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

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Le Calvez, Patrice and Romero Ruiz del Portal, Francisco and Salazar, J. M. (2010) Indices of the iterates of R-3-homeomorphisms at fixed points which are isolated invariant sets. Journal of the london mathematical society-second series, 82 (3). pp. 683-696. ISSN 0024-6107

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Official URL: http://jlms.oxfordjournals.org/content/82/3/683.full.pdf



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.


Item Type:Article
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Dedicated to Professor Jose M. Montesinos on the occasion of his 65th birthday

Uncontrolled Keywords:Fixed point index; Dold’s congruences; Conley index; homeomorphism
Subjects:Sciences > Mathematics > Differential equations
Sciences > Mathematics > Topology
ID Code:18099
Deposited On:31 Jan 2013 09:46
Last Modified:19 Feb 2019 11:58

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