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

Abstract

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.

27 pages, 9 figures

References in corpus (1)

• Continuum-wise hyperbolic homeomorphisms on surfaces

Cited by in corpus (2)

• Continuum-wise hyperbolicity, periodic shadowing, and measures of maximal entropy
• Stable/unstable continua of cw-expansive flows