Extensions with shrinking fibers
| Autor: | Kloeckner, Benoit |
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| Rok vydání: | 2018 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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 $Y$. Comment: v3 - an error is corrected in Theorem A(ii): a continuity assumption is needed to lift physicality, as shown in Remark 4.8. Other small modifications |
| Databáze: | arXiv |
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