Smale regular and chaotic A-homeomorphisms and A-diffeomorphisms

Abstract

We introduce Smale A-homeomorphisms that includes regular, semi-chaotic, chaotic, and super chaotic homeomorphisms of topological n-manifold M n , n ≥ 2. Smale A-homeomorphisms 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-diffeomorphisms with trivial and nontrivial basic sets. Super chaotic A-homeomorphisms contain A-diffeomorphisms whose basic sets are nontrivial. We describe invariant sets that determine completely dynamics of regular, semi-chaotic, and chaotic Smale A-homeomorphisms. This allows us to get necessary and sufficient conditions of conjugacy for these Smale A-homeomorphisms. We apply this necessary and sufficient conditions for structurally 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.