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A Spectral Approach for Quenched Limit Theorems for Random Expanding Dynamical Systems

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Abstract

We prove quenched versions of (i) a large deviations principle (LDP), (ii) a central limit theorem (CLT), and (iii) a local central limit theorem for non-autonomous dynamical systems. A key advance is the extension of the spectral method, commonly used in limit laws for deterministic maps, to the general random setting. We achieve this via multiplicative ergodic theory and the development of a general framework to control the regularity of Lyapunov exponents of twisted transfer operator cocycles with respect to a twist parameter. While some versions of the LDP and CLT have previously been proved with other techniques, the local central limit theorem is, to our knowledge, a completely new result, and one that demonstrates the strength of our method. Applications include non-autonomous (piecewise) expanding maps, defined by random compositions of the form \({T_{\sigma^{n-1} \omega} \circ\cdots\circ T_{\sigma\omega}\circ T_\omega}\). An important aspect of our results is that we only assume ergodicity and invertibility of the random driving \({\sigma:\Omega\to\Omega}\) ; in particular no expansivity or mixing properties are required.

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Authors and Affiliations

  1. School of Mathematics and Statistics, University of New South Wales, Sydney, NSW, 2052, Australia

    D. Dragičević & G. Froyland

  2. Department of Mathematics, University of Rijeka, Rijeka, Croatia

    D. Dragičević

  3. School of Mathematics and Physics, The University of Queensland, St. Lucia, QLD, 4072, Australia

    C. González-Tokman

  4. Université de Toulon, CNRS, CPT, UMR 7332, 83957, La Garde, France

    S. Vaienti

Authors
  1. D. Dragičević
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  2. G. Froyland
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  3. C. González-Tokman
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  4. S. Vaienti
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Correspondence to D. Dragičević.

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Communicated by C. Liverani

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Dragičević, D., Froyland, G., González-Tokman, C. et al. A Spectral Approach for Quenched Limit Theorems for Random Expanding Dynamical Systems. Commun. Math. Phys. 360, 1121–1187 (2018). https://doi.org/10.1007/s00220-017-3083-7

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  • DOI: https://doi.org/10.1007/s00220-017-3083-7

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