On the topological classification of structurally stable diffeomorphisms of surfaces with one-dimensional attractors and repellers

Necessary and sufficient conditions for topological conjugacy are established in the case of structurally stable, orientation-preserving diffeomorphisms of a two-dimensional smooth closed oriented manifold $M$ that belong to the class $S(M)$, that is, satisfy the following conditions: 1) all the non-trivial basic sets of each $f\in S(M)$ are one-dimensional attractors or repellers; 2) there exist only finitely many heteroclinic trajectories lying in the intersections of stable and unstable manifolds of saddle periodic points belonging to trivial basic sets.