Limit theorems for parabolic systems

While hyperbolic dynamical systems obey the Sinai Central Limit Theorem, parabolic systems do not: limit distributions have compact support, exist only along subsequences of times and essentially depend on the subsequence: for instance, for translation flows, the delta mass at zero typically arises as a limit distribution along a subsequence.
The survey talk, aimed at undergraduate students, will aim to give a gentle introduction to limit theorems for parabolic dynamical systems. The main step is a construction of a special family of finitely-additive measures on the arcs of trajectories of our flows.
Specific examples include translation flows, horocycle flows (joint work with Giovanni Forni) and tiling flows (joint work with Boris Solomyak).