Ai-Hua Fan: Convergence and divergence of ergodic series

Abstract: By ergodic series, we mean \(\sum_{n=1}^\infty a_n f(T^n x)\) where T preserves a measure \(\mu\) and \(f\) is integrable function of zero mean. For Gibbs measures $\mu$ of hyperbolic systems T, the square-summability of \((a_n)\) and the Holder continuity of \(f\) implies the almost everywhere convergence of the series. But it is far from the case when T is an irrational rotation on the circle. Let us only consider the one sided ergodic Hilbert transform \(\sum_{n=1}^\infty f(x + n a)/n\) associated to the irrational rotation \(x \mapsto x + a \mod 1\). It is proved that (1) For any \(C^2\) function \(f\) which admits a Taylor-Fourier series but non-polynomial, there exists an irrational number \(a\) (actually a residual set of \(a\)) such that the transform diverges everywhere; (2) For any irrational number \(a\), there is a continuous function \(f\) such that  the transform diverges everywhere. The work on the one-sided ergodic Hilbert transform is jointly done with Joerg Schmeling.