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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Let f be a homeomorphism of A¯, the closed annulus, isotopic to the identity and let X be a closed f‐invariant subset of A¯ whose complement is homeomorphic to the half‐open annulus. The dynamics in the circle of prime ends of the complement of X has an associated rotation number, ρ. We prove that either the rotation number of all forward semi‐orbits of accessible points of X are well defined and equal to ρ or the rotation number of all backward semi‐orbits of accessible points of X is well defined and equal to ρ. The result generalizes to the open and half‐open annulus provided the semi‐orbits under consideration are relatively compact.

We prove that any small enough neighborhood of a closed contact submanifold is always tight under a mild assumption on its normal bundle. The non-existence of C 0 -small positive loops of contactomorphisms in general overtwisted manifolds is shown as a corollary.

Matsumoto proved in [M12] that the prime end rotation numbers associated to an invariant annular continuum are contained in its rotation set. An alternative proof of this fact using only simple planar topology is presented.