Rigidity of Mather's $β$-function on a KAM set for analytic billiards-like maps and unique quasi-analytic continuation

In his seminal paper, Kac famously asked whether "one can hear the shape of a drum" - that is whether the isometry class of a bounded domain in a Euclidean space is uniquely determined by the spectrum of its Laplace spectrum. The Laplace spectrum is closely related to the length spectrum of the associated billiard. For a convex bounded planar domain, to each billiard periodic orbit one can associate not only its length but also its rotation number. The set of pairs of length and rotation number of each periodic orbit is called the marked length spectrum. Using the marked length spectrum of the domain one can associate with it its minimal action function also known as Mathers $β$-function denoted by $β_Ω$. Via unique quasianalytic continuation, we prove the following rigidity problem: knowing that Mather's $β$-functions of two billiards in two analytic planar domains coincide on a set of positive measure of KAM diophantine numbers implies that they coincide on a set of all KAM diophantine numbers. In particular, knowing that Mather's $β$-functions of two domains coincide on a positive measure set of KAM diophantine numbers implies that the Marvizi-Melrose invariants of these domains coincide.