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Inverse problems and rigidity questions in billiard dynamics
Part of:
Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems
Low-dimensional dynamical systems
Partial differential equations on manifolds; differential operators
Published online by Cambridge University Press: 31 May 2021
Abstract
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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.
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- Ergodic Theory and Dynamical Systems , Volume 42 , Issue 3: Anatole Katok Memorial Issue Part 2: Special Issue of Ergodic Theory and Dynamical Systems , March 2022 , pp. 1023 - 1056
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- © The Author(s), 2021. Published by Cambridge University Press
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To the memory of Anatole Katok (1944–2018)
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