Renormalization and central limit theorem for critical dynamical systems with weak external random noise

We study the effect on weak random noise on one dimensional critical dynamical systems, that is, maps with a renormalization theory. We establish a one dimensional central limit theorem for weak noises and obtain Berry-Esseen estimates for the rate of this convergence. We analyze in detail maps near the accumulation of period doubling and critical maps of the circle with golden mean rotation number. We derive scaling relations for several features of the effective noise after long times. These scaling relations are used to show that the central limit theorem for weak noise holds in both examples. We perform several numerical experiments that confirm our results and that suggest several conjectures.