2021/05/31 by Alastair Fletcher, Fletcher, Alastair N., Vyron Vellis +1
Mathematics · #Analytic and geometric function theory #Complex Variables (math.CV) #FOS: Mathematics #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Metric Geometry (math.MG)
paper · pdf · doi:10.48550/arxiv.2106.00054
openalex publication_date 2021/05/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The Decomposition Problem in the class LIP(\mathbbS2) is to decompose any bi-Lipschitz map f:\mathbbS2 → \mathbbS2 as a composition of finitely many maps of arbitrarily small isometric distortion. In this paper, we construct a decomposition for certain bi-Lipschitz maps which spiral around every point of a Cantor set X of Assouad dimension strictly smaller than one. These maps are constructed by considering a collection of Dehn twists on the Riemann surface \mathbbS2 ∖ X. The decomposition is then obtained via a bi-Lipschitz path which simultaneously unwinds these Dehn twists. As part of our construction, we also show that X ⊂ \mathbbS2 is uniformly disconnected if and only if the Riemann surface \mathbbS2 ∖ X has a pants decomposition whose cuffs have hyperbolic length uniformly bounded above, which may be of independent interest.