On the Centralizer and Conjugacy of Pseudo-Anosov Homeomorphisms

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.