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Extensions with shrinking fibers. (arXiv:1812.08437v1 [math.DS])
来源于:arXiv
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 $Y$. 查看全文>>