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 T². More precisely, for every irrational α and every 1-periodic real analytic g : R → R of zero mean, let T_(α,g) : T² → T² be defined by (x,y) → x+α ,y+g(x)). We prove that if T_(α,g) is uniquely ergodic then, for every (x,y) ∈ T², the sequence {T^p_(α,g)(x,y)} is equidistributed on T² 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 T².