Fixed Point Theorem for Non-Self Maps of Regions in the Plane

Abstract

Let X and Y be compact, simply connected and locally connected subsets of R^2, and let f : X -> Y be a homeomorphism isotopic to the identity on X. Generalizing Brouwer's plane translation theorem for self-maps of the plane, we prove that f has no recurrent (in particular, no periodic) points if it has no fixed points.

11 pages, 5 figures; significant improvement and generalization of the result in the first version ("Jordan domains" replaced by "compact, simply connected, locally connected sets" in the assumptions of the main result)