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Transactions of the American Mathematical Society

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Large deviations for nonuniformly hyperbolic systems

Authors: Ian Melbourne and Matthew Nicol
Journal: Trans. Amer. Math. Soc. 360 (2008), 6661-6676
MSC (2000): Primary 37D25, 37A50, 60F10
Published electronically: June 4, 2008
MathSciNet review: 2434305
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Abstract | References | Similar Articles | Additional Information

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

Matthew Nicol
Affiliation: Department of Mathematics, University of Houston, Houston, Texas 77204-3008

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.
Article copyright: © Copyright 2008 American Mathematical Society

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