  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.
