American Institute of Mathematical Sciences

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

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  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.

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.
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. 

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 $q_0$.
Figure Options

Download full-size image

Download as PowerPoint slide

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.
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Corentin Boissy. Classification of Rauzy classes in the moduli space of Abelian and quadratic differentials. Discrete and Continuous Dynamical Systems, 2012, 32 (10) : 3433-3457. doi: 10.3934/dcds.2012.32.3433
[2]

Chi Po Choi, Xianfeng Gu, Lok Ming Lui. Subdivision connectivity remeshing via Teichmüller extremal map. Inverse Problems and Imaging, 2017, 11 (5) : 825-855. doi: 10.3934/ipi.2017039
[3]

Boris Andreianov, Mohamed Maliki. On classes of well-posedness for quasilinear diffusion equations in the whole space. Discrete and Continuous Dynamical Systems - S, 2021, 14 (2) : 505-531. doi: 10.3934/dcdss.2020361
[4]

S. Aubry, G. Kopidakis, V. Kadelburg. Variational proof for hard Discrete breathers in some classes of Hamiltonian dynamical systems. Discrete and Continuous Dynamical Systems - B, 2001, 1 (3) : 271-298. doi: 10.3934/dcdsb.2001.1.271
[5]

Jiaxi Huang, Youde Wang, Lifeng Zhao. Equivariant Schrödinger map flow on two dimensional hyperbolic space. Discrete and Continuous Dynamical Systems, 2020, 40 (7) : 4379-4425. doi: 10.3934/dcds.2020184
[6]

Ihsane Bikri, Ronald B. Guenther, Enrique A. Thomann. The Dirichlet to Neumann map - An application to the Stokes problem in half space. Discrete and Continuous Dynamical Systems - S, 2010, 3 (2) : 221-230. doi: 10.3934/dcdss.2010.3.221
[7]

Alex Eskin, Maryam Mirzakhani. Counting closed geodesics in moduli space. Journal of Modern Dynamics, 2011, 5 (1) : 71-105. doi: 10.3934/jmd.2011.5.71
[8]

Jonathan Meddaugh. Shadowing as a structural property of the space of dynamical systems. Discrete and Continuous Dynamical Systems, 2022, 42 (5) : 2439-2451. doi: 10.3934/dcds.2021197
[9]

Martin Möller. Shimura and Teichmüller curves. Journal of Modern Dynamics, 2011, 5 (1) : 1-32. doi: 10.3934/jmd.2011.5.1
[10]

Fabrizio Colombo, Irene Sabadini, Frank Sommen. The inverse Fueter mapping theorem. Communications on Pure and Applied Analysis, 2011, 10 (4) : 1165-1181. doi: 10.3934/cpaa.2011.10.1165
[11]

Jeremy Kahn, Alex Wright. Hodge and Teichmüller. Journal of Modern Dynamics, 2022, 18: 149-160. doi: 10.3934/jmd.2022007
[12]

H.T. Banks, S. Dediu, H.K. Nguyen. Sensitivity of dynamical systems to parameters in a convex subset of a topological vector space. Mathematical Biosciences & Engineering, 2007, 4 (3) : 403-430. doi: 10.3934/mbe.2007.4.403
[13]

John Banks. Topological mapping properties defined by digraphs. Discrete and Continuous Dynamical Systems, 1999, 5 (1) : 83-92. doi: 10.3934/dcds.1999.5.83
[14]

Mads Kyed. On a mapping property of the Oseen operator with rotation. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1315-1322. doi: 10.3934/dcdss.2013.6.1315
[15]

Chihiro Matsuoka, Koichi Hiraide. Special functions created by Borel-Laplace transform of Hénon map. Electronic Research Announcements, 2011, 18: 1-11. doi: 10.3934/era.2011.18.1
[16]

Kathy Horadam, Russell East. Partitioning CCZ classes into EA classes. Advances in Mathematics of Communications, 2012, 6 (1) : 95-106. doi: 10.3934/amc.2012.6.95
[17]

Mike Boyle, Sompong Chuysurichay. The mapping class group of a shift of finite type. Journal of Modern Dynamics, 2018, 13: 115-145. doi: 10.3934/jmd.2018014
[18]

Dawei Chen. Strata of abelian differentials and the Teichmüller dynamics. Journal of Modern Dynamics, 2013, 7 (1) : 135-152. doi: 10.3934/jmd.2013.7.135
[19]

Ursula Hamenstädt. Bowen's construction for the Teichmüller flow. Journal of Modern Dynamics, 2013, 7 (4) : 489-526. doi: 10.3934/jmd.2013.7.489
[20]

Ursula Hamenstädt. Dynamics of the Teichmüller flow on compact invariant sets. Journal of Modern Dynamics, 2010, 4 (2) : 393-418. doi: 10.3934/jmd.2010.4.393

2020 Impact Factor: 0.848

Tools

Metrics

  • PDF downloads (47)
  • HTML views (170)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy