Prime number theorem for analytic skew products

We establish a prime number theorem for all uniquely ergodic, analytic skew products on the 2-torus T². More precisely, for every irrational a and every 1-periodic real analytic g: R → R of zero mean, let Tα,_g: T² → T² be defined by (x,y) → (x + α, y + g(x)). We prove that if Tα,g is uniquely ergodic then, for every (x,y) Є T², the sequence {Tpa,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 T.