Mather β-Function for Ellipses and Rigidity

Area of Science:

• Dynamical Systems
• Geometric Mechanics

Background:

• The Mather β-function is crucial in understanding dynamical systems, particularly in celestial mechanics and Hamiltonian systems.
• Calculating the Mather β-function and rotation number for elliptical systems traditionally involves complex mathematical approaches.

Purpose of the Study:

• To derive an explicit formula for the rotation number and Mather β-function specifically for elliptical billiards.
• To demonstrate a simplified derivation method using a non-standard generating function.
• To explore the application of the Mather β-function in rigidity problems.

Main Methods:

• Utilized a non-standard generating function derived from the billiard problem.
• Applied this generating function to obtain explicit formulas for the rotation number and Mather β-function.
• Investigated the implications of these functions in the context of rigidity theory.

Main Results:

• An explicit formula for the rotation number of elliptical billiards was successfully obtained.
• A simplified method for deriving the Mather β-function for ellipses was established.
• The study provides a foundation for applying these results to rigidity problems.

Conclusions:

• The non-standard generating function offers a more straightforward approach to calculating key dynamical invariants for elliptical systems.
• The derived formulas and methods enhance the analytical tractability of problems in dynamical systems and rigidity.
• This work contributes to a deeper understanding of the Mather β-function and its applications.