### On classification of gradient-like diffeomorphisms on surfaces by means automorphisms of three-color graphs.

#### V. Z. Grines^{1}, S. H. Kapkaeva^{2}

Annotation | This article is a continuation of the paper \cite{kapkaeva6}, in which the conditions of topological conjugacy of gradient-like diffeomorphisms are found, under suggestion that wandering set consists of only fixed points. In this paper we consider the class of orientation preserving gradient-like diffeomorphisms whose nonwandering set admits an existence of periodic orbits of period greater than one. To each diffeomorphism we appreciate three-color graph equipped by an automorphism given on the set of vertices of the graph. It is stated that all vertices of the graph have the same period under action of the automorphism. It is proved that the three-color graph equipped with the automorphism, is a complete topological invariant in the considered class of diffeomorphisms |
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Keywords | Morse-Smale diffeomorphisms, gradient-like diffeomorphisms, topological conjugate diffeomorphisms, three-color graph. |

^{1}Professor, Lobachevsky State University of Nizhni Novgorod, Nizhni Novgorod; vgrines@yandex.ru.^{2}Student, Mordovian State University after N.P. Ogarev, Saransk; kapkaevasvetlana@yandex.ru.

**Citation**: V. Z. Grines, S. H. Kapkaeva, "[On classification of gradient-like diffeomorphisms on surfaces by means automorphisms of three-color graphs.]", Zhurnal Srednevolzhskogo matematicheskogo obshchestva,15:2 (2013) 12–22 (In Russian)