# Marked Length Spectral determination of analytic chaotic billiards with axial symmetries

## TL;DR

This paper demonstrates that, under certain symmetry and genericity conditions, the marked length spectrum uniquely determines the shape of analytic chaotic billiards with convex obstacles, linking spectral data to geometric structure.

## Contribution

It establishes the spectral rigidity of analytic chaotic billiards with axial symmetries, showing the marked length spectrum determines the billiard table's geometry.

## Key findings

- Marked length spectrum determines billiard geometry
- Conjugation to a subshift labels periodic orbits
- Results hold under symmetry and genericity assumptions

## Abstract

We consider billiards obtained by removing from the plane finitely many strictly convex analytic obstacles satisfying the non-eclipse condition. The restriction of the dynamics to the set of non-escaping orbits is conjugated to a subshift, which provides a natural labeling of periodic orbits. We show that under suitable symmetry and genericity assumptions, the Marked Length Spectrum determines the geometry of the billiard table.