Hamiltonian Perturbation Theory for Ultra-Differentiable Functions

Hamiltonian Perturbation Theory for Ultra-Differentiable Functions
Title Hamiltonian Perturbation Theory for Ultra-Differentiable Functions PDF eBook
Author Abed Bounemoura
Publisher American Mathematical Soc.
Pages 89
Release 2021-07-21
Genre Education
ISBN 147044691X

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Some scales of spaces of ultra-differentiable functions are introduced, having good stability properties with respect to infinitely many derivatives and compositions. They are well-suited for solving non-linear functional equations by means of hard implicit function theorems. They comprise Gevrey functions and thus, as a limiting case, analytic functions. Using majorizing series, we manage to characterize them in terms of a real sequence M bounding the growth of derivatives. In this functional setting, we prove two fundamental results of Hamiltonian perturbation theory: the invariant torus theorem, where the invariant torus remains ultra-differentiable under the assumption that its frequency satisfies some arithmetic condition which we call BRM, and which generalizes the Bruno-R¨ussmann condition; and Nekhoroshev’s theorem, where the stability time depends on the ultra-differentiable class of the pertubation, through the same sequence M. Our proof uses periodic averaging, while a substitute for the analyticity width allows us to bypass analytic smoothing. We also prove converse statements on the destruction of invariant tori and on the existence of diffusing orbits with ultra-differentiable perturbations, by respectively mimicking a construction of Bessi (in the analytic category) and MarcoSauzin (in the Gevrey non-analytic category). When the perturbation space satisfies some additional condition (we then call it matching), we manage to narrow the gap between stability hypotheses (e.g. the BRM condition) and instability hypotheses, thus circumbscribing the stability threshold. The formulas relating the growth M of derivatives of the perturbation on the one hand, and the arithmetics of robust frequencies or the stability time on the other hand, bring light to the competition between stability properties of nearly integrable systems and the distance to integrability. Due to our method of proof using width of regularity as a regularizing parameter, these formulas are closer to optimal as the the regularity tends to analyticity

HAMILTONIAN PERTURBATION THEORY FOR ULTRA-DIFFERENTIABLE FUNCTIONS.

HAMILTONIAN PERTURBATION THEORY FOR ULTRA-DIFFERENTIABLE FUNCTIONS.
Title HAMILTONIAN PERTURBATION THEORY FOR ULTRA-DIFFERENTIABLE FUNCTIONS. PDF eBook
Author ABED. BOUNEMOURA
Publisher
Pages
Release 2021
Genre
ISBN 9781470465261

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Spectral Expansions of Non-Self-Adjoint Generalized Laguerre Semigroups

Spectral Expansions of Non-Self-Adjoint Generalized Laguerre Semigroups
Title Spectral Expansions of Non-Self-Adjoint Generalized Laguerre Semigroups PDF eBook
Author Pierre Patie
Publisher American Mathematical Society
Pages 182
Release 2021-11-16
Genre Mathematics
ISBN 1470449366

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Intense Automorphisms of Finite Groups

Intense Automorphisms of Finite Groups
Title Intense Automorphisms of Finite Groups PDF eBook
Author Mima Stanojkovski
Publisher American Mathematical Society
Pages 117
Release 2021-12-09
Genre Mathematics
ISBN 1470450038

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The Yang-Mills Heat Equation with Finite Action in Three Dimensions

The Yang-Mills Heat Equation with Finite Action in Three Dimensions
Title The Yang-Mills Heat Equation with Finite Action in Three Dimensions PDF eBook
Author Leonard Gross
Publisher American Mathematical Society
Pages 111
Release 2022-02-02
Genre Mathematics
ISBN 1470450534

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Decoupling on the Wiener Space, Related Besov Spaces, and Applications to BSDEs

Decoupling on the Wiener Space, Related Besov Spaces, and Applications to BSDEs
Title Decoupling on the Wiener Space, Related Besov Spaces, and Applications to BSDEs PDF eBook
Author Stefan Geiss
Publisher American Mathematical Society
Pages 112
Release 2021-11-16
Genre Mathematics
ISBN 1470449358

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Naturality and Mapping Class Groups in Heegard Floer Homology

Naturality and Mapping Class Groups in Heegard Floer Homology
Title Naturality and Mapping Class Groups in Heegard Floer Homology PDF eBook
Author András Juhász
Publisher American Mathematical Society
Pages 174
Release 2021-12-09
Genre Mathematics
ISBN 1470449722

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