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P. Albers, S. Tabachnikov (2018)
Introducing symplectic billiardsAdv. Math., 333
S. Artstein-Avidan, R. Karasev, Y. Ostrover (2014)
From symplectic measurements to the Mahler conjectureDuke Math. J., 163
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.
International Mathematics Research Notices – Oxford University Press
Published: Jan 3, 2022
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