New York Journal of Mathematics
Volume 9 (2003) 55-68


David Richeson and Jim Wiseman

Bounded homeomorphisms of the open annulus

Published: April 2, 2003
Keywords: Annulus, Poincaré-Birkhoff theorem, twist map, fixed point, nonwandering set, periodic point, rotation number
Subject: Primary 37E40; Secondary 37E45, 54H25

We prove a generalization of the Poincaré-Birkhoff theorem for the open annulus showing that if a homeomorphism satisfies a certain twist condition and the nonwandering set is connected, then there is a fixed point. Our main focus is the study of bounded homeomorphisms of the open annulus. We prove a fixed point theorem for bounded homeomorphisms and study the special case of those homeomorphisms possessing at most one fixed point. Lastly we use the existence of rational rotation numbers to prove the existence of periodic orbits.


The second author was partially supported by the Swarthmore College Research Fund.

Author information

David Richeson:
Dickinson College, Carlisle, PA 17013

Jim Wiseman:
Swarthmore College, Swarthmore, PA 19081