Marked Length Spectrum, homoclinic orbits and the geometry of open dispersing billiards

TL;DR

This paper investigates whether the Marked Length Spectrum uniquely determines the geometry of open dispersing billiards with three convex obstacles, showing it can recover curvature and Lyapunov exponents of periodic orbits.

Contribution

It demonstrates that the Marked Length Spectrum allows for the recovery of curvature at period-two points and Lyapunov exponents in open dispersing billiards.

Findings

Marked Length Spectrum determines curvature at period-two points.

Marked Length Spectrum determines Lyapunov exponents of periodic orbits.

The inverse problem is solvable for the given billiard configuration.

Abstract

We consider billiards obtained by removing three strictly convex obstacles satisfying the non-eclipse condition on the plane. The restriction of the dynamics to the set of non-escaping orbits is conjugated to a subshift on three symbols that provides a natural labeling of all periodic orbits. We study the following inverse problem: does the Marked Length Spectrum (i.e., the set of lengths of periodic orbits together with their labeling), determine the geometry of the billiard table? We show that from the Marked Length Spectrum it is possible to recover the curvature at periodic points of period two, as well as the Lyapunov exponent of each periodic orbit.