2024/10/14 by Hrant Hakobyan, Hakobyan, Hrant, Michael Pandazis +3 · 1 citation
Engineering · Materials Science · #30F15 #30F25 #37A25 #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #Fluid Dynamics and Thin Films #Fluid Dynamics and Turbulent Flows #Geometric Topology (math.GT) #Pickering emulsions and particle stabilization
paper · pdf · doi:10.48550/arxiv.2410.10057
openalex publication_date 2024/10/14 · openalex created_date 2024/10/21 · openalex updated_date 2026/07/28
A Riemann surface X is parabolic if and only if the geodesic flow (for the hyperbolic metric) on the unit tangent bundle of X is ergodic. Consider a Riemann surface X with a single topological end and a sequence αn of pairwise disjoint, simple closed geodesics converging to the end, called \it cuffs. Basmajian, the first and the third author, proved that when the lengths ℓ (αn) of cuffs are at most 2log n, the surface X is parabolic. One could expect that having arbitrary large cuff lengths ℓ (αn) (think of ℓ (αn)=n!n!) would allow the geodesic flow to escape to infinity, thus making X not parabolic. Contrary to this and motivated by their proof of the Surface Subgroup Theorem, Kahn and Marković conjectured that for every choice of lengths ℓ (αn), there is a choice of twists that would make X parabolic. We show that their conjecture is essentially true. Namely, for any sequence of positive numbers \ an\, there is a choice of lengths ℓ (αn)≥ an such that the (relative) twists by 1/2 make X parabolic. This result extends to the surfaces with countably many ends while it does not hold for uncountably many ends.