  We introduce Smale A-homeomorphisms that includes regular, semi-chaotic,
chaotic, and super chaotic homeomorphisms of topo\-lo\-gi\-cal $n$-manifold
$M^n$, $n\geq 2$. Smale A-homeo\-mor\-p\-hisms contain A-diffeomorphisms (in
particular, structurally stable diffeomorphisms) provided $M^n$ admits a smooth
structure. Regular A-homeomorphisms contain all Morse-Smale diffeomorphisms,
while semi-chaotic and chaotic A-homeomorphisms contain A-diffeo\-mor\-p\-hisms
with trivial and nontrivial basic sets. Super chaotic A-homeo\-mor\-p\-hisms
contain A-diffeomorphisms whose basic sets are nontrivial. We describe
invariant sets that determine completely dynamics of regular, semi-chaotic, and
chaotic Smale A-homeo\-mor\-p\-hisms. This allows us to get necessary and
sufficient conditions of conjugacy for these Smale A-homeomorp\-hisms. We apply
this necessary and sufficient conditions for structu\-ral\-ly stable surface
diffeomorphisms with arbitrary number of one-dimensional expanding attractors.
We also use this conditions to get the complete classification of Morse-Smale
diffeomorphisms on projective-like $n$-manifolds for $n=2,8,16$.
