E. Y. Gurevich, E. K. Rodionova

National Research University «Higher School of Economics» (Nizhny Novgorod, Russian Federation)

Abstract. The purpose of this study is to single out a class of Morse-Smale cascades (diffeomorphisms) with a three-dimensional phase space that allow topological classification using combinatorial invariants. In the general case, an obstacle to such a classification is the possibility of wild embedding of separatrix closures in the ambient manifold, which leads to a countable set of topologically nonequivalent systems. To solve the problem, we study the orbit space of a cascade. The ambient manifold of a diffeomorphism can be represented as a union of three pairwise disjoint sets: a connected attractor and a repeller whose dimension does not exceed one, and their complement consisting of wandering points of a cascade called the characteristic set. It is known that the topology of the orbit space of the restriction of the Morse-Smale diffeomorphism to the characteristic set and the embedding of the projections of two-dimensional separatrices into it is a complete topological invariant for Morse-Smale cascades on three-dimensional manifolds. Moreover, a criterion for the inclusion of Morse-Smale cascades in the topological flow was obtained earlier.These results are used in this paper to show that the topological conjugacy classes of Morse-Smale cascades that are included in a topological flow and do not have heteroclinic curves admit a combinatorial description. More exactly, the class of Morse-Smale diffeomorphisms without heteroclinic intersections, defined on closed three-dimensional manifolds included in topological flows and not having heteroclinic curves, is considered. Each cascade from this class is associated with a two-color graph describing the mutual arrangement of two-dimensional separatrices of saddle periodic points. It is proved that the existence of an isomorphism of two-color graphs that preserves the color of edges is a necessary and sufficient condition for the topological conjugacy of cascades. It is shown that the speed of the algorithm that distinguishes two-color graphs depends polynomially on the number of its vertices. An algorithm for constructing a representative of each topological conjugacy class is described.

Key Words: Morse-Smale diffeomorphisms, topological classification, structurally stable diffeomorphisms, be-color graph, topological conjugacy

For citation: E. Y. Gurevich, E. K. Rodionova. Bicolor Graph of Morse-Smale Cascades on Manifolds of Dimension Three. Zhurnal Srednevolzhskogo matematicheskogo obshchestva. 25:2(2023), 37–52. DOI: https://doi.org/10.15507/2079-6900.25.202302.37-52

Information about the authors:

Elena Ya. Gurevich, Associate Professor, Department of Fundamental Mathematics, Senior Researcher, Laboratory "Dynamical Systems and Applications"(25/12 Bolshaya Pecherskaya St., Nizniy Novgorod 603150, Russia), ORCID: https://orcid.org/0000-0003-1815-3120, egurevich@hse.ru

Elena K. Rodionova, Student of the Faculty of Informatics, Mathematics and Computer Science, Trainee Researcher, Laboratory "Dynamical Systems and Applications" (25/12 B. Pecherskaya St., Nizhny Novgorod 603150, Russia), ORCID: https://orcid.org/0009-0004-2449-521X, ekrodionova@edu.hse.ru