Reducibility of quasiperiodic cocycles in linear Lie groups

Abstract

Let G be a linear Lie group. We define the G

Strong almost reducibility for analytic and Gevrey quasi-periodic cocycles

“…In comparison with the real framework, the symplectic framework does not introduce any new constraints in the elimination of resonances; therefore there is no more loss of periodicity here than in the case when G = GL(n, R). As before in [2], a single period doubling is sufficient in the case when G is a real symplectic group.…”

“…In dimension 2 or if G is gl(n, C) or u(n), these results can be rephrased as density of reducible cocycles in the neighbourhood of constant cocycles: Theorem 1.3 Let G = gl(n, C), u(n), gl(2, R), sl(2, R) or o (2). Let 0 < r ′ < r ≤ 1 2 and A ∈ G, F ∈ C ω r (T d , G).…”

Reducibility of ultra-differentiable quasi-periodic cocycles under an adapted arithmetic condition

“…Since the matrices actually take values in sl(2, R), a perhaps more natural definition would be to require that the conjugacy Y takes value in the corresponding Lie group SL(2, R): it follows from the work in [3] that these two definitions are the same. In order to get such a reducibility, one need to impose regularity assumptions on F and an arithmetic condition on ω (we will also impose a similar arithmetic condition on the fibered rotation number).…”

“…Argue by contradiction that (ω, A + F ) is reducible. Since it takes values in so(2, R), it follows from [3] that it is reducible by a transformation that takes values in SO(2, R), therefore there exists v :…”

Continuous Spectrum or Measurable Reducibility for Quasiperiodic Cocycles in $${\mathbb{T} ^{d} \times SU(2)}$$ T d × S U ( 2 )

“…Further, and certainly non-exhaustive, literature in the subject includes the works of Chavaudret [Cha11,Cha12,Cha13], also in collaboration with St. Marmi [CM12] and with Stolovich [CS] HP13], Zhou [YZ13], and the paper of Avila-FayadKocsard [AFK12], which triggered this finer study that we took up in our recent papers.…”