# Extensions with shrinking fibers

Abstract : 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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Preprints, Working Papers, ...

Cited literature [47 references]

https://hal.archives-ouvertes.fr/hal-01961053
Contributor : Benoît Kloeckner <>
Submitted on : Friday, February 7, 2020 - 3:01:15 PM
Last modification on : Thursday, February 13, 2020 - 1:35:09 AM

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### Identifiers

• HAL Id : hal-01961053, version 2
• ARXIV : 1812.08437

### Citation

Benoit Kloeckner. Extensions with shrinking fibers. 2020. ⟨hal-01961053v2⟩

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