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

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

2007-03-22

arxiv.org/abs/math/0703670v1

Mathematics

Dynamical Systems

55 pages

Scientific paper

We study skew-products of the form (x,\omega)\mapsto (Tx, \omega+\phi(x)) where T is a nonuniformly expanding map on a space X, preserving a (possibly singular) probability measure \tilde\mu, and \phi:X\to S^1 is a C^1 function. Under mild assumptions on \tilde\mu and \phi, we prove that such a map is exponentially mixing, and satisfies the central 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.

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