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.