Monotone Families of Circle Diffeomorphisms Driven by Expanding Circle Maps

Abstract We consider monotone families of circle diffeomorphisms forced by the strongly chaotic circle endomorphisms $$x\mapsto bx \mod 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>↦</mml:mo> <mml:mi>b</mml:mi> <mml:mi>x</mml:mi> <mml:mspace/> <mml:mo>mod</mml:mo> <mml:mspace/> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> , 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 .