Continuum-wise hyperbolicity and periodic points
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arXiv
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| Format: | Preprint |
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2025
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| author | Carvalho, Bernardo Oprocha, Piotr Rego, Elias |
| author_facet | Carvalho, Bernardo Oprocha, Piotr Rego, Elias |
| contents | 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 $\mathbb{S}^2$, 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 [12], which does not satisfy the specification property. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_08991 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Continuum-wise hyperbolicity and periodic points Carvalho, Bernardo Oprocha, Piotr Rego, Elias Dynamical Systems Primary 37D20, Secondary 37C05 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 $\mathbb{S}^2$, 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 [12], which does not satisfy the specification property. |
| title | Continuum-wise hyperbolicity and periodic points |
| topic | Dynamical Systems Primary 37D20, Secondary 37C05 |
| url | https://arxiv.org/abs/2503.08991 |