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.
- Eric Bedford and John Smillie, Polynomial diffeomorphisms of $\mathbf C^2$. VIII. Quasi-expansion, Amer. J. Math. 124 (2002), no. 2, 221–271. MR 1890993, DOI 10.1353/ajm.2002.0008
- Lipman Bers, Simultaneous uniformization, Bull. Amer. Math. Soc. 66 (1960), 94–97. MR 111834, DOI 10.1090/S0002-9904-1960-10413-2
- Carlos Cabrera, On the classification of laminations associated to quadratic polynomials, J. Geom. Anal. 18 (2008), no. 1, 29–67. MR 2365667, DOI 10.1007/s12220-007-9009-4
- C. Cabrera and T. Kawahira. Topology of the regular part for infinitely renormalizable quadratic polynomials. Preprint, 2007. (arXiv:math.DS/0706.4225)
- Alberto Candel and Lawrence Conlon, Foliations. I, Graduate Studies in Mathematics, vol. 23, American Mathematical Society, Providence, RI, 2000. MR 1732868, DOI 10.1090/gsm/023
- M. Denker and M. Urbański, Hausdorff and conformal measures on Julia sets with a rationally indifferent periodic point, J. London Math. Soc. (2) 43 (1991), no. 1, 107–118. MR 1099090, DOI 10.1112/jlms/s2-43.1.107
- Peter Haïssinsky, Rigidity and expansion for rational maps, J. London Math. Soc. (2) 63 (2001), no. 1, 128–140. MR 1802762, DOI 10.1112/S0024610700001563
- John H. Hubbard and Ralph W. Oberste-Vorth, Hénon mappings in the complex domain. II. Projective and inductive limits of polynomials, Real and complex dynamical systems (Hillerød, 1993) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 464, Kluwer Acad. Publ., Dordrecht, 1995, pp. 89–132. MR 1351520
- Vadim A. Kaimanovich and Mikhail Lyubich, Conformal and harmonic measures on laminations associated with rational maps, Mem. Amer. Math. Soc. 173 (2005), no. 820, vi+119. MR 2111096, DOI 10.1090/memo/0820
- Jeremy Kahn and Mikhail Lyubich, A priori bounds for some infinitely renormalizable quadratics. II. Decorations, Ann. Sci. Éc. Norm. Supér. (4) 41 (2008), no. 1, 57–84 (English, with English and French summaries). MR 2423310, DOI 10.24033/asens.2063
- Tomoki Kawahira, On the regular leaf space of the cauliflower, Kodai Math. J. 26 (2003), no. 2, 167–178. MR 1993672, DOI 10.2996/kmj/1061901060
- Tomoki Kawahira, Semiconjugacies between the Julia sets of geometrically finite rational maps. II, Dynamics on the Riemann sphere, Eur. Math. Soc., Zürich, 2006, pp. 131–138. MR 2348959, DOI 10.4171/011-1/7
- T. Kawahira. Tessellation and Lyubich–Minsky laminations associated with quadratic maps, I: Pinching semiconjugacies. Ergodic Theory Dynam. Systems 29 (2009), no. 9.
- M. Lyubich. Laminations and holomorphic dynamics. Lecture notes in “New Direction in Dynamical Systems”, Kyoto, 2002, available at his web-page.
- Mikhail Lyubich and Yair Minsky, Laminations in holomorphic dynamics, J. Differential Geom. 47 (1997), no. 1, 17–94. MR 1601430
- Katsuhiko Matsuzaki and Masahiko Taniguchi, Hyperbolic manifolds and Kleinian groups, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1998. Oxford Science Publications. MR 1638795
- Curtis T. McMullen, Renormalization and 3-manifolds which fiber over the circle, Annals of Mathematics Studies, vol. 142, Princeton University Press, Princeton, NJ, 1996. MR 1401347, DOI 10.1515/9781400865178
- Curtis T. McMullen, Thermodynamics, dimension and the Weil-Petersson metric, Invent. Math. 173 (2008), no. 2, 365–425. MR 2415311, DOI 10.1007/s00222-008-0121-2
- John Milnor, Periodic orbits, externals rays and the Mandelbrot set: an expository account, Astérisque 261 (2000), xiii, 277–333 (English, with English and French summaries). Géométrie complexe et systèmes dynamiques (Orsay, 1995). MR 1755445
- Welington de Melo and Sebastian van Strien, One-dimensional dynamics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 25, Springer-Verlag, Berlin, 1993. MR 1239171, DOI 10.1007/978-3-642-78043-1
- R. Mañé, P. Sad, and D. Sullivan, On the dynamics of rational maps, Ann. Sci. École Norm. Sup. (4) 16 (1983), no. 2, 193–217. MR 732343, DOI 10.24033/asens.1446
- Dennis Sullivan, Linking the universalities of Milnor-Thurston, Feigenbaum and Ahlfors-Bers, Topological methods in modern mathematics (Stony Brook, NY, 1991) Publish or Perish, Houston, TX, 1993, pp. 543–564. MR 1215976
- 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