Jacopo de Simoi, University of Toronto
Given a (convex or not) billiard table, we can record the lengths of all periodic orbits in a set called the “Length spectrum”; we can then ask how much of the Geometry of the domain is encoded in the Length Spectrum. This question is tightly related to the analogous (quantum) question for the Spectrum of the Laplace operator, that is known as “Can one hear the shape of a drum?”.
It is known that a marking of the length spectrum (i.e. knowing “which” orbit corresponds to “which” length) allows to gather lots of dynamical and geometrical information. In the first part of this talk I will show what is possible to do for some class of dispersing billiards (three obstacles systems). We will see how to retrieve Lyapunov exponents of periodic orbits and, in some cases, the whole geometry of the domain. This part is joint work with M. Leguil and V. Kaloshin.
In the second part of the talk we try to understand to which extent can such results be obtained without a marking. Does the length spectrum has any structure that can be used to recover a marking? The task seems to be quite intricate: we will construct (a dense set of) smooth convex billiard domains with a very degenerate (uncountable) Length Spectrum.