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Extensions with shrinking fibers

Part of: Ergodic theory Dynamical systems with hyperbolic behavior

Published online by Cambridge University Press:  24 April 2020

BENOÎT R. KLOECKNER Open the ORCID record for BENOÎT R. KLOECKNER [Opens in a new window]
BENOÎT R. KLOECKNER*
Affiliation:
LAMA, Univ. Paris-Est Créteil, Univ. Gustave Eiffel, UPEM, CNRS, F-94010, Créteil, France email [email protected]
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Abstract

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 $Y$.

Keywords

classical ergodic theorysmooth dynamicstopological dynamics

MSC classification

Primary: 37D35: Thermodynamic formalism, variational principles, equilibrium states
Secondary: 37D25: Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.) 37A25: Ergodicity, mixing, rates of mixing 37A30: Ergodic theorems, spectral theory, Markov operators 37A35: Entropy and other invariants, isomorphism, classification
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 6 , June 2021 , pp. 1795 - 1834
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Copyright
© The Author(s) 2020. Published by Cambridge University Press

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