  We consider the question of exponential mixing for random dynamical systems
on arbitrary compact manifolds without boundary. We put forward a robust,
dynamics-based framework that allows us to construct space-time smooth,
uniformly bounded in time, universal exponential mixers. The framework is then
applied to the problem of proving exponential mixing in a classical example
proposed by Pierrehumbert in 1994, consisting of alternating periodic shear
flows with randomized phases. This settles a longstanding open problem on
proving the existence of a space-time smooth (universal) exponentially mixing
incompressible velocity field on a two-dimensional periodic domain while also
providing a toolbox for constructing such smooth universal mixers in all
dimensions.
