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

Abstract

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