## Preprint ims05-02

Jeremy Kahn, Mikhail Lyubich
The Quasi-Additivity Law in Conformal Geometry

Abstract: We consider a Riemann surface $S$ of finite type containing a family of $N$ disjoint disks $D_i$, and prove the following Quasi-Additivity Law: If the total extremal width $\sum \WW(S\sm D_i)$ is big enough (depending on $N$) then it is comparable with the extremal width $\WW (S,\cup D_i)$ (under a certain separation assumption'') . We also consider a branched covering $f: U\ra V$ of degree $N$ between two disks that restricts to a map $\La\ra B$ of degree $d$ on some disk $\La \Subset U$. We derive from the Quasi-Additivity Law that if $\mod(U\sm \La)$ is sufficiently small, then (under a collar assumption'') the modulus is quasi-invariant under $f$, namely $\mod(V\sm B)$ is comparable with $d^2 \mod(U\sm \La)$. This Covering Lemma has important consequences in holomorphic dynamics which will be addressed in the forthcoming notes.
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