Inverse problems and rigidity questions in billiard dynamics

Abstract

A Birkhoff billiard is a system describing the inertial motion of a point mass inside a strictly convex planar domain, with elastic reflections at the boundary. The study of the associated dynamics is profoundly intertwined with the geometric properties of the domain: while it is evident how the shape determines the dynamics, a more subtle and difficult question is the extent to which the knowledge of the dynamics allows one to reconstruct the shape of the domain. This translates into many intriguing inverse problems and unanswered rigidity questions, which have been the focus of very active research in recent decades. In this paper we describe some of these questions, along with their connection to other problems in analysis and geometry, with particular emphasis on recent results obtained by the authors and their collaborators.

On the persistence of periodic tori for symplectic twist maps and the rigidity of integrable twist maps

“…Let us mention that there are different possible notions of integrability: one that allows singularities of the invariant foliation, that we will refer to as integrability, and the other that requires a regular foliation everywhere, that we call complete integrability. Among the many instances of questions for integrability and complete integrability, we recall: the Birkhoff conjecture in billiard dynamics (i.e., the only 2-dimensional billiard tables for which the corresponding dynamics is integrable, are elliptic ones, see [22] for a survey and for more references; this question in the case of complete integrability has been solved in [7]), the problem of characterizing integrable Riemannian geodesic flows on the d-dimensional torus (see for example [8,10,25] and references therein for the integrability problem and [12] for complete integrability, which is related to the so-called Hopf conjecture), or the problem of characterizing (unbounded) domains of the plane where the N-vortex problem is integrable (see [16] for a first step in the case of 2-vortex).…”

Finiteness theorems on elliptical billiards and a variant of the dynamical Mordell–Lang conjecture

“…One could ask what would happen by imposing further assumptions on such billiards, for example, integrability conditions. However, this would easily lead again to elliptical billiards, according to a conjecture of Birkhoff (see [33]). Diophantine content.…”