# American Institute of Mathematical Sciences

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

## A disconnected deformation space of rational maps

 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  November 2016 Revised  February 10, 2017 Published  November 2017

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

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.

Citation: Eriko Hironaka, Sarah Koch. A disconnected deformation space of rational maps. Journal of Modern Dynamics, 2017, 11: 409-423. doi: 10.3934/jmd.2017016
##### References:
 [1] J. Birman, Braids, Links and Mapping Class Groups, Annals of Math. Studies, No. 82, Princeton University Press, Princeton N. J., 1974. [2] X. Buff, G. Cui and L. Tan, Teichmüller spaces and holomorphic dynamics, in Handbook of Teichmüller Theory, Vol. Ⅳ (ed. A. Papadopoulos), IRMA Lect. Math. Theor. Phys., 19, Eur. Math. Soc., Zürich, 2014,717–756. doi: 10.4171/117-1/17. [3] A. Douady and J. H. Hubbard, A Proof of Thurston's characterization of rational functions, Acta Math., 171 (1993), 263-297. doi: 10.1007/BF02392534. [4] A. Epstein, Transversality in holomorphic dynamics, manuscript available at http://www.warwick.ac.uk/~mases. [5] T. Firsova, J. Kahn and N. Selinger, On deformation spaces of quadratic rational maps, preprint, 2016. [6] S. Koch, Teichmüller theory and critically finite endomorphisms, Adv. Math., 248 (2013), 573-617. doi: 10.1016/j.aim.2013.08.019. [7] S. Koch, K. Pilgrim and N. Selinger, Pullback invariants of Thurston maps, Trans. Amer. Math. Soc., 368 (2016), 4621-4655. doi: 10.1090/tran/6482. [8] J. Milnor, On Lattès maps, in Dynamics on the Riemann Sphere, a Bodil Branner Festschrift (eds. P. Hjorth and C. L. Petersen), Eur. Math. Soc., Zürich, 2006, 9–43. doi: 10.4171/011-1/1. [9] J. Milnor, Geometry and dynamics of quadratic rational maps, with an appendix by the author and Lei Tan, Experiment. Math., 2 (1993), 37-83. doi: 10.1080/10586458.1993.10504267. [10] M. Rees, Views of parameter space: Topographer and Resident, Astérisque, 288 (2003), 1-418.

show all references

##### References:
 [1] J. Birman, Braids, Links and Mapping Class Groups, Annals of Math. Studies, No. 82, Princeton University Press, Princeton N. J., 1974. [2] X. Buff, G. Cui and L. Tan, Teichmüller spaces and holomorphic dynamics, in Handbook of Teichmüller Theory, Vol. Ⅳ (ed. A. Papadopoulos), IRMA Lect. Math. Theor. Phys., 19, Eur. Math. Soc., Zürich, 2014,717–756. doi: 10.4171/117-1/17. [3] A. Douady and J. H. Hubbard, A Proof of Thurston's characterization of rational functions, Acta Math., 171 (1993), 263-297. doi: 10.1007/BF02392534. [4] A. Epstein, Transversality in holomorphic dynamics, manuscript available at http://www.warwick.ac.uk/~mases. [5] T. Firsova, J. Kahn and N. Selinger, On deformation spaces of quadratic rational maps, preprint, 2016. [6] S. Koch, Teichmüller theory and critically finite endomorphisms, Adv. Math., 248 (2013), 573-617. doi: 10.1016/j.aim.2013.08.019. [7] S. Koch, K. Pilgrim and N. Selinger, Pullback invariants of Thurston maps, Trans. Amer. Math. Soc., 368 (2016), 4621-4655. doi: 10.1090/tran/6482. [8] J. Milnor, On Lattès maps, in Dynamics on the Riemann Sphere, a Bodil Branner Festschrift (eds. P. Hjorth and C. L. Petersen), Eur. Math. Soc., Zürich, 2006, 9–43. doi: 10.4171/011-1/1. [9] J. Milnor, Geometry and dynamics of quadratic rational maps, with an appendix by the author and Lei Tan, Experiment. Math., 2 (1993), 37-83. doi: 10.1080/10586458.1993.10504267. [10] M. Rees, Views of parameter space: Topographer and Resident, Astérisque, 288 (2003), 1-418.
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 $q_0$.
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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