Continuum-wise hyperbolicity.

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 S 2 that are induced by topologically hyperbolic homeomorphisms of T 2 are continuum-wise-hyperbolic and topologically conjugate to linear cw-Anosov diffeomorphisms of S 2 , being in particular cw2-hyperbolic. [ABSTRACT FROM AUTHOR]