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 projective-like manifolds.