Ekaterina Chilina

In this paper, following J. Nielsen, we introduce a complete characteristic of orientationpreserving periodic maps on the two-dimensional torus. All admissible complete characteristics were found and realized. In particular, each of the classes of orientation-preserving periodic homeomorphisms on the 2-torus that are nonhomotopic to the identity is realized by an algebraic automorphism. Moreover, it is shown that the number of such classes is finite. According to V. Z. Grines and A.Bezdenezhnykh, any gradient-like orientation-preserving diffeomorphism of an orientable surface is represented as a superposition of the time-1 map of a gradient-like flow and some periodic homeomorphism. Thus, the results of this work are directly related to the complete topological classification of gradient-like diffeomorphisms on surfaces.

The present paper is devoted to the study of the dynamics of mappings commuting with pseudo-Anosov surface homeomorphisms. It is proved that the centralizer of a pseudo-Anosov homeomorphism $P$ consists of pairwise nonhomotopic mappings, each of which is a composition of a power of the pseudo-Anosov mapping and a periodic homeomorphism. For periodic mappings commuting with $P$, it is proved that their number is finite and does not exceed the number $N_P^{}$, which is equal to the minimum among the number of all separatrices related to saddles of the same valency of $P$-invariant foliations. For a periodic homeomorphism $J$ lying in the centralizer of $P$, it is also shown that, if the period of a point is equal to half the period of the homeomorphism $J$, then this point is located in the complement of the separatrices of saddle singularities. If the period of the point is less than half the period of $J$, then this point is contained in the set of saddle singularities. In addition, it is proved that there exists a monomorphism from the group of periodic maps commuting with a pseudo-Anosov homeomorphism to the symmetric group of degree $N_P^{}$. Each permutation from the image of the monomorphism is represented as a product of disjoint cycles of the same length. Furthermore, it is deduced that a pseudo-Anosov homeomorphism with the trivial centralizer exists on each orientable closed surface of genus greater than $2$. As an application of the results related to the structure of the centralizer of pseudo-Anosov homeomorphisms to their topological classification, it is proved that there are no pairwise distinct homotopic conjugating mappings for topologically conjugated pseudo-Anosov homeomorphisms.