2019/06/18 by Andrew Clarke, Clarke, Andrew, Dmitry Turaev +1
Mathematics · Physics and Astronomy · #Dynamical Systems (math.DS) #FOS: Mathematics #Geometry and complex manifolds #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.1906.07778
openalex publication_date 2019/06/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30
Consider billiard dynamics in a strictly convex domain, and consider a trajectory that begins with the velocity vector making a small positive angle with the boundary. Lazutkin proved that in two dimensions, it is impossible for this angle to tend to zero along trajectories. We prove that such trajectories can exist in higher dimensions. Namely, using the geometric techniques of Arnold diffusion, we show that in three or more dimensions, assuming the geodesic flow on the boundary of the domain has a hyperbolic periodic orbit and a transverse homoclinic, the existence of trajectories asymptotically approaching the billiard boundary is a generic phenomenon in the real-analytic topology.