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 circle $\mathbb{T}^{1}$. It is supposed that the transformation $A: \mathbb{T}^{1} \to SL(2, \mathbb{R})$ being $C^{1}$-map has form $A(x) R(\varphi(x)) Z(\lambda(x))$, where $R(\varphi)$ in $\mathbb{R}^{2}$ angle $\varphi$ and $Z(\lambda)= diag\{\lambda, \lambda^{-1}\}$ diagonal matrix. Assuming $\lambda(x) \ge \lambda_{0} > 1$ with sufficiently large constant $\lambda_{0}$ function be such $\cos \varphi(x)$ possesses only simple zeroes, we study hyperbolic properties cocycle generated by $F_{A}$. apply critical set method to show that, under some additional requirements on derivative $\varphi$, secondary collisions compensate weakening hyperbolicity due primary $F_{A}$ becomes contrary case when can partially eliminated.