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Large deviation principles for non-uniformly hyperbolic rational maps

Published online by Cambridge University Press:  10 May 2010

HENRI COMMAN  and
JUAN RIVERA-LETELIER
HENRI COMMAN
Affiliation:
Institute of Mathematics, Pontifical Catholic University of Valparaiso, Chile (email: henri.comman@ucv.cl)
JUAN RIVERA-LETELIER
Affiliation:
Facultad de Matemáticas, Campus San Joaquín, P. Universidad Católica de Chile, Avenida Vicuña Mackenna 4860, Santiago, Chile (email: riveraletelier@mat.puc.cl)
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Abstract

We show some level-2 large deviation principles for rational maps satisfying a strong form of non-uniform hyperbolicity, called ‘Topological Collet–Eckmann’. More precisely, we prove a large deviation principle for the distribution of iterated preimages, periodic points, and Birkhoff averages. For this purpose we show that each Hölder continuous potential admits a unique equilibrium state, and that the pressure function can be characterized in terms of iterated preimages, periodic points, and Birkhoff averages. Then we use a variant of a general result of Kifer.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 31 , Issue 2 , April 2011 , pp. 321 - 349
Copyright
Copyright © Cambridge University Press 2010

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