Off-Center Reflections: Caustics and Chaos

Abstract

We study the possible link between the dynamics of a certain family of circle maps and the caustics of their iterates. The maps are defined by off-center reflections in a mirrored circle; they can calso be regarded as perturbed rotations. Some of our experimental observations can be justified rigorously; for example, a lower bound is given for the number of cusps and the mode-locking behavior are studied. Symplectic topology is a particularly useful tool in this study.