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 $Y$.

中文翻译：

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带有收缩纤维的延长线

我们考虑动力系统$T:X\右箭头 X$是一个因素的延伸$S:Y\右箭头 Y$通过投影$\unicode[STIX]{x1D70B}:X\rightarrow Y$和缩小纤维，也就是说，这样$T$沿纤维均匀连续$\unicode[STIX]{x1D70B}^{-1}(y)$以及纤维迭代图像的直径$T^{n}(\unicode[STIX]{x1D70B}^{-1}(y))$一致地归零为$n\rightarrow \infty$. 我们证明每$新元- 不变测度$\check{\unicode[STIX]{x1D707}}$有一个独特的$T$- 不变的提升$\unicode[STIX]{x1D707}$，并证明许多性质$\check{\unicode[STIX]{x1D707}}$提升至$\unicode[STIX]{x1D707}$：遍历性、弱混合和强混合、相关性和统计特性的衰减（可能随着速率的减弱）。基本工具是 Wasserstein 距离的变体，通过将最优传输范式约束为沿纤维的位移而获得。我们将经典论证扩展到一般环境，使我们能够在两者之间来回转换势能和可观察值$X$和$Y$.