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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Instability, Index Theorem, and Exponential Trichotomy for Linear Hamiltonian PDEs

Instability, Index Theorem, and Exponential Trichotomy for Linear Hamiltonian PDEs
Title Instability, Index Theorem, and Exponential Trichotomy for Linear Hamiltonian PDEs PDF eBook
Author Zhiwu Lin
Publisher American Mathematical Society
Pages 136
Release 2022-02-02
Genre Mathematics
ISBN 1470450445

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Non-Semisimple Extended Topological Quantum Field Theories

Non-Semisimple Extended Topological Quantum Field Theories
Title Non-Semisimple Extended Topological Quantum Field Theories PDF eBook
Author Marco De Renzi
Publisher American Mathematical Society
Pages 161
Release 2022-05-24
Genre Mathematics
ISBN 1470452693

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On the Asymptotics to all Orders of the Riemann Zeta Function and of a Two-Parameter Generalization of the Riemann Zeta Function

On the Asymptotics to all Orders of the Riemann Zeta Function and of a Two-Parameter Generalization of the Riemann Zeta Function
Title On the Asymptotics to all Orders of the Riemann Zeta Function and of a Two-Parameter Generalization of the Riemann Zeta Function PDF eBook
Author Athanassios S. Fokas
Publisher American Mathematical Society
Pages 114
Release 2022-02-02
Genre Mathematics
ISBN 1470450984

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Elliptic Theory for Sets with Higher Co-Dimensional Boundaries

Elliptic Theory for Sets with Higher Co-Dimensional Boundaries
Title Elliptic Theory for Sets with Higher Co-Dimensional Boundaries PDF eBook
Author Guy David
Publisher American Mathematical Society
Pages 123
Release 2021-12-30
Genre Mathematics
ISBN 1470450437

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Cell Complexes, Poset Topology and the Representation Theory of Algebras Arising in Algebraic Combinatorics and Discrete Geometry

Cell Complexes, Poset Topology and the Representation Theory of Algebras Arising in Algebraic Combinatorics and Discrete Geometry
Title Cell Complexes, Poset Topology and the Representation Theory of Algebras Arising in Algebraic Combinatorics and Discrete Geometry PDF eBook
Author Stuart Margolis
Publisher American Mathematical Society
Pages 135
Release 2021-12-30
Genre Mathematics
ISBN 1470450429

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