Remote Access Conformal Geometry and Dynamics
Green Open Access

Conformal Geometry and Dynamics

ISSN 1088-4173



Landing property of stretching rays for real cubic polynomials

Authors: Yohei Komori and Shizuo Nakane
Journal: Conform. Geom. Dyn. 8 (2004), 87-114
MSC (2000): Primary 37F45; Secondary 37F30
Published electronically: March 29, 2004
MathSciNet review: 2060379
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The landing property of the stretching rays in the parameter space of bimodal real cubic polynomials is completely determined. Define the Böttcher vector by the difference of escaping two critical points in the logarithmic Böttcher coordinate. It is a stretching invariant in the real shift locus. We show that stretching rays with non-integral Böttcher vectors have non-trivial accumulation sets on the locus where a parabolic fixed point with multiplier one exists.

References [Enhancements On Off] (What's this?)

  • [B] B. Branner: Turning around the connectedness locus. In: ``Topological methods in modern Mathematics.'' pp. 391-427. Houston, Publish or Perish, 1993. MR 94c:58168
  • [B-D] B. Branner and A. Douady: Surgery on complex polynomials. In: ``Holomorphic Dynamics.'' Lect. Notes in Math. 1345, pp. 11-72, 1988. MR 90e:58114
  • [B-H1] B. Branner and J. Hubbard: The iteration of cubic polynomials. Part I: The global topology of parameter space. Acta Math. 160, 143-206 (1988). MR 90d:30073
  • [B-H2] -: The iteration of cubic polynomials. Part II: Patterns and parapatterns. Acta Math. 169, 229-325 (1992). MR 94d:30044
  • [Bu-He] X. Buff and C. Henriksen: Julia sets in parameter spaces. Comm. Math. Phys. 220, 333-375 (2001). MR 2002d:37075
  • [C-G] L. Carleson and T. Gamelin: Complex Dynamics. Springer-Verlag, 1993. MR 94h:30033
  • [D] A. Douady: Does a Julia set depend continuously on the polynomials? Proceedings of Symposia in Applied Mathematics 49, pp. 91-138, 1994.
  • [D-H] A. Douady and J. Hubbard: On the dynamics of polynomial-like mappings. Ann. Sci. Ec. Norm. Sup. 18, 287-343 (1985). MR 87f:58083
  • [E-Y] A. Epstein and M. Yampolsky: Geography of the cubic connectedness locus I: Intertwining surgery. Ann. Sci. Ec. Norm. Sup. 32, 151-185 (1999). MR 2000i:37067
  • [G-S] J. Graczyk and G. Swiatek: Generic hyperbolicity in the logistic family. Annals Math. 146, 1-52 (1997). MR 99b:58079
  • [K] J. Kiwi: Rational rays and critical portraits of complex polynomials. Stony Brook IMS Preprint 1997/15, 1997.
  • [La] P. Lavaurs: Systèmes dynamiques holomorphes: explosion de points périodiques para- boliques. These Univ. Paris-Sud, 1989.
  • [Ly] M. Lyubich: Dynamics of quadratic polynomials, I-II. Acta Math. 178, 185-297 (1997). MR 98e:58145
  • [Mc] C. McMullen: Hausdorff dimension and conformal dynamics II: Geometrically finite rational maps. Comment. Math. Helv. 75, 535-593 (2000). MR 2001m:37089
  • [Mi1] J. Milnor: Remarks on iterated cubic maps. Experimental Math. 1, 5-23 (1992). MR 94c:58096
  • [Mi2] -: Dynamics in one complex variable: Introductory Lectures. Vieweg (1999). MR 2002i:37057
  • [P] C. Pommerenke: Boundary Behaviour of Conformal Maps. Springer-Verlag, 1991. MR 95b:30008
  • [P-U] F. Przytycki and M. Urbanski: Porosity of Julia sets of non-recurrent and parabolic Collet-Eckmann rational functions. Ann. Acad. Sci. Fenn. Math. 26, 125-154 (2001). MR 2002b:37063
  • [Sh] M. Shishikura: The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets. Annals Math. 147, 225-267 (1998). MR 2000f:37056
  • [W] P. Willumsen: Holomorphic Dynamics: On accumulation of stretching rays. Ph.D. thesis Tech. Univ. Denmark, 1997.
  • [Y] Y. Yin: Geometry and dimension of Julia sets. In: ``Tan Lei (eds.) The Mandelbrot set, Theme and Variations.'' Lecture Notes Series 274, pp. 281-287. London Math. Soc., 2000. MR 2001d:37062

Similar Articles

Retrieve articles in Conformal Geometry and Dynamics of the American Mathematical Society with MSC (2000): 37F45, 37F30

Retrieve articles in all journals with MSC (2000): 37F45, 37F30

Additional Information

Yohei Komori
Affiliation: Department of Mathematics, Osaka City University, Sugimoto 3-3-138, Sumiyoshi-ku, Osaka 558-8585, Japan

Shizuo Nakane
Affiliation: Tokyo Polytechnic University, 1583 Iiyama, Atsugi, Kanagawa 243-0297, Japan

Keywords: Stretching rays, parabolic implosion, radial Julia set
Received by editor(s): May 15, 2003
Received by editor(s) in revised form: November 20, 2003
Published electronically: March 29, 2004
Article copyright: © Copyright 2004 American Mathematical Society

American Mathematical Society