The authors investigate billiards on convex polygons wherein collisions with the boundary of the billiard table are not simply specular, but instead the angle of reflection contracts toward the normal to the boundary. Previous work has shown that this leads to a finite number of ergodic Sinai-Ruelle-Bowen (SRB) measures whose support lies on generalized hyperbolic attractors.
In this paper the authors consider specifically the so-called slap map where the angle of reflection (with respect to the normal) is zero. In particular, they explore the relationship between those other SRB measures and ergodic absolutely continuous invariant probabilities of the one-dimensional slap map. Billiard dynamics with contacting reflection differ significantly from the dynamics with regular specular reflection. Ordinary specular reflection leads to non-hyperbolic systems.
When the angle of reflection \(f\) is identically zero, the billiard map is no longer injective; its image instead is a one-dimensional set. The restriction of the billiard map to this set, the slap map, is a piecewise affine map of the circle. For a polygon \(P\) and a contacting reflection \(f\), the authors study the function \(\Phi_{f,P}\), the billiard map of the reflection \(f\) on the polygon \(P\), to determine how close \(\Phi_{f,P}\) is to the slap map \(\Phi_{0,P}\) (as measured by a Lipschitz constant \(\lambda(f)\). The authors consider the question of the relationship between \(\Phi_{f, Q}\) and \(\Phi_{0, P}\) when \(\lambda(f)\) is small and polygon \(Q\) is close to polygon \(P\).
If the polygon \(P\) does not have parallel sides, then the corresponding slap map is uniformly expanding and admits a finite number of ergodic absolutely continuous invariant probabilities. Indeed the authors identify a broader condition on non-acute vertices that is generic in the space of polygons that guarantees the same results.