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 [16] 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∈N and obtain further dynamical properties of these systems such as finiteness of periodic points with the same period. We prove that homeomorphisms of S2 that are induced by topologically hyperbolic homeomorphisms of T2 are continuum-wise-hyperbolic and topologically conjugate to linear cw-Anosov diffeomorphisms of S2, being in particular cw2-hyperbolic.