Monotone families of circle diffeomorphisms driven by expanding circle maps (Q6593159)

The authors consider monotone families \(f_x(x\in\mathbb T)\) of circle diffeomorphisms forced by the strongly chaotic map \(T:x\longmapsto bx\) on \(\mathbb T\), where the integer \(b\ge 2\) is very large. More specifically, assume that \(f:\mathbb T^2\to \mathbb T^2\) is \(C^2\) and such that for all \((x,y)\in \mathbb T^2\),\N\[\N\frac{\partial f}{\partial x}(x,y)>0 \hbox{ and } \frac{\partial f}{\partial y}(x,y)>0.\N\]\NMoreover, assume that \(f_x:y\longmapsto f(x,y)\) and \(u_y:x\longmapsto f(x,y)\) both have degree \(1\).\N\NThe authors obtain estimates of the fibered Lyapunov exponents. Namely, for any \(\epsilon>0\), there exists an integer \(b_0>2\) such that for every \(b\ge b_0\),\N\[\NL_0-\epsilon<\underline{L}(x,y)\le \bar{L}(x,y)<L_0+\epsilon \hbox{ for a.e. } (x,y)\in \mathbb T^2.\N\]\NMoreover, the authors show that as \(b\) tends to infinity, the limit approach to the value of the Lyapunov exponent for the corresponding random case. The strategy is to control the distribution of the orbits of a.e. \((x,y)\) under the skew-product \(\mathcal F\), up to a fixed scale.