Prime number theorem for analytic skew products

Abstract

We establish a prime number theorem for all uniquely ergodic, analytic skew products on the $2$-torus ${\mathbb{T}}^2$. More precisely, for every irrational $\alpha$ and every $1$-periodic real analytic $g:\mathbb{R}\to \mathbb{R}$ of zero mean, let $T_{\alpha ,g}:{\mathbb{T}}^2\to {\mathbb{T}}^2$ be defined by $(x,y)\mapsto (x+\alpha ,y+g(x))$. We prove that if $T_{\alpha ,g}$ is uniquely ergodic then, for every $(x,y)\in {\mathbb{T}}^2$, the sequence $\{T_{\alpha ,g}^p(x,y)\}$ is equidistributed on ${\mathbb{T}}^2$ as $p$ traverses prime numbers. This is the first example of a class of natural, non-algebraic and smooth dynamical systems for which a prime number theorem holds. We also show that such a prime number theorem does not necessarily hold if $g$ is only continuous on ${\mathbb{T}}^2$.