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.