On perturbations of algebraic periodic automorphisms of a two-dimensional torus

Abstract

According to the results of V. Z. Grines and A. N. Bezdenezhnykh, for each gradient-like diffeomorphism of a closed orientable surface M2 there exist a gradient-like flow and a periodic diffeomorphism of this surface such that the original diffeomorphism is a superposition of a diffeomorphism that is a shift per unit time of the flow and the periodic diffeomorphism. In the case when M2 is a two-dimensional torus, there is a topological classification of periodic maps. Moreover, it is known that there is only a finite number of topological conjugacy classes of periodic diffeomorphisms that are not homotopic to identity one. Each such class contains a representative that is a periodic algebraic automorphism of a two-dimensional torus. Periodic automorphisms of a two-dimensional torus are not structurally stable maps, and, in general, it is impossible to predict the dynamics of their arbitrarily small perturbations. However, in the case when a periodic diffeomorphism is algebraic, we constructed a one-parameter family of maps consisting of the initial periodic algebraic automorphism at zero parameter value and gradient-like diffeomorphisms of a twodimensional torus for all non-zero parameter values. Each diffeomorphism of the constructed one-parameter families inherits, in a certain sense, the dynamics of a periodic algebraic automorphism being perturbed.