Prime number theorem for analytic skew products


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 $T_{\alpha,g} : \mathbb{T}^2 \rightarrow \mathbb{T}^2$ 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 $\mathbb{T}$.


Adam Kanigowski

Department of Mathematics, University of Maryland at College Park, College Park, MD 20740, USA and Faculty of Mathematics and Computer Science, Jagiellonian University, Łojasiewicza 6, Kraków, Poland

Mariusz Lemańczyk

Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopin street 12/18, 87-100 Toruń, Poland

Maksym Radziwiłł

Department of Mathematics, California Institute of Technology, 1200 E California Blvd, Pasadena, CA, 91125, USA

Current address:

Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208, USA