On parallelizability of flows of free mappings

Summary.

We consider a flow of free mappings defined on the plane. We prove that the boundary of every equivalence class of a certain equivalence relation is a union of orbits and that at most two of the boundary orbits of a class can be contained in this class. The main result says that a flow of free mappings restricted to each equivalence class of the relation is conjugate with a flow of translations. As a corollary we obtain the fact that the iterates of each arc contained in an equivalence class tend to infinity.