Ergodic Theory and Dynamical Systems Seminar: Peter Kosenko

Given an arbitrary probability measure $\mu$ on $PSU(1,1)$, understanding the structure of $\mu$-stationary measures is a notoriously difficult problem, in particular, due to the number of different settings one can work in. The answer is known to depend on the moment conditions of $\mu$, finiteness of the support, and which subgroup in $PSU(1,1)$ the support generates -- dense or discrete. In the discrete case the answer should also heavily depend on the limit set in $S^1$ and co-compactness of the action.

Moreover, there is no single approach that works for every setting, and the existing literature in incredibly vast and diverse, which makes it very difficult to comprehend and appreciate the sheet scope of this problem. I am going to discuss a recently developed complex-analytic approach to the classification of positive stationary measures on $S^1$ which reframes many existing results in the context of complex analytic properties of the Cauchy transforms of stationary measures, and works for basically any countably supported $\mu$. We reduce the global classification of studying the action of a very concrete weighted composition operator on the family of backward shift-invariant subspaces in Hardy spaces $H^p(D)$.

In partuclar, we will demonstrate that several important results and conjectures by Furstenberg, Bourgain and Kaimanovish are "predicted" by classical results of Douglas-Shapiro-Shields and Brown-Shields-Zeller, and approximation theorems of Walsh. This is work in progress, based on https://arxiv.org/pdf/2403.11065.pdf.