# Arnold Diffusion in Multi-Dimensional Convex Billiards

## TL;DR

This paper demonstrates that in higher-dimensional convex billiards, trajectories can asymptotically approach the boundary, contrasting with the two-dimensional case where Lazutkin proved this is impossible.

## Contribution

It extends Lazutkin's result by showing that Arnold diffusion techniques imply such boundary-approaching trajectories exist in three or more dimensions.

## Key findings

- Existence of boundary-approaching trajectories in higher dimensions.
- Use of Arnold diffusion to prove genericity of these trajectories.
- Contrast with two-dimensional billiard dynamics.

## Abstract

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.