Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

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Locally maximizing orbits for multi-dimensional twist maps and Birkhoff billiards
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by Misha Bialy and Daniel Tsodikovich;
Trans. Amer. Math. Soc. 378 (2025), 4077-4108
DOI: https://doi.org/10.1090/tran/9327
Published electronically: March 25, 2025

Abstract:

In this work, we consider the variational properties of exact symplectic twist maps $T$ that act on the cotangent bundle of a torus or on a ball bundle over a sphere. An example of such a map is the well-known Birkhoff billiard map corresponding to smooth convex hypersurfaces. In this work, we will focus on the important class $\mathcal {M}$ of orbits of $T$ which are locally maximizing with respect to the variational principle associated with a generating function of the symplectic twist map. Our first goal is to give a geometric and variational characterization for the orbits in the class $\mathcal M$.

The billiard map is known to have two different generating functions, and this invokes the following question: how to compare the properties of these two generating functions? While our motivation comes from billiards, we will work in general, and assume that a general twist map $T$ has two generating functions. Thus, we consider the orbits of $T$ which are locally maximizing with respect to either of the generating functions. We formulate a geometric criterion guaranteeing that two generating functions of the same twist map have the same class of locally maximizing orbits, and we will show that the two generating functions for the Birkhoff billiard map do, in fact, satisfy this criterion. The proof of this last property will rely on the Sinai-Chernov formula from geometric optics and billiard dynamics.

References
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Bibliographic Information
  • Misha Bialy
  • Affiliation: School of Mathematical Sciences, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Israel
  • MR Author ID: 259382
  • ORCID: 0000-0001-7566-5443
  • Email: bialy@tauex.tau.ac.il
  • Daniel Tsodikovich
  • Affiliation: School of Mathematical Sciences, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Israel
  • MR Author ID: 1546898
  • Email: tsodaniel314@gmail.com
  • Received by editor(s): May 21, 2023
  • Received by editor(s) in revised form: August 11, 2024
  • Published electronically: March 25, 2025
  • Additional Notes: The first author was partially supported by ISF grant 580/20, the second author was supported by ISF grants 580/20, 667/18 and DFG grant MA-2565/7-1 within the Middle East Collaboration Program.
  • © Copyright 2025 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 378 (2025), 4077-4108
  • MSC (2020): Primary 37C83; Secondary 37E40
  • DOI: https://doi.org/10.1090/tran/9327