Interfering circle combs and uniform hyperbolicity of cocycles

We study interference of two circle combs shifted one to another. Assuming the number of jags of the combs to be large, we estimate the Lebesgue measure of those values of the shift parameter which correspond to overlapping of jags with some given precision. We apply the obtained results to study hyperbolic properties of cocycles generated by a smooth family of skew products F
A,t
= (σ
ω
, A
t
) over irrational rotation σ
ω
(x) = x + ω of a circle ${\mathbb{T}^1}$ with a real parameter t. It is supposed that the transformation ${A_t}:{\mathbb{T}^1} \to SL(2,\mathbb{R})$ is of the form A
t
(x) = R(φ (x))Z(λ(x)), where R(φ) stands for a rotation in ℝ
²
over an angle φ and Z(λ) = diag{λ, λ
⁻¹
} is a diagonal matrix. Under assumption that cos(x) possesses only two simple zeroes and λ(x) ≥ λ
0
≫ 1, we obtain a low bound for the Lebesgue measure of values of the parameter t which correspond to the uniform hyperbolicity for the cocycle.