An integrable deformation of an ellipse of small eccentricity is an ellipse

Ávila, Artur; Simoi, Jacopo De; Kaloshin, Vadim

doi:10.4007/annals.2016.184.2.5

Cited by 69 publications

(120 citation statements)

References 25 publications

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“…This lemma is proven for periodic orbits in [2], but the same proof applies to orbits glancing only at a part of the boundary of a C 5 strictly convex domain.…”

Section: Consider Now the Billiard Map In Lazutkin Coordinatesmentioning

confidence: 93%

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Nonisometric Domains with the Same Marvizi – Melrose Invariants

Buhovsky

Kaloshin

2018

Regul. Chaot. Dyn.

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For any strictly convex planar domain Ω ⊂ R 2 with a C ∞ boundary one can associate an infinite sequence of spectral invariants introduced by Marvizi-Merlose [5]. These invariants can generically be determined using the spectrum of the Dirichlet problem of the Laplace operator. A natural question asks if this collection is sufficient to determine Ω up to isometry. In this paper we give a counterexample, namely, we present two non-isometric domains Ω andΩ with the same collection of Marvizi-Melrose invariants. Moreover, each domain has countably many periodic orbits {S n } n 1 (resp. {S n } n 1 ) of period going to infinity such that S n andS n have the same period and perimeter for each n.Consider a C ∞ smooth strictly convex planar domain Ω ⊂ R 2 . Let us start by introducing the Length Spectrum of a domain Ω. The length spectrum of Ω is given by the set of lengths of its periodic orbits, counted with multiplicity:L(Ω) := N{ lengths of closed geodesics in Ω} ∪ N |∂Ω|, where |∂Ω| denotes the length of the boundary of Ω. Generically this collection can be determined from the spectrum of the Laplace operator in Ω with Dirichlet boundary condition (similarly for Neumann boundary one):(1) ∆f = λf in Ω f | ∂Ω = 0.From the physical point of view, the eigenvalues λ's are the eigenfrequencies of the membrane Ω with a fixed boundary. There is the following relation between the Laplace spectrum and the length spectrum (see e.g. [1,6]). Call the functionthe wave trace. Then, the wave trace w(t) is a well-defined generalized function (distribution) of t, smooth away from the length spectrum, namely,(2) sing. supp. w(t) ⊆ ±L(Ω) ∪ {0}.So if l > 0 belongs to the singular support of this distribution, then there exists either a closed billiard trajectory of length l, or a closed geodesic of length l in the boundary of the billiard table.

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“…This lemma is proven for periodic orbits in [2], but the same proof applies to orbits glancing only at a part of the boundary of a C 5 strictly convex domain.…”

Section: Consider Now the Billiard Map In Lazutkin Coordinatesmentioning

confidence: 93%

“…Generically, equality holds in (2). More precisely, if no two distinct orbits have the same length and the Poincaré map of any periodic orbit is non-degenerate, then the singular support of the wave trace coincides with ±L(Ω) ∪ {0} (see e.g.…”

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confidence: 99%

Nonisometric Domains with the Same Marvizi – Melrose Invariants

Buhovsky

Kaloshin

2018

Regul. Chaot. Dyn.

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“…Recently, substantial progress toward Birkhoff's conjecture was made by V. Kaloshin and his co-authors [5,127,117]: they proved the Birkhoff conjecture for small perturbations of ellipses. The general case remains open, and it is one of the foremost problems in this field.…”

Section: Around the Birkhoff Conjecturementioning

confidence: 99%

Open problems, questions and challenges in finite- dimensional integrable systems

Bolsinov

Matveev

Miranda

et al. 2018

Phil. Trans. R. Soc. A.

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The paper surveys open problems and questions related to different aspects of integrable systems with finitely many degrees of freedom. Many of the open problems were suggested by the participants of the conference 'Finite-dimensional Integrable Systems, FDIS 2017' held at CRM, Barcelona in July 2017.This article is part of the theme issue 'Finite dimensional integrable systems: new trends and methods'.

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“…Now, by noting that ε/v = 1/ √ 2E where E is the non-rescaled energy, we immediately obtain the statement of the theorem from estimates (55),(56). Now we proceed to obtain a first order expression in perturbation parameters ε, δ for S Γ in terms of E and t. Let (Ē 0 ,t 0 ) ∈Λ and (Ẽ 0 ,t 0 ) ∈Λ be two points in the domain of S Γ given by (2). We will obtain a perturbative expression up to εO(ε + |δ|) for the scattering map S Γ :Λ →Λ:…”

Section: T)mentioning

confidence: 99%

Splitting of separatrices, scattering maps, and energy growth for a billiard inside a time-dependent symmetric domain close to an ellipse

2018

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We study billiard dynamics inside an ellipse for which the axes lengths are changed periodically in time and an O(δ)-small quartic polynomial deformation is added to the boundary. In this situation the energy of the particle in the billiard is no longer conserved. We show a Fermi acceleration in such system: there exists a billiard trajectory on which the energy tends to infinity. The construction is based on the analysis of dynamics in the phase space near a homoclinic intersection of the stable and unstable manifolds of the normally hyperbolic invariant cylinder Λ, parameterised by the energy and time, that corresponds to the motion along the major axis of the ellipse. The proof depends on the reduction of the billiard map near the homoclinic channel to an iterated function system comprised by the shifts along two Hamiltonian flows defined on Λ. The two flows approximate the so-called inner and scattering maps, which are basic tools that arise in the studies of the Arnold diffusion; the scattering maps defined by the projection along the strong stable and strong unstable foliations W ss,uu of the stable and unstable invariant manifolds W s,u (Λ) at the homoclinic points. Melnikov type calculations imply that the behaviour of the scattering map in this problem is quite unusual: it is only defined on a small subset of Λ that shrinks, in the large energy limit, to a set of parallel lines t = const as δ → 0. *

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An integrable deformation of an ellipse of small eccentricity is an ellipse

Cited by 69 publications

References 25 publications

Nonisometric Domains with the Same Marvizi – Melrose Invariants

Nonisometric Domains with the Same Marvizi – Melrose Invariants

Open problems, questions and challenges in finite- dimensional integrable systems

Splitting of separatrices, scattering maps, and energy growth for a billiard inside a time-dependent symmetric domain close to an ellipse

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