Combinatorial invariant of Morse-Smale diffeomorphisms on surfaces with orientable heteroclinic

In this paper we consider class of orientation-preserving Morse-Smale diffeomorphisms $f$, given on orientable surface $M^{2}$. In their articles A.A. Bezdenezhnich and V.Z. Grines has shown, that such diffeomorfisms contain finite number of heteroclinic orbits. Moreover, the problem of classification for such diffeomorphisms is reduced to the problem of distinguishing orientable graphs with substitutions describing the geometry of heteroclinic intersections. Howewer, these graphs generally do not allow polynomial distinguishing algorithms. In this paper, we propose a new approach to the classification of such cascades. To this end, each considered diffeomorphism $f$ is associated with a graph whose embeddablility in the ambient surface makes it possible to construct an effective algoritm for distinguishing such graphs.