You can hear the shape of a billiard table: Symbolic dynamics and rigidity for flat surfaces

TL;DR

This paper characterizes how the shape of a Euclidean polygon influences its billiard flow dynamics, showing that only right-angled tables related by affine maps share the same bounce spectrum, using a new flat cone metric theorem.

Contribution

It introduces a complete characterization linking polygon shape and billiard dynamics, and proves flat cone metrics are determined by their Liouville current support.

Findings

Only right-angled tables related by affine maps share the same bounce spectrum

Flat cone metrics are uniquely determined by the support of their Liouville current

Provides a new theorem connecting flat cone metrics and Liouville currents

Abstract

We give a complete characterization of the relationship between the shape of a Euclidean polygon and the symbolic dynamics of its billiard flow. We prove that the only pairs of tables that can have the same bounce spectrum are right-angled tables that differ by an affine map. The main tool is a new theorem that establishes that a flat cone metric is completely determined by the support of its Liouville current.