On $SL(2,\mathbb{R})$-Cocycles over Irrational Rotations with Secondary Collisions

We consider a skew product $F_{A} = (\sigma_{\omega}, A)$ over irrational rotation $\sigma_{\omega}(x) = x + \omega$ of a circle $\mathbb{T}^{1}$. It is supposed that the transformation $A: \mathbb{T}^{1} \to SL(2, \mathbb{R})$ which is a $C^{1}$-map has the form $A(x) = R\big(\varphi(x)\big) Z\big(\lambda(x)\big)$, where $R(\varphi)$ is a rotation in $\mathbb{R}^{2}$ through the angle $\varphi$ and $Z(\lambda)= \text{diag}\{\lambda, \lambda^{-1}\}$ is a diagonal matrix. Assuming that $\lambda(x) \geqslant \lambda_{0} > 1$ with a sufficiently large constant $\lambda_{0}$ and the function $\varphi$ is such that $\cos \varphi(x)$ possesses only simple zeroes, we study hyperbolic properties of the cocycle generated by $F_{A}$. We apply the critical set method to show that, under some additional requirements on the derivative of the function $\varphi$, the secondary collisions compensate weakening of the hyperbolicity due to primary collisions and the cocycle generated by $F_{A}$ becomes uniformly hyperbolic in contrast to the case where secondary collisions can be partially eliminated.