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Tessellation and Lyubich-Minsky laminations associated with quadratic maps, II: Topological structures of $ 3$-laminations

Author: Tomoki Kawahira
Journal: Conform. Geom. Dyn. 13 (2009), 6-75
MSC (2000): Primary 37F45; Secondary 37F99
Published electronically: February 3, 2009
MathSciNet review: 2476656
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Abstract: According to an analogy to quasi-Fuchsian groups, we investigate the topological and combinatorial structures of Lyubich and Minsky's affine and hyperbolic $ 3$-laminations associated with hyperbolic and parabolic quadratic maps.

We begin by showing that hyperbolic rational maps in the same hyperbolic component have quasi-isometrically the same $ 3$-laminations. This gives a good reason to regard the main cardioid of the Mandelbrot set as an analogue of the Bers slices in the quasi-Fuchsian space. Then we describe the topological and combinatorial changes of laminations associated with hyperbolic-to-parabolic degenerations (and parabolic-to-hyperbolic bifurcations) of quadratic maps. For example, the differences between the structures of the quotient $ 3$-laminations of Douady's rabbit, the Cauliflower, and $ z \mapsto z^2$ 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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Additional Information

Tomoki Kawahira
Affiliation: Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya, 464-8602 Japan

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 Grant-in-Aid 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

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