Numerical Evidence of Robust Dynamical Spectral Rigidity of Ellipses Among Smooth -Symmetric Domains

Abstract

We present numerical evidence for robust spectral rigidity among ${\mathbb{Z}}_2$-symmetric domains of ellipses of eccentricity smaller than 0.30.

Keywords:

• Ellipses
• convex billiards
• inverse problem
• length spectrum
• spectral rigidity

Acknowledgments

Both authors are grateful to the anonymous referees for their most useful and relevant comments.

Declaration of Interest

No potential conflict of interest was reported by the author(s).

Funding

SA acknowledges support from the NSERC USRA, reference number 541714-2019; JDS acknowledges partial support from the NSERC Discovery grant, reference number 502617-2017.

Supplementary material

Tables are available in the online version of this article. Link to the code: GitHub repository

Notes

1 Historically, the majority results in the field have been obtained with Dirichlet boundary conditions, although other type of boundary conditions can be treated and are equally relevant. In this paper we will follow this long established tradition and consider only Dirichlet boundary conditions.

2 This is in fact a local version of the celebrated Birkhoff–Poritski conjecture.

3 That is to say: an ellipse is completely identified by its perimeter and the length of its minor axis.

4 Note that an ellipse with e = 0 is a circle.

5 The subscripts in $X_j^q$ are considered to be modulo q.

6 Continuity is proved for $\Omega =\mathbb{D}$ in [3], but in fact holds for any sufficiently smooth Ω.

7 This nonstandard choice of origin for the parametrization $\phi$ simplifies some formulae in Lazutkin coordinates.

8 This fact is peculiar for elliptical billiards and follows from Poncelet’s Porism

9 A more precise definition of the winding number can be given, but we avoid giving it here, since this definition will suffice for our uses below.

10 This may be due to the system’s sensitivity to initial conditions. If the conditions were highly accurate, we would see the terms to decay continuously.

11 The computation with $\gamma =3.1$ was run for $e\in [0.25,0.4]$ , and with $\gamma =3.01$ was run for $e\in [0.32,0.4]$.