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Conformal fitness and uniformization of holomorphically moving disks


Author: Saeed Zakeri
Journal: Trans. Amer. Math. Soc. 368 (2016), 1023-1049
MSC (2010): Primary 37Fxx, 30C85
DOI: https://doi.org/10.1090/tran/6362
Published electronically: May 6, 2015
MathSciNet review: 3430357
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Abstract: Let $ \{U_t\}_{t \in \mathbb{D}}$ be a family of topological disks on the Riemann sphere containing the origin 0 whose boundaries undergo a holomorphic motion over the unit disk $ \mathbb{D}$. We study the question of when there exists a family of Riemann maps $ g_t:(\mathbb{D},0) \to (U_t,0)$ which depends holomorphically on the parameter $ t$. We give five equivalent conditions which provide analytic, dynamical and measure-theoretic characterizations for the existence of the family $ \{ g_t \}_{t \in \mathbb{D}}$, and explore the consequences.


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  • [1] Lars V. Ahlfors, Lectures on quasiconformal mappings, 2nd ed., University Lecture Series, vol. 38, American Mathematical Society, Providence, RI, 2006. With supplemental chapters by C. J. Earle, I. Kra, M. Shishikura and J. H. Hubbard. MR 2241787 (2009d:30001)
  • [2] Lars Ahlfors and Lipman Bers, Riemann's mapping theorem for variable metrics, Ann. of Math. (2) 72 (1960), 385-404. MR 0115006 (22 #5813)
  • [3] Xavier Buff and Arnaud Chéritat, A new proof of a conjecture of Yoccoz, Ann. Inst. Fourier (Grenoble) 61 (2011), no. 1, 319-350 (English, with English and French summaries). MR 2828132 (2012e:37096), https://doi.org/10.5802/aif.2603
  • [4] Xavier Buff and Carsten L. Petersen, On the size of linearization domains, Math. Proc. Cambridge Philos. Soc. 145 (2008), no. 2, 443-456. MR 2442136 (2009i:37113), https://doi.org/10.1017/S0305004108001436
  • [5] Curtis T. McMullen, Complex dynamics and renormalization, Annals of Mathematics Studies, vol. 135, Princeton University Press, Princeton, NJ, 1994. MR 1312365 (96b:58097)
  • [6] John Milnor, Dynamics in one complex variable, 3rd ed., Annals of Mathematics Studies, vol. 160, Princeton University Press, Princeton, NJ, 2006. MR 2193309 (2006g:37070)
  • [7] Raghavan Narasimhan, Several complex variables, The University of Chicago Press, Chicago, Ill.-London, 1971. Chicago Lectures in Mathematics. MR 0342725 (49 #7470)
  • [8] Ch. Pommerenke, Boundary behaviour of conformal maps, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 299, Springer-Verlag, Berlin, 1992. MR 1217706 (95b:30008)
  • [9] Thomas Ransford, Potential theory in the complex plane, London Mathematical Society Student Texts, vol. 28, Cambridge University Press, Cambridge, 1995. MR 1334766 (96e:31001)
  • [10] Reinhold Remmert, Classical topics in complex function theory, Graduate Texts in Mathematics, vol. 172, Springer-Verlag, New York, 1998. Translated from the German by Leslie Kay. MR 1483074 (98g:30002)
  • [11] Zbigniew Slodkowski, Holomorphic motions and polynomial hulls, Proc. Amer. Math. Soc. 111 (1991), no. 2, 347-355. MR 1037218 (91f:58078), https://doi.org/10.2307/2048323
  • [12] Stanislav K. Smirnov, On supports of dynamical laminations and biaccessible points in polynomial Julia sets, Colloq. Math. 87 (2001), no. 2, 287-295. MR 1814670 (2001m:37092), https://doi.org/10.4064/cm87-2-11
  • [13] D. Sullivan, Quasiconformal homeomorphisms and dynamics III: Topological conjugacy classes of analytic endomorphisms, unpublished manuscript.
  • [14] Saeed Zakeri, Biaccessibility in quadratic Julia sets, Ergodic Theory Dynam. Systems 20 (2000), no. 6, 1859-1883. MR 1804961 (2001k:37068), https://doi.org/10.1017/S0143385700001024
  • [15] Saeed Zakeri, Dynamics of cubic Siegel polynomials, Comm. Math. Phys. 206 (1999), no. 1, 185-233. MR 1736986 (2000m:37073), https://doi.org/10.1007/s002200050702
  • [16] Saeed Zakeri, On Siegel disks of a class of entire maps, Duke Math. J. 152 (2010), no. 3, 481-532. MR 2654221 (2011e:37102), https://doi.org/10.1215/00127094-2010-017
  • [17] Anna Zdunik, On biaccessible points in Julia sets of polynomials, Fund. Math. 163 (2000), no. 3, 277-286. MR 1758329 (2001f:37058)

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Additional Information

Saeed Zakeri
Affiliation: Department of Mathematics, Queens College and Graduate Center of CUNY, Queens, New York 11367
Email: saeed.zakeri@qc.cuny.edu

DOI: https://doi.org/10.1090/tran/6362
Received by editor(s): February 11, 2013
Received by editor(s) in revised form: December 14, 2013
Published electronically: May 6, 2015
Article copyright: © Copyright 2015 American Mathematical Society

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