Stable/unstable holonomies, density of periodic points, and transitivity for continuum-wise hyperbolic homeomorphisms

• Published in: arXiv.org 2023-06
• Main Authors: Carvalho, Bernardo, Rego, Elias
• Format: Article
• Language: English
• Subjects: Self-similarity
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.2306.00524

We discuss different regularities on stable/unstable holonomies of cw-hyperbolic homeomorphisms and prove that if a cw-hyperbolic homeomorphism has continuous joint stable/unstable holonomies, then it has a dense set of periodic points in its non-wandering set. For that, we prove that the hyperbolic cw-metric (introduced in [9]) can be adapted to be self-similar (as in [6]) and, in this case, continuous joint stable/unstable holonomies are pseudo-isometric. We also prove transitivity of cw-hyperbolic homeomorphisms assuming that the stable/unstable holonomies are isometric. In the case the ambient space is a surface, we prove that a cw\(_F\)-hyperbolic homeomorphism has continuous joint stable/unstable holonomies when every bi-asymptotic sector is regular.