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.

我们介绍 Smale A-同胚，包括拓扑\(n\) -流形\(M^{n}\) , \(n\geqslant 2\)的规则、半混沌、混沌和超混沌同胚。Smale A-同胚包含公理 A 微分同胚（简称 A-微分同胚），前提是\(M^{n}\)承认光滑的结构。正则 A-同胚包含所有 Morse – Smale 微分同胚，而半混沌和混沌 A-同胚包含具有平凡和非平凡基集的 A-微分同胚。超混沌 A-同胚包含 A-微分同胚，其基集是非平凡的。考虑 Smale A-同胚而不是 A-微分同胚的原因可能是因为它很好地削弱了非均匀双曲性和伪双曲性，这是一个已经有大量应用的主题。

我们描述了完全决定规则、半混沌和混沌 Smale A-同胚动力学的不变集。这使我们能够获得这些 Smale A-同胚（特别是所有 Morse - Smale 微分同胚）的充分必要条件。我们将这些充分必要条件应用于具有任意数量的扩展吸引子的结构稳定的表面微分同胚。我们还使用这些条件来获得 Morse 的完整分类 - 射影流形上的 Smale 微分同胚。