Exponential law for random maps on compact manifoldsThis work was partially supported by FAPESB, CNPq, CAPES, FCT project PTDC/MAT-PUR/28177/2017, with national funds, and by CMUP (UIDB/00144/2020), which is funded by FCT with national (MCTES) and European structural funds through the programs FEDER, under the partnership agreement PT2020. N Haydn is supported by Simons Foundation: Collaboration Grants for Mathematicians: ID 526571

We consider random dynamical systems on manifolds modelled by a skew product which have certain geometric properties and whose measures satisfy quenched decay of correlations at a sufficient rate. We prove that the limiting distribution for the hitting and return times to geometric balls are both exponential for almost every realisation. We then apply this result to random C2 maps of the interval, random parabolic maps on the unit interval and random perturbation of partially hyperbolic attractors on a compact Riemannian manifold.