Anosov–Katok Constructions for Quasi-Periodic $\textrm{SL}(2,{{\mathbb {R}}})$ Cocycles

Abstract

We prove that if the frequency of the quasi-periodic $\textrm{SL}(2,{{\mathbb {R}}})$ cocycle is Diophantine, then each of the following properties is 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 with a subcritical potential has power-law decaying eigenfunctions. The proof is based on fibered Anosov–Katok constructions for quasi-periodic $\textrm{SL}(2,{{\mathbb {R}}})$ cocycles.

Funding

National Key R &D Program of China(2020YFA0713300)