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

Let U be an open subset of R2 and let f : U→R2 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.