Local limit theorem for nonuniformly partially hyperbolic skew-products, and Farey sequences
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:X S1 is a C1 function. Under mild assumptions on μ and φ, 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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