Stationary measures on the circle from hyperbolic surfaces with cusps cannot be straightened by quasi-symmetries

Azemar, A. and Gadre, V. (2023) Stationary measures on the circle from hyperbolic surfaces with cusps cannot be straightened by quasi-symmetries. arXiv, (doi: 10.48550/arXiv.2311.09973) (Unpublished)

Abstract

Stationary measures on the circle that arise from a large class of random walks on the fundamental group of a finite-area complete hyperbolic surface with cusps are singular with respect to the Lebesgue measure. In particular, it is sufficient for singularity that a stationary measure satisfies an exponential decay for cusp excursions with excursions being measured in the path metric on horocycles bounding cusps. In this note, we settle a conjecture of McMullen by proving that the singularity of stationary measures satisfying such exponential decay is quasi-symmetrically stable, that is under push-forward by any quasi-symmetry of the circle the measure remains singular.

• Item Type: Articles
• Status: Unpublished
• Refereed: No
• Glasgow Author(s) Enlighten ID: Azemar, Mr Aitor and Gadre, Dr Vaibhav
• Authors: Azemar, A., and Gadre, V.
• Subjects: Q Science > QA Mathematics
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: arXiv