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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Large deviations for nonuniformly hyperbolic systems
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by Ian Melbourne and Matthew Nicol PDF
Trans. Amer. Math. Soc. 360 (2008), 6661-6676 Request permission

Abstract:

We obtain large deviation estimates for a large class of nonuniformly hyperbolic systems: namely those modelled by Young towers with summable decay of correlations. In the case of exponential decay of correlations, we obtain exponential large deviation estimates given by a rate function. In the case of polynomial decay of correlations, we obtain polynomial large deviation estimates, and exhibit examples where these estimates are essentially optimal.

In contrast with many treatments of large deviations, our methods do not rely on thermodynamic formalism. Hence, for Hölder observables we are able to obtain exponential estimates in situations where the space of equilibrium measures is not known to be a singleton, as well as polynomial estimates in situations where there is not a unique equilibrium measure.

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Additional Information
  • Ian Melbourne
  • Affiliation: Department of Mathematics, University of Surrey, Guildford, Surrey GU2 7XH, United Kingdom
  • MR Author ID: 123300
  • Email: ism@math.uh.edu
  • Matthew Nicol
  • Affiliation: Department of Mathematics, University of Houston, Houston, Texas 77204-3008
  • MR Author ID: 350236
  • Email: nicol@math.uh.edu
  • Received by editor(s): December 7, 2006
  • Received by editor(s) in revised form: March 14, 2007
  • Published electronically: June 4, 2008
  • Additional Notes: The research of the first author was supported in part by EPSRC Grant EP/D055520/1 and a Leverhulme Research Fellowship.
    The research of the second author was supported in part by NSF grants DMS 0600927 and DMS-0607345
    The authors would like to thank the Universities of Houston and Surrey respectively for hospitality during part of this research.
  • © Copyright 2008 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 360 (2008), 6661-6676
  • MSC (2000): Primary 37D25, 37A50, 60F10
  • DOI: https://doi.org/10.1090/S0002-9947-08-04520-0
  • MathSciNet review: 2434305