2025/02/04 by Andrey E. Mironov, Mironov, Andrey E., Siyao Yin +1 · 1 citation
Physics and Astronomy · #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.2502.01997
openalex publication_date 2025/02/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Recently it was proved that every billiard trajectory inside a C3 convex cone has a finite number of reflections. Here, by a C3 convex cone, we mean a cone whose section with some hyperplane is a strictly convex closed C3 submanifold of the hyperplane with nondegenerate second fundamental form. In this paper, we prove the existence of C2 convex cones admitting billiard trajectories with infinitely many reflections in finite time. We also estimate the number of reflections of billiard trajectories in elliptic cones in ℝ3 using two first integrals.