We discuss a common framework for studying twists of Riemann surfaces coming from earthquakes, Teichmüller theory and Schiffer variations, and use it to analyze geodesics in the moduli space of isoperiodic 1-forms.
In this paper we study branched coverings of metrized, simplicial trees which arise from polynomial maps with disconnected Julia sets. We show that the collection of all such trees, up to scale, forms a contractible space compactifying the moduli space of polynomials of degree ; that records the asymptotic behavior of the multipliers of ; and that any meromorphic family of polynomials over can be completed by a unique tree at its central fiber. In the cubic case we give a combinatorial...
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