Chaotic lensed billiards

Abstract

Lensed billiards are an extension of the notion of billiard dynamical systems obtained by adding a potential function of the form $C1_{\mathcal{A}}$, where $C$ is a real valued constant and $1_{\mathcal{A}}$ is the indicator function of an open subset $\mathcal{A}$ of the billiard table whose boundaries (of $\mathcal{A}$ and the table) are piecewise smooth. Trajectories are polygonal lines that undergo either reflection or refraction at the boundary of $\mathcal{A}$ depending on the angle of incidence. After laying out some basic concepts and general facts, in particular reviewing the optical/mechanical analogy that motivates these billiard models, we explore how their dynamical properties depend on the potential parameter $C$ using a number of families of examples. In particular, we explore numerically the Lyapunov exponents for these parametric families and highlight the more salient common properties that distinguish them from standard billiard systems. We further justify some of these properties by characterizing lensed billiards in terms of switching dynamics between two open (standard) billiard subsystems and obtaining mean values associated to orbit sojourn in each subsystem.