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Indices of the iterates of R3-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 R3-homeomorphisms at fixed points which are isolated invariant sets. Journal of the London Mathematical Society. Second Series, 82 (2). pp. 683-696. ISSN 0024-6107

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



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.


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

Subjects:Sciences > Mathematics > Differential equations
Sciences > Mathematics > Topology
ID Code:18167
Deposited On:07 Feb 2013 11:24
Last Modified:20 Feb 2019 10:25

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