abstract: In this paper, we give a precise meaning to the following fact, and we prove it: $ C^1$-open and densely, all the non-hyperbolic ergodic measures generated by a robust cycle are approximated by periodic measures.

We apply our technique to the global setting of partially hyperbolic diffeomorphisms with one-dimensional center. When both strong stable and unstable foliations are minimal, we get that the closure of the set of ergodic measures is the union of two convex sets corresponding to the two possible $ s$-indices; these two convex sets intersect along the closure of the set of non-hyperbolic ergodic measures. That is the case for robustly transitive perturbations of a time-one map of a transitive Anosov flow, or of the skew product of an Anosov torus diffeomorphism by a rotation of the circle.


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Additional Information

Christian Bonatti
Affiliation: Institut de Mathématiques de Bourgogne, UMR 5584 du CNRS, Université de Bourgogne, 21004 Dijon, France
Email: bonatti@u-bourgogne.fr

Jinhua Zhang
Affiliation: School of Mathematical Sciences, Peking University, Beijing 100871, People’s Republic of China – and – Institut de Mathématiques de Bourgogne, UMR 5584 du CNRS, Université de Bourgogne, 21004 Dijon, France
Address at time of publication: Laboratoire de Mathématiques d’Orsay, CNRS-Université Paris-Sud, Orsay 91405, France
Email: zjh200889@gmail.com, jinhua.zhang@u-bourgogne.fr

DOI: https://doi.org/10.1090/tran/7252
Keywords: Blender, robust cycle, partial hyperbolicity, ergodic measure, periodic measure, non-hyperbolic measure, Lyapunov exponent, quasi-hyperbolic string
Received by editor(s): September 28, 2016
Received by editor(s) in revised form: March 30, 2017
Published electronically: April 18, 2019
Additional Notes: Jinhua Zhang is the corresponding author
Article copyright: © Copyright 2019 American Mathematical Society