Tessellation and LyubichMinsky laminations associated with quadratic maps, II: Topological structures of laminations
Author:
Tomoki Kawahira
Journal:
Conform. Geom. Dyn. 13 (2009), 675
MSC (2000):
Primary 37F45; Secondary 37F99
Published electronically:
February 3, 2009
MathSciNet review:
2476656
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Abstract: According to an analogy to quasiFuchsian groups, we investigate the topological and combinatorial structures of Lyubich and Minsky's affine and hyperbolic laminations associated with hyperbolic and parabolic quadratic maps. We begin by showing that hyperbolic rational maps in the same hyperbolic component have quasiisometrically the same laminations. This gives a good reason to regard the main cardioid of the Mandelbrot set as an analogue of the Bers slices in the quasiFuchsian space. Then we describe the topological and combinatorial changes of laminations associated with hyperbolictoparabolic degenerations (and parabolictohyperbolic bifurcations) of quadratic maps. For example, the differences between the structures of the quotient laminations of Douady's rabbit, the Cauliflower, and are described. The descriptions employ a new method of tessellation inside the filled Julia set introduced in Part I [Ergodic Theory Dynam. Systems 29 (2009), no. 2] that works like external rays outside the Julia set.
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 T. Kawahira. On the regular leaf space of the cauliflower. Kodai Math. J. 26 (2003), 167178. MR 1993672 (2004c:37089)
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 T. Kawahira. Semiconjugacies between the Julia sets of geometrically finite rational maps II. Dynamics on the Riemann Sphere: A Bodil Branner Festschrift, P. G. Hjorth (ed.) et al., European Math. Soc., 2006, 131138. MR 2348959 (2008h:37048)
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 T. Kawahira. Tessellation and LyubichMinsky laminations associated with quadratic maps, I: Pinching semiconjugacies. Ergodic Theory Dynam. Systems 29 (2009), no. 9.
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 M. Lyubich. Laminations and holomorphic dynamics. Lecture notes in ``New Direction in Dynamical Systems'', Kyoto, 2002, available at his webpage.
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 K. Matsuzaki and M. Taniguchi. Hyperbolic Manifolds and Kleinain Groups. Oxford Univ. Press, 1998. MR 1638795 (99g:30055)
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 C. McMullen. Renormalization and 3manifold which fiber over the circle. Annals of Math Studies 142, Princeton University Press, 1996. MR 1401347 (97f:57022)
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 C. McMullen. Thermodynamics, dimension and the WeilPetersson metric. Invent. Math. 173 (2008), 365  425. MR 2415311
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Additional Information
Tomoki Kawahira
Affiliation:
Graduate School of Mathematics, Nagoya University, Chikusaku, Nagoya, 4648602 Japan
DOI:
http://dx.doi.org/10.1090/S1088417309001866
PII:
S 10884173(09)001866
Received by editor(s):
May 9, 2007
Published electronically:
February 3, 2009
Additional Notes:
Research partially supported by JSPS Research Fellowships for Young Scientists, JSPS GrantinAid for Young Scientists, the Circle for the Promotion of Science and Engineering, Inamori Foundation, and the IHÉS, in chronological order. I sincerely appreciate their support.
Article copyright:
© Copyright 2009 American Mathematical Society
