Journal of Modern Dynamics (JMD)

A disconnected deformation space of rational maps

Pages: 409 - 423, Volume 11, 2017      doi:10.3934/jmd.2017016

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Eriko Hironaka - Department of Mathematics, Florida State University, 1017 Academic Way, 208 LOV, Tallahassee, FL 32306-4510, United States (email)
Sarah Koch - Department of Mathematics, University of Michigan, East Hall, 530 Church Street, Ann Arbor, MI 48109, United States (email)

Abstract: The deformation space of a branched cover $f:(S^2,A)\to (S^2,B)$ is a complex submanifold of a certain Teichmüller space, which consists of classes of marked rational maps $F:(\mathbb{P}^1,A')\to (\mathbb{P}^1,B')$ that are combinatorially equivalent to $f$. In the case $A=B$, under a mild assumption on $f$, William Thurston gave a topological criterion for which the deformation space of $f:(S^2,A)\to (S^2,B)$ is nonempty, and he proved that it is always connected. We show that if $A ⊊ B$, then the deformation space need not be connected. We exhibit a family of quadratic rational maps for which the associated deformation spaces are disconnected; in fact, each has infinitely many components.

Keywords:  Dynamical Teichmüller space of a rational map, dynamical moduli space of a rational map, liftable mapping classes, special liftable mapping classes.
Mathematics Subject Classification:  Primary: 37F45, 37F20, 37F10; Secondary: 20F36.

Received: November 2016;      Revised: February 2017;      Available Online: June 2017.