Duke Mathematical Journal

Local limit theorem for nonuniformly partially hyperbolic skew-products and Farey sequences

Sébastien Gouëzel

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Abstract

We study skew-products of the form (x,ω)(Tx,ω+φ(x)), where T is a nonuniformly expanding map on a space X, preserving a (possibly singular) probability measure μ~, and φ:XS1 is a C1 function. Under mild assumptions on μ~ and φ, we prove that such a map is exponentially mixing and satisfies both the central limit and local limit theorems. These results apply to a random walk related to the Farey sequence, thereby answering a question of Guivarc'h and Raugi [GR, Section 5.3]

Article information

Source
Duke Math. J., Volume 147, Number 2 (2009), 193-284.

Dates
First available in Project Euclid: 17 March 2009

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1237295909

Digital Object Identifier
doi:10.1215/00127094-2009-011

Mathematical Reviews number (MathSciNet)
MR2495076

Zentralblatt MATH identifier
1170.37006

Subjects
Primary: 37A25: Ergodicity, mixing, rates of mixing 37A50: Relations with probability theory and stochastic processes [See also 60Fxx and 60G10] 37D30: Partially hyperbolic systems and dominated splittings
Secondary: 37A30: Ergodic theorems, spectral theory, Markov operators {For operator ergodic theory, see mainly 47A35} 37D25: Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)

Citation

Gouëzel, Sébastien. Local limit theorem for nonuniformly partially hyperbolic skew-products and Farey sequences. Duke Math. J. 147 (2009), no. 2, 193--284. doi:10.1215/00127094-2009-011. https://projecteuclid.org/euclid.dmj/1237295909


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