We consider dynamical systems $T: X to X$ that are extensions of a factor $S: Y to Y$ through a projection $pi: X to Y$ with shrinking fibers, i.e. such that $T$ is uniformly continuous along fibers $pi^{-1}(y)$ and the diameter of iterate images of fibers $T^n(pi^{-1}(y))$ uniformly go to zero as $n to infty$.We prove that every $S$-invariant measure has a unique $T$-invariant lift, and prove that many properties of the original measure lift: ergodicity, weak and strong mixing, decay of correlations and statistical properties (possibly with weakening in the rates).The basic tool is a variation of the Wasserstein distance, obtained by constraining the optimal transportation paradigm to displacements along the fibers. We extend to a general setting classical arguments, enabling to translate potentials and observables back and forth between $X$ and $Y$.
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