Indices of the iterates of R-3-homeomorphisms at fixed points which are isolated invariant sets

Abstract

Let U subset of R-3 be an open set and f : U -> f(U) subset of R-3 be a homeomorphism. Let p is an element of U be a fixed point. It is known that if {p} is not an isolated invariant set, then the sequence of the fixed-point indices of the iterates of f at p, (i(f(n), p))(n >=) (1), is, in general, unbounded. The main goal of this paper is to show that when {p} is an isolated invariant set, the sequence (i(f(n), p))(n >= 1) is periodic. Conversely, we show that, for any periodic sequence of integers (I-n)(n >= 1) satisfying Dold's necessary congruences, there exists an orientation-preserving homeomorphism such that i(f(n), p) = I-n for every n >= 1. Finally we also present an application to the study of the local structure of the stable/unstable sets at p.