Outer Billiards and Symplectic Geometry
Open Access
- Author:
- Sharipova, Anastasiia
- Graduate Program:
- Mathematics
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- February 04, 2025
- Committee Members:
- Pierre-Emmanuel Jabin, Program Head/Chair
Luen-Chau Li, Major Field Member
Anton Petrunin, Major Field Member
Sergei Tabachnikov, Chair & Dissertation Advisor
Mark Latash, Outside Unit & Field Member - Keywords:
- symplectic geometry
closed characteristics
outer billiards
Viterbo’s conjecture
Ivrii's conjecture
periodic orbits
symplectic billiards - Abstract:
- In this dissertation we study questions in the intersection of symplectic geometry, convex geometry and billiard dynamical systems. We get a new characterization of a ball in terms of Hamiltonian and outer billiard dynamics: we show that smooth and strongly convex bodies in the symplectic R^2n for n > 1 with all characteristics planar, or all outer billiard trajectories planar are affine symplectic images of balls. We also study questions about periodic billiard orbits and give a proof for (2n + 1, n) and (2n, n − 1)-periodic Ivrii’s conjecture for planar outer billiards. In addition, we give new simple geometric proofs for the 3 and 4-periodic cases for outer and symplectic billiards, and generalize for higher dimensions in case of symplectic billiards.
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