Invariants of Isospectral Deformations and Spectral Rigidity

Popov, Georgi; Topalov, Peter

doi:10.1080/03605302.2011.641051

Cited by 16 publications

(33 citation statements)

References 56 publications

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“…A similar result has been obtained for the ellipse in [4] and more generally for L.B.T.s of classical type in dimension n = 2 in [10] and [12]. It is always interesting to find a smaller set of data Λ for which the Radon transform is one-to-one.…”

Section: Introductionsupporting

confidence: 76%

“…where m ≥ 1 is the minimal power of B that leaves Λ c invariant, i.e., B m (Λ c ) = Λ c . Note that (2.3) appears as a spectral invariant of (1.2) in [12]. We show in Sect.…”

Section: Invariant Manifolds Leray Form and Radon Transformmentioning

confidence: 80%

“…We consider here a weaker notion of isospectrality which has been introduced in [12]. Fix two positive constants c and d > 1/2, and consider the union of infinitely many disjoint intervals…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On the Integral Geometry of Liouville Billiard Tables

Popov

Topalov

2011

Commun. Math. Phys.

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The notion of a Radon transform is introduced for completely integrable billiard tables. In the case of Liouville billiard tables of dimension 3 we prove that the Radon transform is one-to-one on the space of continuous functions K on the boundary which are invariant with respect to the corresponding group of symmetries. We prove also that the frequency map associated with a class of Liouville billiard tables is non-degenerate. This allows us to obtain spectral rigidity of the corresponding Laplace-Beltrami operator with Robin boundary conditions.

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Section: Introductionsupporting

confidence: 76%

“…where m ≥ 1 is the minimal power of B that leaves Λ c invariant, i.e., B m (Λ c ) = Λ c . Note that (2.3) appears as a spectral invariant of (1.2) in [12]. We show in Sect.…”

Section: Invariant Manifolds Leray Form and Radon Transformmentioning

confidence: 80%

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On the Integral Geometry of Liouville Billiard Tables

Popov

Topalov

2011

Commun. Math. Phys.

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“…Let 0 be bounded by an ellipse E. Then any one-parameter isospectral C ∞ -deformation ( τ ) |τ |≤1 which additionally preserves the Z 2 × Z 2 symmetry group of the ellipse is necessarily flat (that is all derivatives have to vanish for τ = 0) (results of this kind are usually referred to as infinitesimal spectral rigidity). Popov and Topalov [78] recently extended these results (see also [79]).…”

Section: Laplace Spectral Rigiditymentioning

confidence: 80%

Inverse problems and rigidity questions in billiard dynamics

Kaloshin

Sorrentino²

2021

Ergod. Th. Dynam. Sys.

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A Birkhoff billiard is a system describing the inertial motion of a point mass inside a strictly convex planar domain, with elastic reflections at the boundary. The study of the associated dynamics is profoundly intertwined with the geometric properties of the domain: while it is evident how the shape determines the dynamics, a more subtle and difficult question is the extent to which the knowledge of the dynamics allows one to reconstruct the shape of the domain. This translates into many intriguing inverse problems and unanswered rigidity questions, which have been the focus of very active research in recent decades. In this paper we describe some of these questions, along with their connection to other problems in analysis and geometry, with particular emphasis on recent results obtained by the authors and their collaborators.

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“…This article is a part of a project (cf. [61]- [64]) 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 the spectral rigidity of the Laplace-Beltrami operator associated with C 1 deformations (X, g t ), 0 ≤ t ≤ 1, of a billiard table (X, g), where X is a C ∞ smooth compact manifold with a connected boundary Γ := ∂X of dimension dim X = n ≥ 2 and t → g t is a C 1 family of smooth Riemannian metric on X.…”

Section: Introductionmentioning

confidence: 99%

From K.A.M. Tori to Isospectral Invariants and Spectral Rigidity of Billiard Tables

Popov¹,

Topalov²

2016

Preprint

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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.

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Invariants of Isospectral Deformations and Spectral Rigidity

Cited by 16 publications

References 56 publications

On the Integral Geometry of Liouville Billiard Tables

On the Integral Geometry of Liouville Billiard Tables

Inverse problems and rigidity questions in billiard dynamics

From K.A.M. Tori to Isospectral Invariants and Spectral Rigidity of Billiard Tables

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