Off-Center Reflections: Caustics and Chaos

We study the properties of a particular one-parameter family of circle maps called offcenter reflections defined in §2. This map, in its 2-dimensional version, is first introduced in an open problem by S. T. Yau, [Y, problem 21], who suggests a cross study of the dynamics and geometry. In this article, we attempt to explore the possible link between the dynamics of this family of circle maps and their caustics. Although our study has not much contents in differential geometry as Yau expected, it reveals some interesting phenomena. For example, we observe and partially prove that in a certain generic range of the parameter, the caustics have exactly 4 cusp points for odd iterations; whereas for even iterations, each caustic is a curve tangential to the circle at exactly four points. This may not be the best result that one could state about the dynamics and the geometry of the map; nevertheless, we still put it forward in the hope that our study may invite better understanding to the subject. The off-center reflection also bares several interesting analytic forms. It is a Blaschke product restricted to the circle. Moreover, it has an infinite series expression highlighting that it is a perturbation of rotation on the circle. Since the work of Arnold, [A2], a standard type of perturbations has attracted much interests in mathematics and physics communities, [BBJ, Di, Z]. This standard type is exactly a reduction of the series of the off-center reflection. This adds more flavor to our study.