# Extensions with shrinking fibers

## TL;DR

This paper studies dynamical systems with shrinking fibers, showing that invariant measures and many statistical properties lift from the base system to the extension, using a specialized Wasserstein distance.

## Contribution

It introduces a method to lift invariant measures and properties in systems with shrinking fibers, extending classical arguments with a new Wasserstein distance variant.

## Key findings

- Unique invariant measure lift for systems with shrinking fibers
- Properties like ergodicity and mixing are preserved in the lift
- Method applicable to general settings using a fiber-constrained Wasserstein distance

## 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 $Y$.