# Polyhedral billiards, eigenfunction concentration and almost periodic control

## TL;DR

This paper investigates the dynamics of billiard flows in convex polyhedra, establishes finiteness and length estimates for periodic tubes, and applies these results to eigenfunction concentration, introducing new control estimates for almost-periodic boundary conditions.

## Contribution

It extends known results on billiard dynamics to higher dimensions, provides quantitative estimates for periodic tubes, and introduces a novel control estimate for almost-periodic boundary conditions.

## Key findings

- Finitely many immersed periodic tubes exist outside pockets.
- Quantitative estimates for lengths of periodic tubes.
- Eigenfunction mass concentrates near pockets in convex polyhedral billiards.

## Abstract

We study dynamical properties of the billiard flow on convex polyhedra away from a neighbourhood of the non-smooth part of the boundary, called ``pockets''. We prove there are only finitely many immersed periodic tubes missing the pockets and moreover establish a new quantitative estimate for the lengths of such tubes. This extends well-known results in dimension $2$. We then apply these dynamical results to prove a quantitative Laplace eigenfunction mass concentration near the pockets of convex polyhedral billiards. As a technical tool for proving our concentration results on irrational polyhedra, we establish a control-theoretic estimate on a product space with an almost-periodic boundary condition. This extends previously known control estimates for periodic boundary conditions, and seems to be of independent interest.