Anderson localization for CMV matrices with Verblunsky coefficients defined by the hyperbolic toral automorphism

TL;DR

This paper establishes Anderson localization for CMV matrices with Verblunsky coefficients generated by hyperbolic toral automorphisms, extending key results from Schrödinger operators to this setting.

Contribution

It introduces large deviation estimates and localization results for CMV matrices with dynamically defined coefficients, bridging techniques from Schrödinger operators.

Findings

Proves large deviation estimates for CMV matrices

Establishes Anderson localization for these matrices

Extends positivity results of Lyapunov exponents to this context

Abstract

In this paper, we prove the large deviation estimates and Anderson localization for CMV matrices on $\ell^2(\mathbb{Z}_+)$ with Verblunsky coefficients defined dynamically by the hyperbolic toral automorphism. Part of positivity results on the Lyapunov exponents of Chulaevsky-Spencer and Anderson localization results of Bourgain-Schlag on Schr\"{o}dinger operators with strongly mixing potentials are extended to CMV matrices.