Классификация меняющих ориентацию периодических гомеоморфизмов двумерного тора

Математика и теоретические компьютерные науки. 2025. Т. 3. № 3. С. 87–109.

Мартынов Т. Д., Pochinka O., Chilina E.

According to J. Nielsen and H. Hang, each class of topological conjugacy of periodic homeomorphisms of orientable compact surfaces is completely described by a finite set of data called the characteristic. For a two-dimensional sphere, exhaustive classification results with the construction of linear representatives in each conjugacy class were obtained by B. Kerekyarto. For a two-dimensional torus, similar results were obtained with the participation of the authors of this article. Here we find all the characteristics of orientation-changing periodic homeomorphisms of a two-dimensional torus. A homeomorphism representing a class of topological conjugacy is constructed for each of them. The classification of periodic homeomorphisms, in addition to being of independent interest, plays a key role in solving the Palis–Pugh problem of constructing stable arcs in the space of discrete dynamical systems, which is included in the list of 50 most important problems of dynamical systems. For all classes of gradient-like diffeomorphisms of surfaces where this problem is solved, the idea of a close connection of such systems with periodic transformations was used. Thus, the obtained result will make it possible to expand the class of systems for which the Palis problem has been solved.

Keywords: топологическая классификация, topological classification, two-dimensional torus, двумерный тор, periodic homeomorphism, периодический гомеоморфизм

Publication based on the results of: Differential-geometric properties of dynamical systems and mathematical modeling of processes (2025)