ISSN 2079-6900 (Print)
ISSN 2587-7496 (Online)

#### Middle Volga Mathematical Society Journal

MSC2020 05C62, 14J80, 37D15

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

#### A. I. Morozov1, O.V. Pochinka2

Annotation 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. Morse-Smale diffeomorphism, orientation-preserving diffeomorphism, topological invariant of diffeomorphism, surface diffeomorphism, orientable heteroclinic

1Andrey I. Morozov, Research Trainee, International Laboratory of Dynamical Systems and Applications, Department of Fundamental Mathematics, Higher School of Economics (25/12 Bolshaya Pecherskaya St., Nizhny Novgorod, 603155, Russia), ORCID: https://orcid.org/0000-0003-3125-1825, morozov-lux@yandex.ru

2Olga V. Pochinka, Laboratory Head, International Laboratory of Dynamical Systems and Applications, Higher School of Economics (25/12 Bolshaya Pecherskaya St., Nizhny Novgorod, 603155, Russia), Dr. Sci. (Physics and Mathematics), ORCID: http://orcid.org/0000-0002-6587-5305, olgapochinka@yandex.ru

Citation: A. I. Morozov, O.V. Pochinka, "[Combinatorial invariant of Morse-Smale diffeomorphisms on surfaces with orientable heteroclinic]", Zhurnal Srednevolzhskogo matematicheskogo obshchestva,22:1 (2020) 71–80 (In Russian)

DOI 10.15507/2079-6900.22.202001.71-80