Can you hear the shape of a chaotic billiard?

In an ongoing project with J. De Simoi, V. Kaloshin, and P. Bálint, we have been studying the question of spectral rigidity for a class of dispersing billiards. For such billiard tables, there is a natural symbolic coding of the set of periodic orbits, and we wonder how much geometric information the Marked Length Spectrum (i.e., the set of lengths of all periodic orbits together with their marking) conveys. One direction we have been investigating is whether it is possible to recover some local geometric information near period two orbits without assuming symmetries. Moreover, we will see in this talk how it is generically possible, in the analytic category, to recover the geometry of such dispersing billiards with some symmetries from the purely dynamical data encoded in their Marked Length Spectrum.