  We show that genuinely higher rank expanding actions of abelian semi-groups
on compact manifolds are $C^{\infty}$-conjugate to affine actions on
infra-nilmanifolds. This is based on the classification of expanding
diffeomorphisms up to \holder conjugacy by Gromov and Shub, and is similar to
recent work on smooth classification of higher rank Anosov actions on tori and
nilmanifolds. To prove regularity of the conjugacy in the higher rank setting,
we establish exponential mixing of solenoid actions induced from semi-group
actions by nilmanifold endomorphisms, a result of independent interest. We then
proceed similar to the case of higher rank Anosov actions.
