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

Abstract

Let U be an open subset of R2 and let f :U → R2 be a local homeomorphism. Let p € U be a non-repeller 4xed point of f suchth at {p} is an isolated invariant set. We introduce a particular class of index pairs for {p} that we call generalized 4ltration pairs. The computation of the 4xed point index of any iteration of f at p is quite easy once one knows a generalized 4ltration pair. The existence of
generalized 4ltration 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 4xed point index of any iteration of arbitrary local homeomorphisms.