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Stability and Approximation of Statistical Limit Laws for Multidimensional Piecewise Expanding Maps

  • Harry CrimminsEmail author
  • Gary Froyland
Article

Abstract

The unpredictability of chaotic nonlinear dynamics leads naturally to statistical descriptions, including probabilistic limit laws such as the central limit theorem and large deviation principle. A key tool in the Nagaev–Guivarc’h spectral method for establishing statistical limit theorems is a “twisted” transfer operator. In the abstract setting of Keller and Liverani (Annali della Scuola Normale Superiore di Pisa-Classe di Scienze, 28:141–152, 1999), we prove that derivatives of all orders of the leading eigenvalues and eigenprojections of the twisted transfer operators with respect to the twist parameter are stable when subjected to a broad class of perturbations. As a result, we demonstrate stability of the variance in the central limit theorem and the rate function from a large deviation principle with respect to deterministic and stochastic perturbations of the dynamics and perturbations induced by numerical schemes. We apply these results to piecewise expanding maps in one and multiple dimensions, including new convergence results for Ulam projections on quasi-Hölder spaces.

Notes

Acknowledgements

HC is supported by an Australian Government Research Training Program Scholarship and the UNSW School of Mathematics and Statistics. GF is partially supported by an Australian Research Council Discovery Project. Both authors thank Davor Dragičević for helpful conversations during the writing of this work, and to an anonymous referee for their suggested strengthening of continuity to Hölder continuity in Theorems 2.6, 3.4 and 3.8.

Supplementary material

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Authors and Affiliations

  1. 1.School of Mathematics and StatisticsUniversity of New South WalesSydney Australia

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