From KAM Tori to Isospectral Invariants and Spectral Rigidity of Billiard Tables

G Popov (1) , P Topalov

1 LMJL - Laboratoire de Mathématiques Jean Leray

This article is a part of a project investigating the relationship between the dynamics of completely integrable or "close" to completely integrable billiard tables, the integral geometry on them, and the spectrum of the corresponding Laplace-Beltrami operators. It is concerned with new isospectral invariants and with the spectral rigidity problem for the Laplace-Beltrami operators ∆ t , t ∈ [0, 1], with Dirichlet, Neumann or Robin boundary conditions , associated with C 1 families of billiard tables (X, g t). We introduce a notion of weak isospectrality for such deformations. The main dynamical assumption on (X, g 0) is that the corresponding billiard ball map B 0 or an iterate P 0 = B m 0 of it posses a Kronecker invariant torus with a Diophantine frequency ω 0 and that the corresponding Birkhoff Normal Form is nondegenerate in Kolmogorov sense. Then we prove that there exists δ 0 > 0 and a set Ξ of Diophantine frequencies containing ω 0 and of full Lebesgue measure around ω 0 such that for each ω ∈ Ξ and 0 < δ < δ 0 there exists a C 1 family of Kronecker tori Λ t (ω) of P t for t ∈ [0, δ]. If the family ∆ t , t ∈ [0, 1], satisfies the weak isospectral condition we prove that the average action β t (ω) on Λ t (ω) and the Birkhoff Normal Form of P t at Λ t (ω) are independent of t ∈ [0, δ] for each ω ∈ Ξ. As an application we obtain infinitesimal spectral rigidity for Liouville billiard tables in dimensions 2 and 3. In particular infinitesimal spectral rigidity for the ellipse and the ellipsoid is obtained under the weak isospectral condition. Applications are obtained also for strictly convex billiard tables in R 2 as well as in the case when (X, g 0) admits an elliptic periodic billiard trajectory with no resonances of order ≤ 4. In particular we obtain spectral rigidity (under the weak isospectral condition) of elliptical billiard tables in the class of analytic and Z 2 × Z 2 symmetric billiard tables in R 2. We prove also that billiard tables with boundaries close to ellipses are spectrally rigid in this class. The results are based on a construction of C 1 families of quasi-modes associated with the Kronecker tori Λ t (ω) and on suitable KAM theorems for C 1 families of Hamiltonians. We propose a new iteration schema (a modified iterative lemma) in the proof of the KAM theorem with parameters, which avoids the Whitney extension theorem for C ∞ jets and allows one to obtain global estimates of the corresponding canonical transformations and Hamiltonians in the scale of all Hölder norms. The classical and quantum Birkhoff Normal Forms for C 1 or analytic families of symplectic mappings (Hamiltonians) obtained here can be used as well in order to investigate problems related to the quantum non-ergodicity of C ∞-smooth KAM systems.

Domaines: Mathématiques [math]; Equations aux dérivées partielles [math.AP]; Systèmes dynamiques [math.DS]; Théorie spectrale [math.SP]

G Popov, P Topalov. From KAM Tori to Isospectral Invariants and Spectral Rigidity of Billiard Tables. 2019. ⟨hal-02012786⟩