Continuum-wise hyperbolicity

TL;DR

This paper introduces continuum-wise hyperbolicity, a new generalization of hyperbolic systems based on continuum theory, and explores its properties, examples, and connections to classical hyperbolic dynamics.

Contribution

It defines continuum-wise hyperbolicity, proves a shadowing lemma and spectral decomposition, and constructs hyperbolic cw-metrics, extending hyperbolic theory to continuum-wise settings.

Findings

Established a shadowing lemma for cw-hyperbolic homeomorphisms.

Constructed hyperbolic cw-metrics using cw-expansivity.

Showed that certain homeomorphisms of the sphere are cw-hyperbolic and conjugate to linear cw-Anosov diffeomorphisms.

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.