Let U ⊂ ℝ3 be an open set and f : U → f(U) ⊂ ℝ3 be a homeomorphism. Let p ∈ U be a fixed point. It is known that if {p} is not an isolated invariant set, then the sequence of the fixed‐point indices of the iterates of f at p, (i(fn, p))n ⩾ 1, is, in general, unbounded. The main goal of this paper is to show that when {p} is an isolated invariant set, the sequence (i(fn, p))n ⩾ 1 is periodic. Conversely, we show that, for any periodic sequence of integers (In)n ⩾ 1 satisfying Dold's necessary congruences, there exists an orientation‐preserving homeomorphism such that i(fn, p) = In for every n ⩾ 1. Finally we also present an application to the study of the local structure of the stable/unstable sets at p.

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Let "Equation missing" be an open subset and "Equation missing" be an arbitrary local homeomorphism with "Equation missing". We compute the fixed point indices of the iterates of "Equation missing" at "Equation missing", 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Let X be a locally compact metric absolute neighbourhood retract for metric spaces, U 9 X be an open subset and f : U ,-X be a continuous map. The aim of the paper is to study the fixed point index of the map that f induces in the hyperspace of X. For any compact isolated invariant set, K 9 U, this fixed point index produces, in a very natural way, a Conley-type (integer valued) index for K. This index is computed and it is shown that it only depends on what is called the attracting part of K. The index is used to obtain a characterization of isolating neighbourhoods of compact invariant sets with non-empty attracting part. This index also provides a characterization of compact isolated minimal sets that are attractors.

Abstract. 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 point. Introduction and preliminary definitionsThe stable and unstable manifold theorem for hyperbolic diffeomorphisms plays a very important role in differential dynamics. At the topological level Baldwin and Slaminka proved, in [1], a stable/unstable manifold theorem for area-and orientationpreserving homeomorphisms of orientable 2-manifolds having isolated fixed points of index less than 1.There are many papers in the literature relating the fixed-point index of a homeomorphism f in a neighborhood of an isolated fixed point, and the local dynamical behavior of f . There are results in both directions, i.e. bounds (or explicit computation) for the fixed-point index from dynamical properties of f and how the knowledge of the fixed-point index can be applied to describe the dynamics locally. We will mention besides and Shub and Sullivan [22], for their relations with the present paper. Their results frequently deal with orientation-preserving homeomorphisms. The main reason for this assumption is the application of some version of Brouwer's lemma on translation arcs, see [5] or [7].The aim of this paper is to use the fixed-point index to obtain information about the dynamical behavior of a planar local homeomorphism in a neighborhood of an