On the other hand, Baldwin and Slaminka, in, dealt with the problem of relating the fixed point index of an orientation and area preserving homeomorphism around an isolated fixed point p and the number of branches in which the stable/unstable "manifold" of p decomposes. It was introduced by Perez-Marco in and it was used more recently by the first author, in, to prove that the index of arbitrary stable planar fixed points is equal to 1. The idea of applying the compactification of Caratheodory to study planar dynamical problems is not new. Le Calvez, in, uses in a very clever way the nice Caratheodory's prime ends theory (see ). Again the fixed point indices of the iterations of the homeomorphism have periodical behavior. Later, Le Calvez extended his theorem with Yoccoz to arbitrary isolated fixed points of orientation preserving planar homeomorphisms. Let U c R2 be an open subset and f : U ^ R2 be an arbitrary local homeomorphism with Fix(f) = and q such that 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. Ruiz del Portal, Received 11 November 2009 Accepted 1 March 2010 Academic Editor: Marlene FrigonĬopyright © 2010 F. Salazar2ġĝepartamento de Geometría y Topología, Facultad de CC.Matematicas, Universidad Complutense de Madrid, Madrid 28040, SpainĢĝepartamento de Matematicas, Universidad de Alcala, Alcala de Henares, Madrid 28871, SpainĬorrespondence should be addressed to Francisco R. Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010, Article ID 323069,31 pages doi:10.1155/2010/323069Ī Poincare Formula for the Fixed Point Indices of the Iterates of Arbitrary Planar Homeomorphismsįrancisco R.
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