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

Petr Kosenko

TL;DR

The paper develops a complex-analytic framework to study μ-stationary measures on S^1 under the PSU(1,1) action, recasting μ-stationarity as a holomorphic condition on the Cauchy transform f_ν via a multiplicative–additive functional equation. By embedding this into Hardy spaces and pseudocontinuation theory, it delivers sharp necessary conditions, nonexistence results for absolutely continuous densities under moment restrictions, and a complete Furstenberg-measure characterization for first-kind Fuchsian groups through the Brown–Shields–Zeller theorem. The approach yields practical singularity criteria for harmonic measures and illuminates structural constraints on Lebesgue stationarity, thereby unifying and extending prior results (e.g., Bourgain) across discrete, dense, and lattice settings. The work offers a versatile toolkit for understanding Poisson boundaries and stationary dynamics in hyperbolic group actions with potential broader impact on harmonic analysis on groups and related dynamical systems.

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 Möbius 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.

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

TL;DR

The paper develops a complex-analytic framework to study μ-stationary measures on S^1 under the PSU(1,1) action, recasting μ-stationarity as a holomorphic condition on the Cauchy transform f_ν via a multiplicative–additive functional equation. By embedding this into Hardy spaces and pseudocontinuation theory, it delivers sharp necessary conditions, nonexistence results for absolutely continuous densities under moment restrictions, and a complete Furstenberg-measure characterization for first-kind Fuchsian groups through the Brown–Shields–Zeller theorem. The approach yields practical singularity criteria for harmonic measures and illuminates structural constraints on Lebesgue stationarity, thereby unifying and extending prior results (e.g., Bourgain) across discrete, dense, and lattice settings. The work offers a versatile toolkit for understanding Poisson boundaries and stationary dynamics in hyperbolic group actions with potential broader impact on harmonic analysis on groups and related dynamical systems.

Abstract

We provide a complex-analytic approach to the classification of stationary probability measures on with respect to the action of on the unit circle via Möbius 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.
Paper Structure (15 sections, 28 theorems, 67 equations)

This paper contains 15 sections, 28 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Main results
  4. Acknowledgements
  5. Preliminaries
  6. Everything you need to know about isometries of the disk model of $\mathbb{H}^2$
  7. Random walks on groups
  8. Complex-analytic prerequisites: Hardy spaces
  9. Complex-analytic prerequisites: pseudocontinuations
  10. Holomorphic stationarity condition
  11. Squeezing water from a stone: a deep dive into \ref{['main equation, simp']}
  12. Functional-analytic necessary condition for existence of absolutely continuous stationary measures
  13. When the Lebesgue measure is stationary?
  14. Criterion for the singularity of the harmonic measure
  15. Open questions

Key Result

Theorem 1.1

Let $\mu$ be a probability measure on $G=PSU(1,1)$.

Theorems & Definitions (54)

  • Definition 1.1
  • Conjecture 1.1: kaimanovich2011matrix
  • Theorem 1.1
  • Corollary 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Theorem 1.2
  • Definition 2.1
  • Lemma 2.1
  • proof
  • ...and 44 more