Let M = G/Γ be a compact nilmanifold endowed with an invariant complex structure. Using a descending series associated to the complex structure and the Borel spectral sequences for the corresponding set of holomorphic fibrations, we prove a version of Nomizu’s Theorem for the Dolbeault cohomology of M .

Motivated by a problem in ergodic Ramsey theory, Furstenberg and Katznelson introduced the notion of strong stationarity, showing that certain recurrence properties hold for arbitrary measure preserving systems if they are valid for strongly stationary ones. We construct some new examples and prove a structure theorem for strongly stationary systems. The building blocks are Bernoulli systems an...

We answer a question posed by Vitaly Bergelson, showing that in a totally ergodic system, the average of a product of functions evaluated along polynomial times, with polynomials of pairwise differing degrees, converges in L to the product of the integrals. Such averages are characterized by nilsystems and so we reduce the problem to one of uniform distribution of polynomial sequences on nilman...