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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 $Y$.
We consider both geometric and measure-theoretic shrinking targets for ergodic maps, investigating when they are visible or invisible. Some Baire category theorems are proved, and particular constructions are given when the underlying map is fixed. O
In this paper we study the existence of positive Lyapunov exponents for three different types of skew products, whose fibers are compact Riemannian surfaces and the action on the fibers are by volume preserving diffeomorphisms. These three types incl
A zero-dimensional (resp. symbolic) flow is a suspension flow over a zero-dimensional system (resp. a subshift). We show that any topological flow admits a principal extension by a zero-dimensional flow. Following [Bur19] we deduce that any topologic
The culmination of the papers (arXiv:0905.0518, arXiv:0910.0909) was a proof of the norm convergence in $L^2(mu)$ of the quadratic nonconventional ergodic averages frac{1}{N}sum_{n=1}^N(f_1circ T_1^{n^2})(f_2circ T_1^{n^2}T_2^n)quadquad f_1,f_2in L^i
Symbolic Extension Entropy Theorem (SEET) describes the possibility of a lossless digitalization of a dynamical system by extending it to a subshift. It gives an estimate on the entropy of symbolic extensions (and the necessary number of symbols). Un