CHAOTIC LENSED BILLIARDS

Lensed billiards are an extension of the notion of billiard dynamical systems obtained by adding a potential function of the form C1_A, where C is a real-valued constant and 1_A is the indicator function of an open subset A of the billiard table whose boundaries (of A and the table) are piecewise smooth. Trajectories are polygonal lines that undergo either reflection or refraction at the boundary of A depending on the angle of incidence. Our main focus is to explore how the dynamical properties of these models 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.