Periodic measures and partially hyperbolic homoclinic classes
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- by Christian Bonatti and Jinhua Zhang PDF
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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
- MR Author ID: 1225510
- Email: zjh200889@gmail.com, jinhua.zhang@u-bourgogne.fr
- 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
- © Copyright 2019 American Mathematical Society
- 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
- MathSciNet review: 3968787