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

HTML articles powered by AMS MathViewer

- 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.

## References

- 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**

## 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