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Almost reducibility of linear quasi-periodic systems. (English) Zbl 1015.34028
Katok, Anatole (ed.) et al., Smooth ergodic theory and its applications. Proceedings of the AMS summer research institute, Seattle, WA, USA, July 26-August 13, 1999. Providence, RI: American Mathematical Society (AMS). Proc. Symp. Pure Math. 69, 679-705 (2001).
The author’s abstract: Reducibility and almost reducibility of linear quasiperiodic systems are discussed. The author proves a result showing that when such a system is analytic and close to constant coefficients and has Diophantine quasiperiodic frequencies it is always almost reducible in the sense that it can be analytically conjugate to a system arbitrarily close to constant coefficients. It is discussed when this approach converges and gives full reducibility, and when it does not. The author studies in particular the quasiperiodic Schrödinger equation and skew-products on \(SO(3,\mathbb{R})\), and the relevance of this result for the perturbation theory of lower-dimensional tori in Hamiltonian systems.
For the entire collection see [Zbl 0973.00044].

34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
34C27 Almost and pseudo-almost periodic solutions to ordinary differential equations
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
70H12 Periodic and almost periodic solutions for problems in Hamiltonian and Lagrangian mechanics
37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
37C55 Periodic and quasi-periodic flows and diffeomorphisms
37H99 Random dynamical systems