On Morse – Smale 3-Diffeomorphisms with a Given Tuple of Sink Points Periods

Abstract

In investigating dynamical systems with chaotic attractors, many aspects of global behavior of a flow or a diffeomorphism with such an attractor are studied by replacing a nontrivial attractor by a trivial one [1, 2, 11, 14]. Such a method allows one to reduce the original system to a regular system, for instance, of a Morse – Smale system, matched with it. In most cases, the possibility of such a substitution is justified by the existence of Morse – Smale diffeomorphisms with partially determined periodic data, the complete understanding of their dynamics and the topology of manifolds, on which they are defined. With this aim in mind, we consider Morse – Smale diffeomorphisms \(f\) with determined periods of the sink points, given on a closed smooth 3-manifold. We have shown that, if the total number of these sinks is \(k\), then their nonwandering set consists of an even number of points which is at least \(2k\). We have found necessary and sufficient conditions for the realizability of a set of sink periods in the minimal nonwandering set. We claim that such diffeomorphisms exist only on the 3-sphere and establish for them a sufficient condition for the existence of heteroclinic points. In addition, we prove that the Morse – Smale 3-diffeomorphism with an arbitrary set of sink periods can be implemented in the nonwandering set consisting of \(2k+2\) points. We claim that any such a diffeomorphism is supported by a lens space or the skew product \(\mathbb{S}^{2}\tilde{\times}\mathbb{S}^{1}\).