H\"older continuity of Lyapunov exponent for quasi-periodic Jacobi operators

Abstract

We consider the quasi-periodic Jacobi operator $H_{x,\omega}$ in $l^2(\mathbb{Z})$ $(H_{x,\omega}\phi)(n) = -b(x+(n+1)\omega)\phi(n+1) - b(x+n\omega)\phi(n-1) + a(x+n\omega)\phi(n) = E\phi(n),\ n\in\mathbb{Z},$ where $a(x),\ b(x)$ are analytic function on $\mathbb{T}$, $b$ is not identically zero, and $\omega$ obeys some strong Diophantine condition. We consider the corresponding unimodular cocycle. We prove that if the Lyapunov exponent $L(E)$ of the cocycle is positive for some $E=E_0$, then there exists $\rho_0=\rho_0(a,b,\omega,E_0)$, $\beta=\beta(a,b,\omega)$ such that $|L(E)-L(E')|<|E-E'|^\beta$ for any $E,E'\in (E_0-\rho_0,E_0+\rho_0)$. If $L(E)>0$ for all $E$ in some compact interval $I$ then $L(E)$ is H\"{o}lder continuous on $I$ with a H\"{o}lder exponent $\beta=\beta(a,b,\omega,I)$. In our derivation we follow the refined version of the Goldstein-Schlag method \cite{GS} developed by Bourgain and Jitomirskaya \cite{BJ}.