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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,), and the relevance of this result for the perturbation theory of lower-dimensional tori in Hamiltonian systems.

MSC:
34C20Transformation and reduction of ODE and systems, normal forms
34C27Almost and pseudo-almost periodic solutions of ODE
81Q05Closed and approximate solutions to quantum-mechanical equations
70H12Periodic and almost periodic solutions (mechanics of particles and systems)
37J40Perturbations, normal forms, small divisors, KAM theory, Arnol’d diffusion
37C55Periodic and quasiperiodic flows and diffeomorphisms
37H99Random dynamical systems