A billiard table close to an ellipse is deformationally spectrally rigid among dihedrally symmetric domains

We prove that a a strongly convex planar domain (Birkhoff table) with dihedral symmetry, which is sufficiently close in a finitely smooth topology to an ellipse, is deformationally spectrally rigid within the class of domains preserving this symmetry. More precisely, any smooth one-parameter family of such domains that preserves the length spectrum (i.e., the set of lengths of periodic billiard orbits) must consist only of rigid motions of the initial domain. The proof combines two types of dynamical data: the asymptotic behavior of certain symmetric periodic orbits, as previously used in the rigidity of nearly circular domains, and new spectral information derived from KAM invariant curves, obtained from Mather’s beta function and its derivatives (in the Whitney sense) at some suitable rotation numbers.

💡 Research Summary

The paper addresses a central inverse problem in planar billiard dynamics: to what extent does the length spectrum (the set of lengths of all periodic billiard trajectories) determine the shape of a convex billiard table? While the global question—whether two tables sharing the same length spectrum must be congruent—is still open, the authors focus on a local, “deformational” version. They consider one‑parameter families of strongly convex planar domains (Birkhoff tables) that preserve the length spectrum and ask whether such families can be non‑trivial, i.e., not just rigid motions.

The main result is a rigidity theorem for domains that are sufficiently close (in a finite‑smooth topology) to an ellipse and that possess full dihedral symmetry (both axial and central symmetry). Precisely, there exist an integer (r>0) and a small (\varepsilon>0), depending only on the eccentricity of the reference ellipse (E), such that any (C^{r})‑smooth, dihedrally symmetric Birkhoff table (\Omega) with (| \partial\Omega - \partial E|{C^{r}} < \varepsilon) is deformationally spectrally rigid: any smooth one‑parameter family ({\Omega{\tau}}) within this class that keeps the length spectrum constant must consist solely of rigid motions of (\Omega).

The proof combines two distinct dynamical ingredients extracted from the length spectrum:

1. Symmetric periodic orbits (D1). For each integer (q\ge 3) there exist symmetric periodic billiard trajectories with rotation number (1/q). Their lengths (L_{q}) are part of the length spectrum. Differentiating the condition (L_{q}(\Omega_{\tau})\equiv L_{q}(\Omega_{0})) with respect to (\tau) yields a linear functional (\Delta_{q}(n)=0) on the infinitesimal deformation (n) of the boundary. Collecting all such equations defines a linear operator (S:H^{1/2}\to\mathbb R^{N}). The kernel of (S) is finite‑dimensional and spanned by a few low‑frequency Fourier modes (essentially the constant mode and the second harmonic), reflecting the fact that symmetric periodic orbits only constrain low‑order components of the deformation.

2. KAM invariant curves (D2). By KAM theory, a sufficiently smooth convex table possesses invariant curves of the billiard map for a full measure set of Diophantine rotation numbers. For each such rotation number (\omega), the Mather (\beta)‑function (\beta_{\Omega}(\omega)) encodes the maximal action (i.e., length) among orbits with that rotation. The length‑preserving hypothesis forces the (\tau)‑derivative of (\beta_{\Omega_{\tau}}(\omega)) to vanish at (\tau=0). Using the Whitney‑smoothness of (\beta) in (\omega), one obtains an infinite family of linear constraints \