Continuum-wise hyperbolicity and periodic points

2025/03/12 by Carvalho, Bernardo, Oprocha, Piotr, Rego, Elias
#Dynamical Systems (math.DS) #FOS: Mathematics #Primary 37D20 #Secondary 37C05

paper · doi:10.48550/arxiv.2503.08991

Abstract

We prove that cw-hyperbolic homeomorphisms with jointly continuous stable/unstable holonomies satisfy the periodic shadowing property and, if they are topologically mixing, the periodic specification property. We discuss difficulties to adapt Bowen's techniques to obtain a measure of maximal entropy for cw-hyperbolic homeomorphisms, exhibit the unique measure of maximal entropy for Walter's pseudo-Anosov diffeomorphism of \mathbbS², and prove it can be obtained, as in the expansive case, as the weak* limit of an average of Dirac measures on periodic orbits. As an application, we exhibit the unique measure of maximal entropy for the homeomorphism on the Sierpiński Carpet defined in [12], which does not satisfy the specification property.

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