Lazutkin coordinates of the maximal symmetric periodic orbits on the ellipse

TL;DR

This paper derives the second order approximation of Lazutkin coordinates for maximal symmetric periodic orbits on ellipses, aiding the understanding of spectral rigidity in symmetric convex domains close to a circle.

Contribution

It introduces a second order approximation of Lazutkin coordinates for symmetric periodic orbits on ellipses using action-angle variables, advancing spectral rigidity analysis.

Findings

Derived second order Lazutkin coordinate approximation for elliptical orbits

Enhanced understanding of spectral rigidity near circular domains

Provided analytical tools for symmetric periodic orbit analysis

Abstract

In De Simoi J., Kaloshin V., Wei Q. "Dynamical spectral rigidity among $\mathbb{Z}_2$-symmetric strictly convex domains close to a circle" (Appendix B coauthored with H. Hezari) Ann. of Math. 186.1 (2017), pp. 277-314 deformational spectral rigidity of $\mathbb{Z}_2$ symmetric domains close to the circle has been shown. One of the steps of the proof was to express the maximal symmetric periodic orbits in the Lazutkin parametrization. Here using the action-angle variables we find the second order approximation of Lazutkin coordinates of the maximal symmetric periodic orbits on the ellipses.