Dynamics of three-dimensional A-diffeomorphisms with two-dimensional attractors and repellers

Abstract

In this paper, we consider a class of A-diffeomorphisms given on a 3-manifold, assuming that all the basic sets of the diffeomorphisms are two dimensional. It is known that such basic sets are either attractors or repellers and they are two types only, surface or expanding (contracting). One of the results of the paper is the proof that different types of two-dimensional basic sets do not coexist in the non-wandering set of the same 3-diffeomorphism. Also, the existence of an energy function is constructively proved for systems of the class under consideration. It is illustrated by examples that the two-dimensionality of the basic sets is essential in this matter and a decrease in the dimension can lead to the absence of the energy function for a diffeomorphism.