Topological conjugacy of disjoint flows on the circle

Abstract

Let {mathcal F}={F^{v}:s longrightarrow s ,:: vin V} and{mathcal G}={G^{v}:s longrightarrow s ,:: vin V} be disjoint flowsdefined on the unit circle s, that is such flows that each theirelement either is the identity mapping or has no fixed point ((V,; +)is a 2-divisible nontrivial abelian group). The aim of this paper isto give a necessary and sufficient condition for topological conjugacyof disjoint flows i.e., the existence of a homeomorphismGamma :s longrightarrow s satisfying[Gamma circ F^{v}=G^{v}circ Gamma ,;;;;; vin V.]Moreover, under some further restrictions, we determine all suchhomeomorphisms.