Compactness of Marked Length Isospectral Sets of Birkhoff Billiard Tables

TL;DR

This paper proves that the set of Birkhoff billiard tables with the same marked length spectrum is compact in the smooth topology, using hierarchical integral invariants and Hamiltonian analysis, with implications for spectral geometry.

Contribution

It establishes the compactness of marked length isospectral Birkhoff billiard tables and introduces a hierarchical structure for integral invariants related to the shape of billiard tables.

Findings

Equivalence classes of isospectral billiard tables are compact in the $C^ abla$ topology.

Hierarchical structure for integral invariants derived from caustic length expansion.

Proof of compactness for Laplace isospectral sets of convex billiard tables.

Abstract

We prove that equivalence classes of marked length isospectral Birkhoff billiard tables are compact in the $C^\infty$ topology, analogous to the Laplace spectral results of Melrose, Osgood, Phillips and Sarnak. To do so, we derive a hierarchical structure for the integral invariants of Marvizi and Melrose, or equivalently the coefficients of a caustic length-Lazutkin parameter expansion, which are in turn algebraically equivalent to the Taylor coefficients of Mather's $\beta$ function (also called the mean minimal action). Under a generically satisfied noncoincidence condition, these are also Laplace spectral invariants and can be used to hear the shape of certain drumheads. As a byproduct, we obtain an independent proof of the compactness of Laplace isospectral sets for strictly convex planar billiard tables. The proof of the structure theorem uses an interpolating Hamiltonian for nearly glancing billiard orbits and some analytic number theory to compute its Taylor coefficients.