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arXiv · 2311.09973

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

Aitor Azemar·Vaibhav Gadre
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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.

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Aitor Azemar, Vaibhav Gadre. 2023-11-16. Stationary measures on the circle from hyperbolic surfaces with cusps cannot be straightened by quasi-symmetries. https://arxiv.org/abs/2311.09973

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