On a complex-analytic approach to stationary measures on $S^1$ with respect to the action of $PSU(1,1)$

Abstract

We provide a complex-analytic approach to the classification of stationary probability measures on $S^1$ with respect to the action of $PSU(1,1)$ on the unit circle via M\"{o}bius transformations by studying their Cauchy transforms from the perspective of generalized analytic continuation. We improve upon results of Bourgain and present a complete characterization of Furstenberg measures for Fuchsian groups of first kind via the Brown-Shields-Zeller theorem.