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

We prove that if the frequency of the quasi-periodic $\mathrm{S}\mathrm{L}(2,\mathbb{R})$ 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 $\kappa$-H"older 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 $\mathrm{S}\mathrm{L}(2,\mathbb{R})$ cocycles.