Outer Billiards with Contraction: Attracting Cantor Sets

TL;DR

This paper studies outer billiards with contraction outside polygons, revealing conditions under which the system exhibits attracting Cantor sets or unique ergodicity, thus advancing understanding of complex dynamical behaviors in these systems.

Contribution

It introduces a parameterized family of outer billiards with contraction, demonstrating the existence of attracting Cantor sets and conditions for unique ergodicity.

Findings

Existence of attracting Cantor sets for certain parameters.

System can be uniquely ergodic with a single attractor.

Dynamics reduce to piecewise contraction on an interval.

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 is 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 existence of an attracting Cantor set for certain parameter values. Moreover, for a different choice of a 1-parameter family, we prove that the system is uniquely ergodic; in particular, the entire domain is asymptotic to a single attractor.