On the dynamics of quasi-periodic Schrödinger cocycles for positive measure sets of frequencies
Kristian Bjerklöv
KTH Royal Institute of Technology, Stockholm, Sweden

Abstract
We consider the family of one-frequency quasi-periodic Schrödinger coycles , parametrized by the energy . For potential functions , where is a Morse function with finitely many critical points and is large, we show that, for any value of and for any phase such that is not too small, there exists a set of frequencies of positive measure such that the following hold: (1) for every , the upper Lyapunov exponent of the cocycle is and is (essentially) a typical point in Oseledets’ theorem; (2) either is uniformly hyperbolic, or there exists a phase such that is an eigenvalue of the corresponding discrete Schrödinger operator .
Cite this article
Kristian Bjerklöv, On the dynamics of quasi-periodic Schrödinger cocycles for positive measure sets of frequencies. J. Spectr. Theory (2025), published online first
DOI 10.4171/JST/548