Periodic measures and partially hyperbolic homoclinic classes

Bonatti, Christian; Zhang, Jinhua

doi:10.1090/tran/7252

Periodic measures and partially hyperbolic homoclinic classes
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by Christian Bonatti and Jinhua Zhang PDF

Trans. Amer. Math. Soc. 372 (2019), 755-802 Request permission

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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