Stefano Baranzini (Università San Raffaele Roma)
In this talk I will discuss recent pointwise spectral rigidity results for several billiard systems, including Birkhoff billiards and symplectic billiards. The main theme is that, in many cases, a single value of Mather’s beta-function is enough to determine whether the domain is a disk (or an ellipse in the affine-invariant setting of symplectic billiards). This provides a “one-frequency” analogue of the classical question “Can one hear the shape of a billiard?”
The results are based on isoperimetric-type inequalities comparing the beta-function of a convex domain with that of a disk having the same perimeter or area, depending on the model. I will explain how equality in these inequalities leads to rigidity, and how this viewpoint gives a dynamical reinterpretation of classical geometric inequalities for extremal polygons. Time permitting, I will also discuss a contrasting phenomenon for outer billiards, where such rigidity fails.
This is a joint work with M. Bialy and A. Sorrentino.