Jacopo De Simoi (University of Toronto)
Given a planar domain with sufficiently regular boundary, one
can study periodic orbits of the associated billiard problem. Periodic
orbits possess a rich and intricate structure: it is then natural to ask
how much information about the domain is encoded in the set of lengths
of such orbits. The quantum analog of this question is the celebrated
Laplace inverse problem, or “Can one hear the shape of a drum?”
We prove Marked Dynamical Spectral Determination among
ℤ₂-symmetric smooth convex domains close to the disk: if any two such
domains have the same Marked Length Spectrum, they must necessarily be
isometric domains. This substantially improves the deformational result
obtained in a prior work with Kaloshin and Wei.