Omega-classification of Surface Diffeomorphisms Realizing Smale Diagrams

The present paper gives a partial answer to Smale's question which diagrams can correspond to $(A,B)$-diffeomorphisms. Model diffeomorphisms of the two-dimensional torus derived by “Smale surgery” are considered, and necessary and sufficient conditions for their topological conjugacy are found. Also, a class $G$ of $(A,B)$-diffeomorphisms on surfaces which are the connected sum of the model diffeomorphisms is introduced. Diffeomorphisms of the class $G$ realize any connected Hasse diagrams (abstract Smale graph). Examples of diffeomorphisms from $G$ with isomorphic labeled Smale diagrams which are not ambiently $\Omega$-conjugated are constructed. Moreover, a subset $G_{*}^{} \subset G$ of diffeomorphisms for which the isomorphism class of labeled Smale diagrams is a complete invariant of the ambient $\Omega$-conjugacy is singled out.