A Diophantine duality applied to the KAM and Nekhoroshev theorems
TLDR
In this paper, a new approach to the perturbation theory for quasi-periodic solutions dealing only with periodic approximations and avoiding classical small divisors estimates was developed.Abstract:
In this paper, we use geometry of numbers to relate two dual Diophantine problems. This allows us to focus on simultaneous approximations rather than small linear forms. As a consequence, we develop a new approach to the perturbation theory for quasi-periodic solutions dealing only with periodic approximations and avoiding classical small divisors estimates. We obtain two results of stability, in the spirit of the KAM and Nekhoroshev theorems, in the model case of a perturbation of a constant vector field on the $$n$$
-dimensional torus. Our first result, which is a Nekhoroshev type theorem, is the construction of a “partial” normal form, that is a normal form with a small remainder whose size depends on the Diophantine properties of the vector. Then, assuming our vector satisfies the Bruno–Russmann condition, we construct an “inverted” normal form, recovering the classical KAM theorem of Kolmogorov, Arnold and Moser for constant vector fields on torus.read more
Citations
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Normal forms, stability and splitting of invariant manifolds I. Gevrey Hamiltonians
TL;DR: In this paper, a new construction of resonant normal forms with a small remainder for near-integrable Hamiltonians at a quasi-periodic frequency was given, based on the special case of a periodic frequency, a Diophantine result concerning the approximation of a vector by independent periodic vectors and a technique of composition of periodic averaging.
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KAM, α -Gevrey regularity and the α -Bruno-Rüssmann condition
Abed Bounemoura,Jacques Féjoz +1 more
TL;DR: In this article, a new invariant torus theorem was proved for α-Gevrey smooth Hamiltonian systems under an arithmetic assumption called the α-Bruno-Russmann condition, which reduces to the classical RB condition in the analytic category.
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Non-degenerate Liouville tori are KAM stable
TL;DR: In this article, it was shown that a quasi-periodic torus, with a non-resonant frequency and invariant to a sufficiently regular Hamiltonian flow, is KAM stable provided it is Kolmogorov non-degenerate.
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Hamiltonian perturbation theory for ultra-differentiable functions
TL;DR: In this paper, the invariant torus theorem and Nekhoroshev's theorem were shown to be true for non-linear functional systems with Gevrey functions in terms of a real sequence M bounding the growth of derivatives.
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On Invariant Tori with Prescribed Frequency in Hamiltonian Systems
TL;DR: In this paper, the persistence of invariant tori with prescribed frequency for analytic nearly integrable Hamiltonian systems under the Brjuno-Rüssmann non-resonant condition is investigated.
References
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Book
An Introduction to the Geometry of Numbers
TL;DR: In this article, the authors introduce the concept of the quotient space and the notion of automorphs for diophantine approximations of diophantas in the Euclidean space.
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