Classifying stationary measures on $S^1$ with respect to $PSU(1,1)$ through the complex-analytic point of view

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 cocompactness of the action.

Moreover, there is no single approach that works for every setting, and the existing literature is incredibly vast and diverse, which makes it very difficult to comprehend and appreciate the sheer 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 bascially any countably supported $\mu$. We reduce the global classification to 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.