Smale Regular and Chaotic A-Homeomorphisms and A-Diffeomorphisms

We introduce Smale A-homeomorphisms that include regular, semichaotic, chaotic, and superchaotic homeomorphisms of a topological $n$-manifold $M^n$, $n\geqslant 2$. Smale A-homeomorphisms contain axiom A diffeomorphisms (in short, A-diffeomorphisms) provided that $M^n$ admits a smooth structure. Regular A-homeomorphisms contain all Morse–Smale diffeomorphisms, while semichaotic and chaotic A-homeomorphisms contain A-diffeomorphisms with trivial and nontrivial basic sets. Superchaotic A-homeomorphisms contain A-diffeomorphisms whose basic sets are nontrivial. The reason to consider Smale A-homeomorphisms instead of A-diffeomorphisms may be attributed to the fact that it is a good weakening of nonuniform hyperbolicity and pseudo-hyperbolicity, a subject which has already seen an immense number of applications.
We describe invariant sets that determine completely the dynamics of regular, semichaotic, and chaotic Smale A-homeomorphisms. This allows us to get necessary and sufficient conditions of conjugacy for these Smale A-homeomorphisms (in particular, for all Morse–Smale diffeomorphisms). We apply these necessary and sufficient conditions for structurally stable surface diffeomorphisms with an arbitrary number of expanding attractors. We also use these conditions to obtain a complete classification of Morse–Smale diffeomorphisms on projectivelike manifolds.