The Quasi-Additivity Law in Conformal Geometry

2005/05/10 by Jeremy Kahn, Kahn, Jeremy, Mikhail Lyubich +1 · 4 citations
Mathematics · #37F45 #Analytic and geometric function theory #Dynamical Systems (math.DS) #FOS: Mathematics #Geometry and complex manifolds #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.math/0505191

openalex publication_date 2005/05/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

On a Riemann surface S of finite type containing a family of N disjoint disks D_i (``islands''), we consider several natural conformal invariants measuring the distance from the islands to \di S and separation between different islands. In a near degenerate situation we establish a relation between them called the Quasi-Additivity Law. We then generalize it to a Quasi-Invariance Law providing us with a transformation rule of the moduli in question under covering maps. This rule (and in particular, its special case called the Covering Lemma) has important applications in holomorphic dynamics which will be addressed in the forthcoming notes.

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