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Conformal Geometry and Dynamics

Published by the American Mathematical Society, the Conformal Geometry and Dynamics (ECGD) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4173

The 2020 MCQ for Conformal Geometry and Dynamics is 0.5.

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Tessellation and Lyubich–Minsky laminations associated with quadratic maps, II: Topological structures of $3$-laminations
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by Tomoki Kawahira PDF
Conform. Geom. Dyn. 13 (2009), 6-75 Request permission

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.
  • © Copyright 2009 American Mathematical Society
  • Journal: Conform. Geom. Dyn. 13 (2009), 6-75
  • MSC (2000): Primary 37F45; Secondary 37F99
  • DOI: https://doi.org/10.1090/S1088-4173-09-00186-6
  • MathSciNet review: 2476656