On linear cocycles over irrational rotations with secondary collisions

We consider a skew product F
A
= (σ
ω
,A) over irrational rotation ${\sigma _\omega }(x) = x + \omega $ of a circle ${\mathbb{T}^1}$. It is supposed that the transformation $A:{\mathbb{T}^1} \to SL(2,\mathbb{R})$, being a C
²
-map, has the form $A(x) = R(\varphi (x))Z(\lambda (x))$, where R(φ) is a rotation in ℝ
²
over the angle φ and Z(λ) = diag{λ,λ
^-1
} is a diagonal matrix. Assuming that λ(x) ≥λ
0
> 1 with a sufficiently large constant λ
o
and the function φ is such that cos φ(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 φ, the secondary collisions compensate weakening of the hyperbolicity due to primary collisions and the cocycle generated by F
A
becomes hyperbolic in contrary to the case when secondary collisions can be partially eliminated.