Seminar 9.6.18 Yang | Ergodic Theory at Ohio State

Title: Random perturbations of predominantly expanding multimodal circle maps

Speaker: Yun Yang (CUNY)

Abstract: In this talk we will study the effects of IID random perturbations of amplitude $ϵ > 0$ on the asymptotic dynamics of one-parameter families ${f_{a} : S^{1} \to S^{1}, a \in [0, 1]}$ of smooth multimodal maps which “predominantly expanding”, i.e., $| f_{a}^{'} | ≫ 1$ away from small neighborhoods of the critical set ${f_{a}^{'} = 0}$ . We will obtain, for any $ϵ > 0$ , 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 $\int_{S^{1}} \log | f_{a}^{'} | d x$ .