Monotone Families of Circle Diffeomorphisms Driven by Expanding Circle Maps

We consider monotone families of circle diffeomorphisms forced by the strongly chaotic circle endomorphisms \(x\mapsto bx \mod 1\) $x\mapsto bxmod1$ , where the integer b is large. We obtain estimates of the fibered Lyapunov exponents and show that in the limit as b tends to infinity, they approach the values of the Lyapunov exponents for the corresponding random case. The estimates are based on a control of the distribution of the iterates of almost every point, up to a fixed (small) scale, depending on b.