Continuum-wise hyperbolicity

Abstract

We introduce continuum-wise hyperbolicity, a generalization of hyperbolicity with respect to the continuum theory. We discuss similarities and differences between topological hyperbolicity and continuum-wise hyperbolicity. A shadowing lemma for cw-hyperbolic homeomorphisms is proved in the form of the L-shadowing property and a Spectral Decomposition is obtained in this scenario. In the proof we generalize the construction of Fathi \cite{Fat89} of a hyperbolic metric using only cw-expansivity, obtaining a hyperbolic cw-metric. We also introduce cwN-hyperbolicity, exhibit examples of these systems for arbitrarily large $N\in \mathbb{N}$ and obtain further dynamical properties of these systems such as finiteness of periodic points with the same period. We prove that homeomorphisms of ${\mathbb{S}}^2$ that are induced by topologically hyperbolic homeomorphisms of ${\mathbb{T}}^2$ are continuum-wise-hyperbolic and topologically conjugate to linear cw-Anosov diffeomorphisms of ${\mathbb{S}}^2$, being in particular cw2-hyperbolic.

References in corpus (2)

• Beyond topological hyperbolicity: the L-shadowing property
• Countably and entropy expansive homeomorphisms with the shadowing property

Cited by in corpus (6)

• Stable/unstable holonomies, density of periodic points, and transitivity for continuum-wise hyperbolic homeomorphisms
• Continuum-wise hyperbolic homeomorphisms on surfaces
• Sensitivity, local stable/unstable sets and shadowing
• Continuum-wise hyperbolicity, periodic shadowing, and measures of maximal entropy
• Shadowing property on hyperspace of continua induced by Morse gradient system
• Stable/unstable continua of cw-expansive flows