Classification of Morse–Smale diffeomorphisms with a finite set of heteroclinic orbits on surfaces

In this paper, we consider orientation-preserving Morse-Smale diffeomorphisms on orientable closed surfaces. Such diffeomorphisms can have infinitely many heteroclinic orbits, which makes their topological classification very difficult. In fact, even in the case of a finite number of heteroclinic orbits, there are no exhaustive classification results. The main problem is that for all currently known complete topological invariants of such systems, the implementation is not described. In this paper, we present a complete topological classification of Morse-Smale diffeomorphisms with a finite number of heteroclinic orbits on surfaces, including a realization.