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Journal of the Institute of Mathematics of Jussieu (1)

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NON-HYPERBOLIC ERGODIC MEASURES AND HORSESHOES IN PARTIALLY HYPERBOLIC HOMOCLINIC CLASSES
Part of
Dawei Yang, Jinhua Zhang
Journal:

Journal of the Institute of Mathematics of Jussieu / Volume 19 / Issue 5 / September 2020

Published online by Cambridge University Press:

07 January 2019, pp. 1765-1792

Print publication:

September 2020
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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.