Positive Lyapunov exponent for random perturbations of predominantly expanding multimodal circle maps (Q2693069)

Summary: We study the effects of independent, identically distributed random perturbations of amplitude \(\varepsilon>0\) on the asymptotic dynamics of one-parameter families \(\{f_a:S^1\to S^1,a\in[0,1]\}\) of smooth multimodal maps which are ``predominantly expanding'', i.e., \(|f'_a|\gg 1\) away from small neighborhoods of the critical set \(\{f'_a=0\}\). We obtain, for any \(\varepsilon>0\), a \textit{checkable, finite-time} criterion on the parameter \(a\) for random perturbations of the map \(f_a\) to exhibit (i) a unique stationary measure and (ii) a positive Lyapunov exponent comparable to \(\int_{S^1}\log|f'_a|dx\). This stands in contrast with the situation for the deterministic dynamics of \(f_a\), the chaotic regimes of which are determined by typically uncheckable, infinite-time conditions. Moreover, our finite-time criterion depends on only \(k\sim\log(\varepsilon^{-1})\) iterates of the deterministic dynamics of \(f_a\). which grows quite slowly as \(\varepsilon\to 0\).