  In this article, we attempt to study the possible link between the dynamics
of a circle map and the caustics of its iterations. The attention is on a
geometrically defined off-center reflections, which, coincidentally, is also a
generalization of standard perturbation of rotation. Some results of the
caustics and dynamics of the family of maps are given. The method of symplectic
topology is particularly exploited to study the caustics because of its
potential in further development. The mode-locking behavior and width of
resonance zone are also studied. Furthermore, there are interesting problems
arised from experimental observations.

Dynamical Systems;Symplectic Geometry