Indices of the iterates of R-3-homeomorphisms at fixed points which are isolated invariant sets
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 gen...
| Authors: | , , |
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| Format: | article |
| Publication Date: | 2010 |
| Country: | España |
| Institution: | Universidad Complutense de Madrid (UCM) |
| Repository: | Docta Complutense |
| Language: | English |
| OAI Identifier: | oai:docta.ucm.es:20.500.14352/42535 |
| Online Access: | https://hdl.handle.net/20.500.14352/42535 |
| Access Level: | Open access |
| Keyword: | 517.9 515.1 Fixed point index Dold’s congruences Conley index homeomorphism Ecuaciones diferenciales Topología 1202.07 Ecuaciones en Diferencias 1210 Topología |
| Summary: | 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. |
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