Francisco Romero Ruiz del Portal

Fixed point index of iterations of local homeomorphisms of the plane: a Conley index approach

Abstract Let U be an open subset of R 2 and let f : U→ R 2 be a local homeomorphism. Let p∈U be a non-repeller fixed point of f such that {p} is an isolated invariant set. We introduce a particular class of index pairs for {p} that we call generalized filtration pairs. The computation of the fixed point index of any iteration of f at p is quite easy once one knows a generalized filtration pair. The existence of generalized filtration pairs provides a short and elementary proof of a theorem of P. Le Calvez and J.C. Yoccoz (Ann. of Meth. 146 (1997) 241–293), and it also allows to compute the fixed point index of any iteration of arbitrary local homeomorphisms.

Attractors with vanishing rotation number

Given an orientation-preserving homeomorphism of the plane, a rotation number can be associated with each locally attracting fixed point. Assuming that the homeomorphism is dissipative and the rotation number vanishes we prove the existence of a second fixed point. The main tools in the proof are Caratheodory prime ends and fixed point index. The result is applicable to some concrete problems in the theory of periodic differential equations.

A note about the shape of attractors of discrete semidynamical systems

We state in a short way a result that improves one of the main theorems in a paper of M. Gobbino concerning the topological properties that the phase space induces in an attractor of a discrete dynamical system.

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

Let be an open subset and be an arbitrary local homeomorphism with . We compute the fixed point indices of the iterates of at , 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.

A stable/unstable manifold theorem for local homeomorphisms of the plane

We use a notion (introduced in Topology 41 (2002), 1119–1212), which is stronger than the concept of filtration pair, to prove a stable/unstable manifold general theorem for local homeomorphisms of the plane in a neighborhood of an isolated fixed.

Fixed point indices of planar continuous maps

We characterize the sequences of fixed point indices

About the homological discrete Conley index of isolated invariant acyclic continua

\{i(f^n, p)\}_{n\ge 1}

Attractors with irrational rotation number

of fixed points that are isolated as an invariant set for a continuous map

Generalized degree in normed spaces

f

Uniqueness of dynamical zeta functions and symmetric products

in the plane. In particular, we prove that the sequence is periodic and