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Isolating blocks of isolated invariant continua and fixed point index

  • Francisco R. Ruiz del PortalEmail author
  • José M. SalazarEmail author
Article
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Abstract

In this paper we study some properties of isolated invariant continua for arbitrary homeomorphisms of \({{\mathbb {R}}}^n\). We study the existence of special isolating blocks for them which allow us to compute the fixed point indices of the iterates of arbitrary homeomorphisms at arbitrary isolated continua in dimension two. Among the consequences we would highlight the following:
  • If \(K \subset {{\mathbb {R}}}^2\) is an isolated invariant continuum that decomposes the plane in more than two connected components, then K contains a periodic orbit.

  • A proper invariant continuum of the 2-sphere containing the set of periodic orbits of a homeomorphism is not isolated.

  • If K is an isolated invariant continuum for \(f: S^n \rightarrow S^n\), then \(S^n {\setminus } K\) has a finite amount of connected components.

Keywords

Isolated invariant continua isolating block fixed point index Conley index 

Mathematics Subject Classification

37C25 37B30 54H20 54H25 

Notes

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Departamento de Geometría y Topología UniversidadComplutense de MadridSpain
  2. 2.Departamento de Física y Matemáticas Universidad de AlcaláAlcalá de HenaresSpain

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