Extensions with shrinking fibers

Abstract

We consider dynamical systems $T:X\rightarrow X$ that are extensions of a factor $S:Y\rightarrow Y$ through a projection $\unicode[STIX]{x1D70B}:X\rightarrow Y$ with shrinking fibers, that is, such that $T$ is uniformly continuous along fibers $\unicode[STIX]{x1D70B}^{-1}(y)$ and the diameter of iterate images of fibers $T^{n}(\unicode[STIX]{x1D70B}^{-1}(y))$ uniformly go to zero as $n\rightarrow \infty$. We prove that every $S$-invariant measure $\check{\unicode[STIX]{x1D707}}$ has a unique $T$-invariant lift $\unicode[STIX]{x1D707}$, and prove that many properties of $\check{\unicode[STIX]{x1D707}}$ lift to $\unicode[STIX]{x1D707}$: 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 classical arguments to a general setting, enabling us to translate potentials and observables back and forth between $X$ and $Y$.