Extensions with shrinking fibers

2018/12/20 by Benoît Kloeckner, Kloeckner, Benoit
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #Complex Systems and Time Series Analysis #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.1812.08437

openalex publication_date 2018/12/20 · openalex created_date 2020/05/01 · openalex updated_date 2026/07/28

Abstract

We consider dynamical systems T: X → X that are extensions of a factor S: Y → Y through a projection π: X → Y with shrinking fibers, i.e. such that T is uniformly continuous along fibers π^-1(y) and the diameter of iterate images of fibers Tⁿ(π^-1(y)) uniformly go to zero as n → ∞.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.

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