Anosov–Katok Constructions for Quasi-Periodic $\textrm{SL}(2,{{\mathbb {R}}})$ Cocycles
Nikolaos Karaliolios , Xu Xu , Qi Zhou
Peking Mathematical Journal ›› 2022, Vol. 7 ›› Issue (1) : 203 -245.
Anosov–Katok Constructions for Quasi-Periodic $\textrm{SL}(2,{{\mathbb {R}}})$ Cocycles
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.
National Key R &D Program of China(2020YFA0713300)
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