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Register a new userKAM Theory. Part II. Kolmogorov spaces
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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 the exponential and therefore relation between Lie algebra and Lie group actions in the infinite dimensional context. Then we introduce a notion of finite dimensional reduction and use it to prove a fixed point theorem for Kolmogorov spaces. We conclude by proving general normal theorems.
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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 Kolmogorovs invariant torus theorem and make it more general allowing discussion of arbitrary invariant Lagrangian varieties over general Poisson algebras. We include it into the general problem of normal forms and group actions. We explain the iteration method used by Kolmogorov by giving a finite dimensional analog. Part I explains in which context we apply the theory of Kolmogorov spaces which will form the core of Part II.
In this paper we reconsider the original Kolmogorov normal form algorithm with a variation on the handling of the frequencies. At difference with respect to the Kolmogorov approach, we do not keep the frequencies fixed along the normalization procedure. Besides we select the frequencies of the final invariant torus and determine a posteriori the corresponding starting one. In particular, we replace the classical translation step with a change of the frequencies. The algorithm is based on classical expansion in a small parameter and particular attention is paid to the constructive aspect: we produce an explicit algorithm that can be effectively applied, e.g., with the aid of an algebraic manipulator, and that we prove to be absolutely convergent.
Lectures given on KAM theory at the University of Ouargla (Algeria). I present a functional analytic treatment of the subject which includes KAM theory into the general framework of deformations and singularity theory.
We show that group actions on many treelike compact spaces are not too complicated dynamically. We first observe that an old argument of Seidler implies that every action of a topological group $G$ on a regular continuum is null and therefore also tame. As every local dendron is regular, one concludes that every action of $G$ on a local dendron is null. We then use a more direct method to show that every continuous group action of $G$ on a dendron is Rosenthal representable, hence also tame. Similar results are obtained for median pretrees. As a related result we show that Hellys selection principle can be extended to bounded monotone sequences defined on median pretrees (e.g., dendrons or linearly ordered sets). Finally, we point out some applications of these results to continuous group actions on dendrites.
By establishing Multiplicative Ergodic Theorem for commutative transformations on a separable infinite dimensional Hilbert space, in this paper, we investigate Pesins entropy formula and SRB measures of a finitely generated random transformations on such space via its commuting generators. Moreover, as an application, we give a formula of Friedlands entropy for certain $C^{2}$ $mathbb{N}^2$-actions.
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