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

**Tomoki Kawahira**

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

DOI:
http://dx.doi.org/10.1090/S1088-4173-09-00186-6

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.

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© Copyright 2009
American Mathematical Society