Fixed Point Theory and Applications
Volume 2010 (2010), Article ID 323069, 31 pages
doi:10.1155/2010/323069
Research Article

A Poincaré Formula for the Fixed Point Indices of the Iterates of Arbitrary Planar Homeomorphisms

Francisco R. Ruiz del Portal1 and José M. Salazar2

1Departamento de Geometría y Topología, Facultad de CC.Matemáticas, Universidad Complutense de Madrid, Madrid 28040, Spain
2Departamento de Matemáticas, Universidad de Alcalá, Alcalá de Henares, Madrid 28871, Spain

Received 11 November 2009; Accepted 1 March 2010

Academic Editor: Marlène Frigon

Copyright © 2010 Francisco R. Ruiz del Portal and José M. Salazar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let U⊂ℝ2 be an open subset and f:U→ℝ2 be an arbitrary local homeomorphism with Fix(f)={p}. We compute the fixed point indices of the iterates of f at p,iℝ2(fk,p), and we identify these indices in dynamical terms. Therefore, we obtain a sort of Poincaré index formula without differentiability assumptions. Our techniques apply equally to both orientation preserving and orientation reversing homeomorphisms. We present some new results, especially in the orientation reversing case.