Deformational spectral rigidity of axially-symmetric symplectic billiards

TL;DR

This paper establishes a rigidity result for symplectic billiards, showing that near an ellipse, axially symmetric domains with similar spectral properties are either trivial or differ by affine transformations, and extends this to a broader setting.

Contribution

It proves a spectral rigidity theorem for symplectic billiards near ellipses and characterizes families of axially symmetric domains preserving area-spectrum in general.

Findings

Near an ellipse, spectral rigidity implies domains differ by affine transformations.

Families preserving area-spectrum are tangent to finite-dimensional spaces.

The results extend classical billiard rigidity to symplectic billiards.

Abstract

Symplectic billiards were introduced by Albers and Tabachnikov as billiards in strictly convex bounded domains of the plane with smooth boundary having a specific law of reflection. This paper proves a rigidity result for symplectic billiards which is similar to a previous result on classical billiards formulated by De Simoi, Kaloshin and Wei. Namely, it states that close to an ellipse, a sufficiently smooth one-parameter family of axially symmetric domains either contains domains with different area-spectra or is trivial, in a sense that the domains differ by area-preserving affine transformations of the plane. The paper also prove that in the general setting - that is even if the domains are not close to an ellipse - any sufficiently smooth one-parameter family of axially symmetric domains which preserves the area-spectrum is tangent to a finite dimensionnal space.