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Dynamical degrees of Hurwitz correspondences

Part of:CurvesProjective and enumerative geometryComplex dynamical systemsSpecial varieties

Published online by Cambridge University Press:  04 December 2018

ROHINI RAMADAS
ROHINI RAMADAS*
Affiliation:
Department of Mathematics, Brown University, Providence, RI, USA email rohini_ramadas@brown.edu
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Abstract

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Let $\unicode[STIX]{x1D719}$ be a post-critically finite branched covering of a two-sphere. By work of Koch, the Thurston pullback map induced by $\unicode[STIX]{x1D719}$ on Teichmüller space descends to a multivalued self-map—a Hurwitz correspondence ${\mathcal{H}}_{\unicode[STIX]{x1D719}}$—of the moduli space ${\mathcal{M}}_{0,\mathbf{P}}$. We study the dynamics of Hurwitz correspondences via numerical invariants called dynamical degrees. We show that the sequence of dynamical degrees of ${\mathcal{H}}_{\unicode[STIX]{x1D719}}$ is always non-increasing and that the behavior of this sequence is constrained by the behavior of $\unicode[STIX]{x1D719}$ at and near points of its post-critical set.

Keywords

dynamical degreescorrespondencesmoduli spaces

MSC classification

Primary:14H10: Families, moduli (algebraic)
Secondary:14N99: None of the above, but in this section14M99: None of the above, but in this section37F05: Relations and correspondences

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 7 , July 2020, pp. 1968 - 1990
Copyright
© Cambridge University Press, 2018

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