Random Young towers and quenched decay of correlations for predominantly expanding multimodal circle maps

In this paper, we study the random dynamical system $f_{\omega }^n$ generated by a family of maps $\{f_{{\omega }_0}:{\mathbb{S}}^1\to {\mathbb{S}}^1{\}}_{{\omega }_0\in [-\epsilon ,\epsilon ]},$ $f_{{\omega }_0}(x)=\alpha \xi (x+{\omega }_0)+a(\mathrm{m}\mathrm{o}\mathrm{d}1),$ where $\xi :{\mathbb{S}}^1\to \mathbb{R}$ is a non-degenerated map, $a\in [0,1)$, and $\alpha ,\epsilon >0$. Fixing a constant $c\in (0,1)$, we show that for $\alpha$ sufficiently large and for $\epsilon >{\alpha }^{-1+c},$ the random dynamical system $f_{\omega }^n$ presents a random Young tower structure and quenched decay of correlations.