For planar billiard tables, the marked length spectrum encodes the lengths
of action (minus the length) minimizing orbits of a given rational rotation
number. For strictly convex tables, a renormalization of these lengths extends
to a continuous function called Mather’s beta function or the mean minimal
action. We show that using the algebraic structure of its Taylor coefficients,
one can prove C infinity compactness of marked length isospectral sets. This
gives a dynamical counterpart to the Laplace spectral results of Melrose,
Osgood, Phillips and Sarnak.