Given an arbitrary probability measure $\mu$ on $PSU(1,1)$, understanding the structure of $\mu$-stationary measures on $S^1$ is a notoriously difficult problem. One reason lies in 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, and even in very concrete cases the structure of the hitting measures is not entirely understood.

Moreover, until very recently there was no single approach that worked 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. In my talk I am going to discuss a recently developed complex-analytic approach to the classification of positive stationary measures on $S^1$ which attempts to unify all mentioned settings and reframes the existing results through theory of Cauchy transforms of measures on $S^1$, and works for any countably supported $\mu$ with the finite first moment.

This method does not suffer from the usual shortcomings of the Fourier-like approaches by allowing us to treat singular measures and absolutely continuous measures using the same toolbox. As it turns out, there is a surprising but deep connection with generalized analytic continuations of holomorphic functions and the well-developed theory of closed backward-shift invariant subspaces of Hardy spaces H^p(D) for 0 < p < \infty. In partuclar, we will demonstrate that a number of fundamental results and conjectures about hitting measures are "predicted" by the classical results of Aleksandrov, Douglas-Shapiro-Shields and Brown-Shields-Zeller, and provide some new insights using our methods.

I will focus on key ideas, (hopefully) without overloading the audience with complex-analytic technicalities.

This is work in progress, based on https://arxiv.org/pdf/2403.11065.pdf.