Abstract
The arithmetics of the frequency and of the rotation number play a fundamental role in the study of reducibility of analytic quasiperiodic cocycles which are sufficiently close to a constant. In this paper we show how to generalize previous works by L.H. Eliasson which deal with the diophantine case so as to implement a Brjuno-Russmann arithmetical condition both on the frequency and on the rotation number. Our approach adapts the Poschel-Russmann KAM method, which was previously used in the problem of linearization of vector fields, to the problem of reducing cocycles.
Highlights
Quasiperiodic cocycles are the fundamental solutions of quasiperiodic linear systems
Where A is a continuous matrix-valued function on a torus Td and ω is a rationally independent vector of some space Rd. The dynamics of such a system can be quite complicated, they are studied in case the cocycle is reducible, i.e when there is a map Z, continuous on the double torus 2Td = Rd /2Zd, taking its values in the group of invertible matrices and such that
In order to obtain an analytic reducibility result, we will have to pick a frequency and a rotation number with good approximation properties, in the sense of Rüssmann ([10]): ω will have to satisfy a strong irrationality condition controlled by an approximation function G, namely
Summary
We will give a reducibility result for analytic cocycles under a weaker arithmetical condition than the diophantine one. In order to obtain an analytic reducibility result, we will have to pick a frequency and a rotation number with good approximation properties, in the sense of Rüssmann ([10]): ω will have to satisfy a strong irrationality condition controlled by an approximation function G, namely For some positive κ (Section 2.1), and ρ will have to satisfy a further arithmetical condition: its approximations by means of linear combinations of the frequencies are controlled by an approximation function g , i.e
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