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Inverse problems and rigidity questions in billiard dynamics

Part of: Partial differential equations on manifolds; differential operators Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems Low-dimensional dynamical systems

Published online by Cambridge University Press:  31 May 2021

VADIM KALOSHIN Open the ORCID record for VADIM KALOSHIN [Opens in a new window]  and
ALFONSO SORRENTINO Open the ORCID record for ALFONSO SORRENTINO [Opens in a new window]
VADIM KALOSHIN*
Affiliation:
Department of Mathematics, University of Maryland, College Park, MD, USA
ALFONSO SORRENTINO
Affiliation:
Dipartimento di Matematica, Università degli Studi di Roma ‘Tor Vergata’, Rome, Italy (e-mail: sorrentino@mat.uniroma2.it)
*
e-mail: vadim.kaloshin@gmail.com
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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.

Keywords

mathematical billiardsintegrable billiardsBirkhoff conjecturespectral rigiditylength spectrum

MSC classification

Secondary: 37E40: Twist maps 37J35: Completely integrable systems, topological structure of phase space, integration methods 58J50: Spectral problems; spectral geometry; scattering theory
Type
Original Article
Information
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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Footnotes

To the memory of Anatole Katok (1944–2018)

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