Indices of the iterates of R3-homeomorphisms at fixed points which are isolated invariant sets

Abstract

Let U ⊂ R3 be an open set and f : U → f(U) ⊂ R3 be a homeomorphism. Let p ∈ U be a fixed point. It is known that if {p} is not an isolated invariant set, then the sequence of the fixedpoint indices of the iterates of f at p, (i(fn, p))n1, is, in general, unbounded. The main goal of this paper is to show that when {p} is an isolated invariant set, the sequence (i(fn, p))n1 is periodic. Conversely, we show that, for any periodic sequence of integers (In)n1 satisfying Dold’s necessary congruences, there exists an orientation-preserving homeomorphism such that i(fn, p) = In for every n 1. Finally we also present an application to the study of the local structure of the stable/unstable sets at p.