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

Abstract

We study the effects of independent, identically distributed random perturbations of amplitude $\epsilon >0$ 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., $|f_a^{\prime }|\gg 1$ away from small neighborhoods of the critical set $\{f_a^{\prime }=0\}$. We obtain, for any $\epsilon >0$, a 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 ${\int }_{S^1}\log |f_a^{\prime }|dx$. 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 $k\sim \log ({\epsilon }^{-1})$ iterates of the deterministic dynamics of $f_a$, which grows quite slowly as $\epsilon \to 0$.