Prime number theorem for analytic skew products

TL;DR

This paper proves a prime number theorem for a class of uniquely ergodic, analytic skew product dynamical systems on the 2-torus, showing primes equidistribute under these transformations, unlike in the continuous case.

Contribution

It establishes the first prime number theorem for non-algebraic, smooth dynamical systems on the torus, expanding understanding of prime distribution in dynamical contexts.

Findings

Prime numbers are equidistributed in the orbits of certain analytic skew products.

The prime number theorem holds for uniquely ergodic systems with analytic, zero-mean functions.

The theorem does not extend to systems where the function is only continuous.

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