A CAT(0)-approach to the marked length spectral rigidity of Sinai billiards

TL;DR

This paper proves that the geometry of Sinai billiards with finite horizon can be uniquely determined from an enriched marked length spectrum, extending spectral rigidity results to non-smooth dynamical systems using CAT(0) space techniques.

Contribution

It introduces an enriched marked length spectrum for Sinai billiards and proves spectral rigidity by adapting methods from negatively curved surfaces to CAT(0) spaces.

Findings

Two Sinai billiards with the same enriched spectrum are isometric.

The approach extends spectral rigidity to discontinuous dynamical systems.

Approximation by geodesic flows on CAT(0) spaces is effective.

Abstract

We study the spectral rigidity problem for Sinai billiards with finite horizon, specifically asking whether the geometry of the billiard table can be recovered from the lengths of its (marked) periodic trajectories. To address this, we introduce an enriched marked length spectrum EL and prove that two Sinai billiards sharing the same EL must be isometric. Our approach involves approximating the billiard flow using geodesic flows on smooth Riemannian surfaces. In the limit, these flows converge to CAT(0) spaces, which encode both the lengths of periodic orbits and the geometry of the boundary. We adapt Otal's original method -- developed for marked length spectrum rigidity in negatively curved surfaces -- to this new setting. Here, the lack of curvature control is offset by metric comparison estimates. By integrating the analysis of geodesic flows with perturbative techniques for periodic orbits, we establish a rigidity theorem for Sinai billiards with finite horizon. These results extend the classical theory of marked length spectrum rigidity beyond the Riemannian setting, demonstrating that even in discontinuous dynamical systems, geometric information is rigidly encoded in spectral data.