Anosov-Katok constructions for quasi-periodic $\mathrm{SL}(2,R)$ cocycles

Abstract

We prove that if the frequency of the quasi-periodic $\mathrm{SL}(2,\mathbb{R})$ cocycle is Diophantine, then the following properties are dense in the subcritical regime: for any $\frac{1}{2}<\kappa <1$, the Lyapunov exponent is exactly $\kappa$-Hölder continuous; the extended eigenstates of the potential have optimal sub-linear growth; and the dual operator associated a subcritical potential has power-law decay eigenfunctions. The proof is based on fibered Anosov-Katok constructions for quasi-periodic $\mathrm{SL}(2,\mathbb{R})$ cocycles.

References in corpus (4)

• Anderson Transitions
• The absolutely continuous spectrum of the almost Mathieu operator
• Almost reducibility and absolute continuity I
• Asymptotics of spectral gaps of quasi-periodic Schrödinger operators