On Length Spectrum Rigidity of Dispersing Billiard Systems

TL;DR

This paper investigates the rigidity of dispersing billiard systems with three convex scatterers, showing that the marked length spectrum uniquely determines the system's geometry under certain conditions.

Contribution

It establishes a rigidity result linking the marked length spectrum to the geometric configuration of scatterers in dispersing billiards.

Findings

Marked length spectrum rigidity is proven for these billiard systems.

Two systems are conjugate if their marked length spectra match near a homoclinic orbit.

The third scatterer is uniquely determined by the other two and the length spectrum.

Abstract

We consider the class of dispersing billiard systems in the plane formed by removing three convex analytic scatterers satisfying the non-eclipse condition. The collision map in this system is conjugated to a subshift, providing a natural labeling of periodic points. We study the problem of marked length spectrum rigidity for this class of systems. We show that two such systems have the same marked length spectrum if and only if their collision maps are analytically conjugate to each other near a homoclinic orbit and that two scatterers and the marked length spectrum together uniquely determine the third scatterer.