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Billiard Tables with Rotational Symmetry

Billiard Tables with Rotational Symmetry We generalize the following simple geometric fact: the only centrally symmetric convex curve of constant width is a circle. Billiard interpretation of the condition of constant width reads: a planar curve has constant width, if and only if, the Birkhoff billiard map inside the planar curve has a rotational invariant curve of $2$-periodic orbits. We generalize this statement to curves that are invariant under a rotation by angle $\frac {2\pi }{k}$, for which the billiard map has a rotational invariant curve of $k$-periodic orbits. Similar result holds true also for outer billiards and symplectic billiards. Finally, we consider Minkowski billiards inside a unit disc of Minkowski (not necessarily symmetric) norm that is invariant under a linear map of order $k\ge 3$. We find a criterion for the existence of an invariant curve of $k$-periodic orbits. As an application, we get rigidity results for all those billiards. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png International Mathematics Research Notices Oxford University Press

Billiard Tables with Rotational Symmetry

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References (2)

Publisher
Oxford University Press
Copyright
© The Author(s) 2022. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permission@oup.com.
ISSN
1073-7928
eISSN
1687-0247
DOI
10.1093/imrn/rnab366
Publisher site
See Article on Publisher Site

Abstract

We generalize the following simple geometric fact: the only centrally symmetric convex curve of constant width is a circle. Billiard interpretation of the condition of constant width reads: a planar curve has constant width, if and only if, the Birkhoff billiard map inside the planar curve has a rotational invariant curve of $2$-periodic orbits. We generalize this statement to curves that are invariant under a rotation by angle $\frac {2\pi }{k}$, for which the billiard map has a rotational invariant curve of $k$-periodic orbits. Similar result holds true also for outer billiards and symplectic billiards. Finally, we consider Minkowski billiards inside a unit disc of Minkowski (not necessarily symmetric) norm that is invariant under a linear map of order $k\ge 3$. We find a criterion for the existence of an invariant curve of $k$-periodic orbits. As an application, we get rigidity results for all those billiards.

Journal

International Mathematics Research NoticesOxford University Press

Published: Jan 3, 2022

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