On hyperbolicity of close to piecewise constant linear cocycles over irrational rotations

We study a family of skew products $\boldsymbol{F_{A,t}=(\sigma_{\omega},\ A_{t})}$ over irrational rotation $\sigma_{\omega}(x)=x+\omega$ of a circle $\boldsymbol{\mathbb{T}^{1}}$, which depend on a real parameter $t$. It is supposed that the transformation $A_{t}\in C(\mathbb{T}^{1},\ SL(2,\mathbb{R}))$ is of the form $A_{t}(x)=R(\varphi(x))Z(\lambda(x))$, where $R(\varphi)$ stands for a rotation in $\mathbb{R}^{2}$ over an angle $\varphi$ and $Z(\lambda)=\text{diag}\{\lambda,\lambda^{-1}\}$ is a diagonal matrix. Assuming $\lambda(x)\geq\lambda_{\mathrm{O}}\gg 1$ and the function $\varphi$ to be piecewise linear such that $\cos(x)$ possesses only simple zeroes, we study the problem of uniform hyperbolicity for the cocycle generated by $\boldsymbol{F_{A,t}}$. We apply the critical set method to formulate sufficient conditions on the parameter values which guarantee the uniform hyperbolicity of the cocycle. Application to the Schrödinger cocycles is also discussed.