Fixed point theorem for non-self maps of regions in the plane

Let X⊂R2 be a compact, simply connected, locally connected set, and let f:X→Y⊂R2 be a homeomorphism isotopic to the identity on X. Generalizing Brouwerʼs 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.