Local Fixed Point Indices of Iterations of Planar Maps

  • Grzegorz Graff
  • Piotr Nowak-Przygodzki
  • Francisco R. Ruiz del Portal
Open Access


Let \({f: U\rightarrow {\mathbb R}^2}\) be a continuous map, where U is an open subset of \({{\mathbb R}^2}\). We consider a fixed point p of f which is neither a sink nor a source and such that {p} is an isolated invariant set. Under these assumption we prove, using Conley index methods and Nielsen theory, that the sequence of fixed point indices of iterations \({\{{\rm ind}(f^n,p)\}_{n=1}^\infty}\) is periodic, bounded from above by 1, and has infinitely many non-positive terms, which is a generalization of Le Calvez and Yoccoz theorem (Annals of Math., 146, 241–293 (1997)) onto the class of non-injective maps. We apply our result to study the dynamics of continuous maps on 2-dimensional sphere.


Fixed point index Conley index Nielsen number Periodic points Iterations 

Mathematics Subject Classification (2000)

Primary 37C25 Secondary 37E30 37B30 


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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Grzegorz Graff
    • 1
  • Piotr Nowak-Przygodzki
    • 1
  • Francisco R. Ruiz del Portal
    • 2
  1. 1.Faculty of Applied Physics and MathematicsGdansk University of TechnologyGdanskPoland
  2. 2.Departamento de Geometría y Topología Facultad de CC.MatemáticasUniversidad Complutense de MadridMadridSpain

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