Minimal sets of periods for torus maps via Nielsen numbers

Abstract

The main results in this paper concern the minimal sets of periods possible in a given homotopy class of torus maps. For maps on the 2—torus, we provide a complete description of these minimal sets. A number of results on higher dimensional tori are also proved; including criteria for every map in a given homotopy class to have all periods, or all but finitely many periods. 1. Introduction. In dynamical systems, it is often the case that topological information can be used to study qualitative properties of the system. This article deals with the problem of determining the set of periods (of the periodic orbits) of a mapping given the homotopy class of the mapping. To fix terminology, suppose / is a continuous self-map on the manifold M. A fixed point of f is a point x in M such that f(x) = x. We will call x a periodic point of period n if x is a fixed point of fn but is not fixed by any /*, for 1 < k < n.