Outer Billiards with Contraction: Attracting Cantor Sets

Abstract

We consider the outer billiards map with contraction outside polygons. We construct a 1-parameter family of systems such that each system has an open set in which the dynamics are reduced to that of a piecewise contraction on the interval. Using the theory of rotation numbers, we deduce that every point inside the open set is asymptotic to either a single periodic orbit (rational case) or a Cantor set (irrational case). In particular, we deduce the existence of an attracting Cantor set for certain parameter values. Moreover, for a different choice of 1-parameter family, we prove that the system is uniquely ergodic; in particular, the entire domain is asymptotic to a single attractor.

2000 AMS Subject Classification::

• 37E05
• 37E45

Keywords:

• rotation number
• unique ergodicity
• dynamics on the interval