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 $Y$.

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

References in corpus (1)

• Rates in almost sure invariance principle for slowly mixing dynamical systems