TY - JOUR

T1 - Billiard Tables with Rotational Symmetry

AU - Bialy, Misha

AU - Tsodikovich, Daniel

N1 - Publisher Copyright:
© 2022 The Author(s). Published by Oxford University Press. All rights reserved.

PY - 2023/3/1

Y1 - 2023/3/1

N2 - 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 2π 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≥ 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.

AB - 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 2π 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≥ 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.

UR - http://www.scopus.com/inward/record.url?scp=85152587877&partnerID=8YFLogxK

U2 - 10.1093/imrn/rnab366

DO - 10.1093/imrn/rnab366

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AN - SCOPUS:85152587877

SN - 1073-7928

VL - 2023

SP - 3970

EP - 4003

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

IS - 5

ER -