Related papers: Kolmogorov-Arnold-Moser Theorem
Among the impressive contributions of Andrej N. Kolmogorov's to mathematics in the 20th century, his 1954 invariant tori theorem is still little understood from a historical point of view [Dumas 2014]. Vladimir I. Arnold, who entered Moscow…
We review V.I. Arnold's 1963 celebrated paper \cite{ARV63} {\sl Proof of A.N. Kolmogorov's theorem on the conservation of conditionally periodic motions with a small variation in the Hamiltonian}, and prove that, optimizing Arnold's scheme,…
In the paper, we prove an abstract KAM (Kolmogorov-Arnold-Moser) theorem for infinite dimensional reversible systems. Using this KAM theorem, we obtain the existence and linear stability of quasi-periodic solutions for a class of reversible…
We study a quantum analogue of the iterative perturbation theory by Kolmogorov used in the proof of the Kolmogorov-Arnold-Moser (KAM) theorem. The method is based on sequent canonical transformations with a "running" coupling constant $…
The purpose of this lecture is to describe the KAM theorem in its most basic form and to give a complete and detailed proof. This proof essentially follows the traditional lines laid out by the inventors of this theory, and the emphasis is…
This paper continues the discussion started in [CK19] concerning Arnold's legacy on classical KAM theory and (some of) its modern developments. We prove a detailed and explicit `global' Arnold's KAM Theorem, which yields, in particular, the…
Regarding the representation theorem of Kolmogorov and Arnold (KA) as an algorithm for representing or <<expressing>> functions, we test its robustness by analyzing its stability to withstand re-parameterizations of the hidden space. One…
It is widespread since the beginning of KAM Theory that, under "sufficiently small" perturbation, of size $\epsilon$, apart a set of measure $O(\sqrt{\epsilon})$, all the KAM Tori of a non-degenerate integrable Hamiltonian system persist up…
The KAM theorem from a combinatorial viewpoint.
This is part I of a book on KAM theory. We start from basic symplectic geometry, review Darboux-Weinstein theorems action angle coordinates and their global obstructions. Then we explain the content of Kolmogorov's invariant torus theorem…
Kolmogorov's invariant torus theorem is proved using a simple fixed point theorem.
Due to M\"{u}ller's theorem, the Kolmogorov complexity of a string was shown to be equal to its quantum Kolmogorov complexity. Thus there are no benefits to using quantum mechanics to compress classical information. The quantitative amount…
We prove an analytic KAM-Theorem, which is used in [1], where the differential part of KAM-theory is discussed. Related theorems on analytic KAM-theory exist in the literature (e. g., among many others, [7], [8], [13]). The aim of the…
The work of Kolmogorov, Arnold and Moser appeared just before the renormalization group approach to statistical mechanics was proposed by Wilson: it can be classified as a multiscale approach which also appeared in works on the convergence…
By using the properties of the uniformly distributed sequences of real numbers on $(0,1)$, a short proof of a certain version of Kolmogorov strong law of large numbers is presented which essentially differs from Kolmogorov's original proof.
We show that for any finite-dimensional quantum systems the conserved quantities can be characterized by their robustness to small perturbations: for fragile symmetries small perturbations can lead to large deviations over long times, while…
In this paper, we investigate the sharp regularity properties of a special weighted Sobolev space defined on the $ n $-dimensional torus, which is of independent interest. As a key application, we show that for almost all $ n $-dimensional…
This is part II of our book on KAM theory. We start by defining functorial analysis and then switch to the particular case of Kolmogorov spaces. We develop functional calculus based on the notion of local operators. This allows to define…
We review Kolmogorov's 1954 fundamental paper {\sl On the Conservation of Conditionally Periodic Motions under Small Perturbation of the Hamiltonian} (Dokl. akad. nauk SSSR,1954, vol. {\bf 98}, pp.527--530), both from the historical and the…
The classical Kolmogorov-Arnold-Moser (KAM) theorem provides the underlying mechanism for the stability of the solar system under some small chaotic perturbations. Despite many previous efforts, any quantum version of the KAM theorem…