In this talk I plan to discuss examples and the dynamics of
topologically hyperbolic homeomorphisms and recent generalizations
considered in joint works with Welington Cordeiro (IMPAN), Alfonso
Artigue and José Vieitez (UDELAR).
As an attempt to classify the cw-expansive homeomorphisms satisfying the
shadowing property we introduce cw-hyperbolicity generalizing the usual
local product structure of the hyperbolic scenario. We prove a shadowing
lemma in the cw-hyperbolic case in the form of the L-shadowing property
and obtain consequences on the dynamics of such systems. We classify
cw-hyperbolic homeomorphisms with respect to the number of intersections
between local stable and unstable continua through the notion of
cwN-hyperbolicity and prove these share additional properties with the
hyperbolic systems.