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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Figure 1.
The space
[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. |