Expansivity of ergodic measures with positive entropy

We prove that for every ergodic invariant measure with positive entropy of a continuous map on a compact metric space there is $δ>0$ such that the dynamical $δ$-balls have measure zero. We use this property to prove, for instance, that the stable classes have measure zero with respect to any ergodic invariant measure with positive entropy. Moreover, continuous maps which either have countably many stable classes or are Lyapunov stable on their recurrent sets have zero topological entropy. We also apply our results to the Li-Yorke chaos.

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