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The quasi-additivity law in conformal geometry. (English) Zbl 1203.30011
Authors’ 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 $$\partial S$$ and the 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.
Reviewer: Pei-Chu Hu (Jinan)

##### MSC:
 30C20 Conformal mappings of special domains 30F45 Conformal metrics (hyperbolic, Poincaré, distance functions) 30F99 Riemann surfaces
##### Keywords:
quasi-additivity law; conformal geometry; Riemann surface
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