Abstract:
We consider a skew product FA=(σω,A) over irrational rotation σω(x)=x+ω of a circle T1. It is supposed that the transformation A:T1→SL(2,R)
which is a C1-map has the form A(x)=R(φ(x))Z(λ(x)), where R(φ) is a rotation in R2 through the angle φ and Z(λ)=diag{λ,λ−1} is a diagonal matrix. Assuming that λ(x)⩾λ0>1 with a sufficiently large constant λ0 and the function φ
is such that cosφ(x) possesses only simple zeroes, we study hyperbolic properties of
the cocycle generated by FA. 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 FA becomes uniformly hyperbolic
in contrast to the case where secondary collisions can be partially eliminated.
Keywords:
linear cocycle, hyperbolicity, Lyapunov exponent, critical set.