  Using optimal transport we study some dynamical properties of expanding
circle maps acting on measures by push-forward.
  Using the definition of the tangent space to the space of measures introduced
by Gigli, their derivative at the unique absolutely continuous invariant
measure is computed. In particular it is shown that 1 is an eigenvalue of
infinite multiplicity, so that the invariant measure admits many deformations
into nearly invariant ones. As a consequence, we obtain counter-examples to an
infinitesimal version of Furstenberg's conjecture.
  We also prove that this action has positive metric mean dimension with
respect to the Wasserstein metric.
