When does the (Marked) Length Spectrum determine geometry of the billiard table?

We study the billiard on the plane ask: does the (Marked) Length Spectrum, i.e., the set of lengths of periodic orbits (together with their labeling), determine the geometry of the billiard table? This question is closely related to the well-known question: "Can you hear the shape of a drum?" We report two results for planar domains having certain symmetry and analytic boundary. First, we consider billiards obtained by removing from the plane three strictly convex analytic obstacles satisfying the non-eclipse condition and a suitable symmetry. We show that under a non-degeneracy assumption, the Marked Length Spectrum determines the geometry of the billiard table. This is a joint work with J. De Simoi and M. Leguil. Second, we consider billiards inside of a strictly convex planar domain having certain symmetry. We show that under a non-degeneracy assumptions, the Length Spectrum determines the geometry of the billiard table. This is a joint work with M. Leguil and K. Zhang. These results are analogous to results of Colin de Verdière, Zelditch and Iantchenko-Sjöstrand-Zworski in terms of the (Marked) Length Spectrum.