Morse – Smale 3-Diffeomorphisms with Saddles of the Same Unstable Manifold Dimension

Abstract

In this paper, we consider a class of Morse – Smale diffeomorphisms defined on a closed 3-manifold (not necessarily orientable) under the assumption that all their saddle points have the same dimension of the unstable manifolds. The simplest example of such diffeomorphisms is the well-known “source-sink” or “north pole – south pole” diffeomorphism, whose non-wandering set consists of exactly one source and one sink. As Reeb showed back in 1946, such systems can only be realized on the sphere. We generalize his result, namely, we show that diffeomorphisms from the considered class also can be defined only on the 3-sphere.