Continuum-wise hyperbolicity, periodic shadowing, and measures of maximal entropy

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 S 2 , 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 Boroński Oprocha (2018 C. R. Math. 356 340–4), which does not satisfy the specification property.