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

Kawahira, Tomoki

doi:10.1090/S1088-4173-09-00186-6

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

Conform. Geom. Dyn. 13 (2009), 6-75

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

Published electronically: February 3, 2009

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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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