Extensions with shrinking fibers

Abstract

We consider dynamical systems : → that are extensions of a factor : → through a projection : → with shrinking fibers, i.e. such that is uniformly continuous along fibers −1 ( ) and the diameter of iterate images of fibers ( −1 ( )) uniformly go to zero as → ∞. We prove that every -invariant measureˇhas a unique -invariant lift , and prove that many properties ofˇlift to : 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 and .

Hölder regularity and exponential decay of correlations for a class of piecewise partially hyperbolic maps

“…Other applications of this metric to obtain limit theorems can be seen in [21,22]. For instance, in [21] the author apply this metric to a more general case of shrinking fibres systems. Definition 3.4.…”

“…These vector spaces are constructed by identifying a measure on the square, M × K, with a path of measures M −→ SM(K) (SM(K) denotes the space of signed measures on K), where SM(K) is endowed with the 'dual of Hölder' norm. This is achieved by generalizing the Wasserstein-Kantorovich-like norm defined in [15] (see also [21] for similar applications of the Wasserstein distance to obtain limit theorems) and this is enough to obtain the limit theorems on a larger Banach space of functions as an application of theorem 4, where we prove that the disintegration of the physical F-invariant measure is ζ-Hölder regular. This sort of result has many other applications.…”

Random‐like properties of chaotic forcing

“…The literature on skew‐products is vast to the extent that there are entire research trends studying particular aspects of these systems (e.g., iterated function systems, random dynamical systems, smoothness of invariant graphs over skew‐products, etc.). Here we focus on those works dealing with statistical properties of skew products that have a “deterministic” base, such as [6–8, 11, 22–26, 28, 29, 34, 38, 41, 44, 47, 51] and references therein. These works usually only require $$g$$ to be a measure preserving ergodic transformation or, at most, to exhibit some uniform hyperbolicity.…”