Hölder continuity of Lyapunov exponent for quasi-periodic Jacobi operators

Abstract

We consider the quasi-periodic Jacobi operator $H_{x,\omega }$ in $l^2(\mathbb{Z})$ $(H_{x,\omega }\varphi )(n)=-b(x+(n+1)\omega )\varphi (n+1)-b(x+n\omega )\varphi (n-1)+a(x+n\omega )\varphi (n)=E\varphi (n),n\in \mathbb{Z},$ 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 $|L(E)-L(E^{\prime })|<|E-E^{\prime }{|}^{\beta }$ for any $E,E^{\prime }\in (E_0-{\rho }_0,E_0+{\rho }_0)$. If $L(E)>0$ for all $E$ in some compact interval $I$ then $L(E)$ is Hölder continuous on $I$ with a Hölder 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}.

Cited by in corpus (3)

• Dynamics and spectral theory of quasi-periodic Schrödinger-type operators
• Dominated splittings and the spectrum of singular quasi-periodic Jacobi operators
• Subcritical behavior for quasi-periodic Schrödinger cocycles with trigonometric potentials