P of the maximal invariant set contained in J. Roughly speaking, if a fixed point p does not lay in arbitrary small isolated compacta, we can consider any disc J containing p in its interior and take Kp, the component containing The main fact to obtain our results is the existence of special classes of filtration pairs in the Caratheodory's prime ends compactification that will allow us to by-pass the technical problem that occurs if the fixed point does not lay in an isolated invariant compactum. We will find with our methods the same formula for orientation preserving homeomorphisms and we shall solve the problem also for orientation reversing homeomorphisms. As we said above, when p does not belong to any isolated invariant compactum and the homeomorphism is orientation preserving, Le Calvez improved a result of Brown, see, showing that the sequence of indices is periodic. The computation of the fixed point index of any iteration of any planar homeomor-phism at an isolated fixed point laying in an isolated invariant compactum was done by the authors in. There are some papers dedicated to the study of the analogous problem in dimension 3. The authors, in, gave a stable/unstable "manifold" theorem for arbitrary planar homeomorphisms near a fixed point admitting nice filtration pairs. The results of Baldwin and Slaminka were improved by Le Roux, in, where the fixed point index is used not only to detect stable/unstable branches but also Leau-Fatou petals around p. 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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