Indices of the iterates of ({\Bbb R}^3)-homeomorphisms at fixed points which are isolated invariant sets

TL;DR

This paper investigates the behavior of fixed point indices of iterates of homeomorphisms in three-dimensional space, establishing conditions under which these indices are periodic and constructing examples with prescribed periodic sequences.

Contribution

The paper proves that fixed point indices are periodic when the fixed point is an isolated invariant set and constructs homeomorphisms realizing any periodic sequence satisfying Dold's congruences.

Findings

Indices are periodic for isolated invariant fixed points.

Existence of homeomorphisms with prescribed periodic index sequences.

Application to local structure of stable/unstable sets.

Abstract

Let (U \subset {\mathbb R}^3) be an open set and (f:U \to f(U) \subset {\mathbb R}^3) be a homeomorphism. Let (p \in U) be a fixed point. It is known that, if (\{p\}) is not an isolated invariant set, the sequence of the fixed point indices of the iterates of (f) at (p), ((i(f^n,p))_{n\geq 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\geq 1}) is periodic. Conversely, we show that for any periodic sequence of integers ((I_n)_{n \geq1}) satisfying Dold's necessary congruences, there exists an orientation preserving homeomorphism such that (i(f^n,p)=I_n) for every (n\geq 1). Finally we also present an application to the study of the local structure of the stable/unstable sets at (p).