A disconnected deformation space of rational maps

Eriko Hironaka; Sarah Koch

doi:10.3934/jmd.2017016

Article Contents

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

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A disconnected deformation space of rational maps

Eriko Hironaka^1, and
Sarah Koch^2,

1.
Department of Mathematics, Florida State University, 1017 Academic Way, 208 LOV, Tallahassee, FL 32306-4510, USA
2.
Department of Mathematics, University of Michigan, East Hall, 530 Church Street, Ann Arbor, MI 48109, USA

Received: October 31, 2016

Revised: February 09, 2017

Published: October 2017

EH: Supported in part by a collaboration grant from the Simons Foundation #209171.
SK: Supported in part by the NSF and the Sloan Foundation.

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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 $A\subsetneq B$, 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.

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Mathematics Subject Classification: Primary: 37F45, 37F20, 37F10; Secondary: 20F36.

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Figure 1. The space $\mathscr X$ (on the left) is the complement in $\mathbb{C}^2$ of the five solid black lines. The dashed diagonal line is $\mathscr E$. The space $\mathscr Q$ (on the right) is the complement in $\mathbb{C}^2$ of the three solid black lines intersecting at $(1, 0)$. The line $\mathscr L$ is the image of $\mathscr K_1$ and $\mathscr K_2$ under the map $\mathfrak{s}$; it is tangent to the dashed conic $\mathfrak{s}(\mathscr E)$ at $q_0$.

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Figure 2. The space $\mathscr{W}_f$ is the complement of the black curves in $\mathbb{C}^2$ (including the axes). The space $\mathscr{V}_f\subseteq \mathscr{W}_f$ is represented by the grey diagonal line.

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