A Kam Scheme for SL(2, $${{\mathbb R}}$$ ) Cocycles with Liouvillean Frequencies

Abstract

We develop a new KAM scheme that applies to SL(2, R) cocycles with one frequency, irrespective of any Diophantine condition on the base dynamics. It gives a generalization of Dinaburg-Sinai's theorem to arbitrary frequencies: under a closeness to constant assumption, the non-Abelian part of the classical reducibility problem can always be solved for a positive measure set of parameters.

On topological genericity of the mode-locking phenomenon

“…Theorem 1 can be seen as a nonlinear version of the reducibility result [4, Theorem 5] for SL(2, R)−cocycles 1 . For SL(2, R)−cocycles with high regularity over torus translations, KAM method and Renormalization (see [13,6,5]) are effective tools for studying reducibility. While in our case, we use a topological method.…”

Upper bounds on the spectral gaps of quasi-periodic Schrödinger operators with Liouville frequencies

“…(2) Under the assumption 0 ≤ |λ| ≤ e −Cβ(α) , Liu and Yuan [31] proved that there is no collapsed spectral gap, i.e., E + m > E − m for any nonzero integer m. 4). We expect the optimal decay in (1.4) to be C|λ| |m| .…”

Full measure reducibility and localization for quasiperiodic Jacobi operators: A topological criterion

“…The key observation is that the argument of Section 2 of [12], while formulated there for discrete one dimensional Schrödinger operators, works verbatim for any dicrete one-dimensional operator with simple eigenvalues. 4 . Thus we get for a.e.…”

“…x a complete set of eigenfunctions {v l,j (x, ·)} l,j with eigenvalues {e l,j (x)} so that (1) for each fixed l, j, v l,j (x, ·) and e l,j (x) are measurable functions of x. 4 The existence of measurable enumeration of eigenfunctions was proved, in great generality in [13]. However, since we need a covariant representation satisfying (3.21) the argument of [12] is better suited to our needs (2) {v l,j (x, ·)} j are attached to j.…”

“…The density of states measure of H c (and thus of H s ,H c andH s ) is absolutely continuous.Proof. By[4] (see Lemma 1.4 therein), if (α,Ã |s|,E ) is L 2 -degree 0 reducible for a.e. E with respect to the density of states measure, then (α,Ã |s|,E ) is C ω rotation-reducible for E ∈ U where U is a set with dN |s| (U ) = 1.…”