In this paper it is proved that every orientable surface admits an orientation-preserving Morse-Smale diffeomorphism with one saddle orbit. It is shown that these diffeomorphisms have exactly three node orbits. In addition, all possible types of periodic data for such diffeomorphisms are established.
Currently, a complete topological classification has been obtained with respect to the topological equivalence of Morse-Smale flows, [9,7], as well as their generalizations of Ω-stable flows on closed surfaces, . Some results on topological conjugacy classification for such systems are also known. In particular, the coincidence of the classes of topological equivalence and conjugacy of gradient-like flows (Morse-Smale flows without periodic orbits) was established in . In the classical paper , it was proved that in the presence of connections (coincidence of saddle separatrices), the topological equivalence class of a Ω-stable flow splits into a continuum of topological conjugacy classes (has moduli). Obviously, each periodic orbit also generates at least one modulus associated with the period of that orbit. In the present work, it was established that the presence of a cell in a flow bounded by two limit cycles leads to the existence of an infinitely many stability moduli. In addition, a criterion for the topological conjugation of flows on such cells was found.