In this talk, we consider a stability of statistical behaviours of dynamical systems with a nonhyperbolic direction ― $U(1)$-extensions of (nonlinear) expanding maps on the circle ― under small random perturbations, which is called stochastic stability. In particular, we investigate whether the exponential decay of quenched correlation functions is preserved under small random perturbations.

This talk is based on a joint work with Masato Tsujii and Jens Wittsten.