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Abstract. 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 Carathéodory prime ends and fixed point index. The result is applicable to some concrete problems in the theory of periodic differential equations.

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.

This article includes an almost self-contained exposition on the discrete Conley index and its duality. We work with a locally defined homeomorphism f in R d and an acyclic continuum X, such as a cellular set or a fixed point, invariant under f and isolated. We prove that the trace of the first discrete homological Conley index of f and X is greater than or equal to -1 and describe its periodical behavior. If equality holds then the traces of the higher homological indices are 0. In the case of orientation-reversing homeomorphisms of R 3 , we obtain a characterization of the fixed point index sequence {i(f n , p)} n≥1 for a fixed point p which is isolated as an invariant set. In particular, we obtain that i(f, p) ≤ 1. As a corollary, we prove that there are no minimal orientation-reversing homeomorphisms in R 3 .

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