Nekhoroshev type estimates for billiard ball maps

Gramchev, Todor; Popov, Georgi

doi:10.5802/aif.1477

Cited by 18 publications

(13 citation statements)

References 16 publications

(26 reference statements)

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“…We cannot quote all the references on this subject, but the interested reader may consult [Ko79], [Gr99], [Th97] and the references therein. Apart from the theory of partial differential equations where they have been widely used, Gevrey functions were also considered in connection with dynamical systems problems, for instance in [El97], [GP95] and [Po00].…”

Section: A1 Elementary Propertiesmentioning

confidence: 99%

Stability and instability for Gevrey quasi-convex near-integrable Hamiltonian systems

Marco

Sauzin²

2003

Publ. math., Inst. Hautes Étud. Sci.

192

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We prove a theorem about the stability of action variables for Gevrey quasi-convex near-integrable Hamiltonian systems and construct in that context a system with an unstable orbit whose mean speed of drift allows us to check the optimality of the stability theorem.Our stability result generalizes those by Lochak-Neishtadt and Pöschel, which give precise exponents of stability in the Nekhoroshev Theorem for the quasi-convex case, to the situation in which the Hamiltonian function is only assumed to belong to some Gevrey class instead of being real-analytic. For n degrees of freedom and Gevrey-α Hamiltonians, α ≥ 1, we prove that one can choose a = 1/2nα as an exponent for the time of stability and b = 1/2n as an exponent for the radius of confinement of the action variables, with refinements for the orbits which start close to a resonant surface (we thus recover the result for the real-analytic case by setting α = 1).On the other hand, for α > 1, the existence of compact-supported Gevrey functions allows us to exhibit for each n ≥ 3 a sequence of Hamiltonian systems with wandering points, whose limit is a quasi-convex integrable system, and where the speed of drift is characterized by the exponent 1/2(n − 2)α. This exponent is optimal for the kind of wandering points we consider, inasmuch as the initial condition is located close to a doubly-resonant surface and the stability result holds with precisely that exponent for such an initial condition. We also discuss the relationship between our example of instability, which relies on a specific construction of a perturbation of a discrete integrable system, and Arnold's mechanism of instability, whose main features (partially hyperbolic tori, heteroclinic connections) are indeed present in our system.

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Section: A1 Elementary Propertiesmentioning

confidence: 99%

Stability and instability for Gevrey quasi-convex near-integrable Hamiltonian systems

Marco

Sauzin²

2003

Publ. math., Inst. Hautes Étud. Sci.

192

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“…that there exist infinitely many resonances approaching the real axis. If the boundary is strictly convex and analytic, the "effective" stability is valid in an exponentially large time interval |(| < Ce b l E (see [3]) with some (7, b > 0 and we believe that there exists a sequence of resonances which tends exponentially fast to the real axis in this case. …”

Section: X-2mentioning

confidence: 99%

“…To do this we make use of the approximate interpolating Hamiltonian C of the corresponding billiard ball map B (see [8], [3], [6] /^/) = ^(.r^r^l+OM).…”

Section: X-6mentioning

confidence: 99%

Resonances for transparent obstacles

Popov¹,

Vodev²

1999

Journées Équations Aux Dérivées Partielles

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“…[2,[21][22][23]29,33]). However, we stress that the study of the Gevrey regularity of the conjugating PDO q(x, D) presents new features and difficulties in comparison with the aforementioned results in dynamical systems.…”

Section: Remark 43mentioning

confidence: 99%

“…This phenomenon resembles a similar one in the effective stability (Nekhoroshev estimates) of normal forms in dynamical systems and their applications (cf. [2, 21, 22, 30, 33]; see also [23] for Nekhoroshev estimates for billiard ball maps in R n , n 3, by means of Gevrey techniques).…”

Section: Introductionmentioning

confidence: 99%

Perturbations of Vector Fields on Tori: Resonant Normal Forms and Diophantine Phenomena

Dickinson¹,

Gramchev²,

Yoshino³

2002

Proceedings of the Edinburgh Mathematical Society

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This paper concerns perturbations of smooth vector fields on T n (constant if n 3) with zeroth-order C ∞ and Gevrey G σ , σ 1, pseudodifferential operators. Simultaneous resonance is introduced and simultaneous resonant normal forms are exhibited (via conjugation with an elliptic pseudodifferential operator) under optimal simultaneous Diophantine conditions outside the resonances. In the C ∞ category the results are complete, while in the Gevrey category the effect of the loss of the Gevrey regularity of the conjugating operators due to Diophantine conditions is encountered. The normal forms are used to study global hypoellipticity in C ∞ and Gevrey G σ . Finally, the exceptional sets associated with the simultaneous Diophantine conditions are studied. A generalized Hausdorff dimension is used to give precise estimates of the 'size' of different exceptional sets, including some inhomogeneous examples.

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Nekhoroshev type estimates for billiard ball maps

Cited by 18 publications

References 16 publications

Stability and instability for Gevrey quasi-convex near-integrable Hamiltonian systems

Stability and instability for Gevrey quasi-convex near-integrable Hamiltonian systems

Resonances for transparent obstacles

Perturbations of Vector Fields on Tori: Resonant Normal Forms and Diophantine Phenomena

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