2012/12/06 by Vaibhav Gadre, Gadre, Vaibhav, Joseph Maher +3 · 1 citation
Mathematics · #20F65 #30F60 #32G15 #37F30 #60J50 #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Topology (math.GT) #Mathematical Dynamics and Fractals #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.1212.1481
openalex publication_date 2012/12/06 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28
Given a measure on the Thurston boundary of Teichmueller space, one can pick\na geodesic ray joining some basepoint to a randomly chosen point on the\nboundary. Different choices of measures may yield typical geodesics with\ndifferent geometric properties. In particular, we consider two families of\nmeasures: the ones which belong to the Lebesgue or visual measure class, and\nharmonic measures for random walks on the mapping class group generated by a\ndistribution with finite first moment in the word metric. We consider the ratio\nbetween the word metric and the relative metric of approximating mapping class\ngroup elements along a geodesic ray, and prove that this ratio tends to\ninfinity along almost all geodesics with respect to Lebesgue measure, while the\nlimit is finite along almost all geodesics with respect to harmonic measure. As\na corollary, we establish singularity of harmonic measure. We show the same\nresult for cofinite volume Fuchsian groups with cusps. As an application, we\nanswer a question of Deroin-Kleptsyn-Navas about the vanishing of the Lyapunov\nexpansion exponent.\n