Martin Leguil, University of Toronto
On the spectral rigidity of chaotic billiards
Abstract: The question of dynamical spectral rigidity has been investigated in a range of settings: for hyperbolic surfaces, convex domains with symmetries... In a joint work with Péter Bálint, Jacopo De Simoi and Vadim Kaloshin, we study a class of dispersing billiards on the plane obtained by removing strictly convex obstacles satisfying the non-eclipse condition. In this case, there is a natural labeling of periodic orbits, and we want to know how much geometric information can be recovered from the Marked Length Spectrum, i.e., the set of lengths of periodic orbits together with their labeling. In particular, we show that for each period two orbit, the curvature at the two bouncing points can be reconstructed. We also show that the Lyapunov exponent of each periodic orbit can be recovered. Our approach is based on a sequence of periodic orbits with symmetries which shadow more and more closely some orbit homoclinic to the periodic orbit that we want to describe. In the case of period two orbits, we will explain how it is possible to extract some asymmetric information to distinguish between the two points in the orbit.