Linked orbits of homeomorphisms of the plane and Gambaudo-Kolev Theorem

Abstract

Let h : R 2 → R 2 be an orientation preserving homeomorphism of the plane. For any bounded orbit O(x) = {h n (x) : n ∈ Z} there exists a fixed point x ∈ R 2 of h linked to O(x) in the sense of Gambaudo: one cannot find a Jordan curve C ⊆ R 2 around O(x), separating it from x , that is isotopic to h(C) in R 2 \ (O(x) ∪ {x }).

Braids and linked periodic orbits of disc homeomorphisms

“…Then Bonino showed that any periodic orbit of period at least 3 of an orientation-reversing homeomorphism of S 2 is linked with an orbit of period 2 [4]. Boroński recently generalized Gambaudo-Kolev Theorem to the case of bounded orbits linked with fixed points of orientation-preserving homeomorphisms of R 2 [5].…”