Structure and evolution of strange attractors in non-elastic triangular billiards

Abstract

We study pinball billiard dynamics in an equilateral triangular table. In such dynamics, collisions with the walls are non-elastic: the outgoing angle with the normal vector to the boundary is a uniform factor $λ< 1$ smaller than the incoming angle. This leads to contraction in phase space for the discrete-time dynamics between consecutive collisions, and hence to attractors of zero Lebesgue measure, which are almost always fractal strange attractors with chaotic dynamics, due to the presence of an expansion mechanism. We study the structure of these strange attractors and their evolution as the contraction parameter $λ$ is varied. For $λ$ in the interval (0, 1/3), we prove rigorously that the attractor has the structure of a Cantor set times an interval, whereas for larger values of $λ$ the billiard dynamics gives rise to nonaccessible regions in phase space. For $λ$ close to 1, the attractor splits into three transitive components, the basins of attraction of which have fractal basin boundaries.