Optimal transport and dynamics of expanding circle maps acting on measures
Résumé
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. We also prove that this action has positive metric mean dimension with respect to the Wasserstein metric.
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