Mather β-Function for Ellipses and Rigidity
Abstract
:1. Introduction
2. Results
3. Preliminaries and Methods
3.1. Non-Standard Generating Function
3.2. Integral for Elliptic Billiard in Various Forms
3.3. The Relations between Conserved Quantities
- 1.
- 2.
- 3.
- In terms of the eccentricities of we have the formulas:
4. Invariant Measure on an Invariant Curve
5. Mather β-Function
6. Mather β-Function and the Lazutkin Parameter
7. Rotation Number ρ
8. Mather β-Function and Rigidity
9. Discussion
- Is it possible to relax symmetry assumption in the main Theorem 1? Our method of proof of Theorem 1 relies on the approach related to the so-called E.Hopf type rigidity phenomenon from [18]. This method is very robust and it is not clear at the moment how it can be generalized.
- Another problem is to adopt our approach to a smaller neighborhood of the boundary of the phase cylinder.
- All known approaches to rigidity in billiards, are based on the properties of orbits near the boundary. We believe there are rigidity results based on the behavior far from the boundary.
- It would be interesting to prove that ellipse is determined by any two values of Mather function where are any two rotation numbers in .
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Bialy, M. Mather β-Function for Ellipses and Rigidity. Entropy 2022, 24, 1600. https://doi.org/10.3390/e24111600
Bialy M. Mather β-Function for Ellipses and Rigidity. Entropy. 2022; 24(11):1600. https://doi.org/10.3390/e24111600
Chicago/Turabian StyleBialy, Michael. 2022. "Mather β-Function for Ellipses and Rigidity" Entropy 24, no. 11: 1600. https://doi.org/10.3390/e24111600
APA StyleBialy, M. (2022). Mather β-Function for Ellipses and Rigidity. Entropy, 24(11), 1600. https://doi.org/10.3390/e24111600