  We discuss the dynamics beyond topological hyperbolicity considering
homeomorphisms satisfying the shadowing property and generalizations of
expansivity. It is proved that transitive countably expansive homeomorphisms
satisfying the shadowing property are expansive in the set of transitive
points. This is in contrast with pseudo-Anosov diffeomorphisms of the
two-dimensional sphere that are transitive, cw-expansive, satisfy the shadowing
property but the dynamical ball in each transitive point contains a Cantor
subset. We exhibit examples of countably expansive homeomorphisms that are not
finite expansive, satisfy the shadowing property and admits an infinite number
of chain-recurrent classes. We further explore the relation between countable
and entropy expansivity and prove that for surface homeomorphisms $f:S\to S$
satisfying the shadowing property and $\Omega(f)=S$, both countably expansive
and entropy cw-expansive are equivalent to being topologically conjugate to an
Anosov diffeomorphism.
