Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On biaccessible points of the Mandelbrot set

Author: Saeed Zakeri
Journal: Proc. Amer. Math. Soc. 134 (2006), 2239-2250
MSC (2000): Primary 37F10, 37F20, 37F35, 35F45
Published electronically: March 14, 2006
MathSciNet review: 2213696
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This paper provides a description for the quadratic polynomials on the boundary of the Mandelbrot set $ \mathcal M$ which are typical in the sense of harmonic measure. In particular, it is shown that a typical point on the boundary of $ \mathcal M$ has a unique parameter ray landing on it. Applications of this result in the study of embedded arcs in $ \mathcal M$ and the lamination associated with $ \mathcal M$ are given.

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

  • [A] L. Ahlfors, Conformal invariants, McGraw-Hill, 1973. MR 0357743 (50:10211)
  • [BS] J. Birman and C. Series, Geodesics with bounded intersection number on surfaces are sparsely distributed, Topology, 24 (1985) 217-225. MR 0793185 (87f:57012)
  • [B] F. Bonahon, Geodesic laminations on surfaces, in ``Laminations and foliations in dynamics, geometry and topology,'' Contemp. Math., 269, Amer. Math. Soc. 2001, 1-37. MR 1810534 (2001m:57023)
  • [BSc] H. Bruin and D. Schleicher, Symbolic dynamics of quadratic polynomials, Mittag-Leffler Institute preprints, 2001/2002, No. 07.
  • [D1] A. Douady, Algorithms for computing angles in the Mandelbrot set, in ``Chaotic Dynamics and Fractals,'' Academic Press, 1986, 155-168. MR 0858013
  • [D2] A. Douady, Descriptions of compact sets in $ \mathbb{C}$, in ``Topological methods in modern mathematics,'' Publish or Perish, 1993, 429-465. MR 1215973 (94g:58185)
  • [DH1] A. Douady and J. Hubbard, Etude dynamique de polynommes complexes, Publications Math. d'Orsay 84-02 (1984) (premiere partie) and 85-04 (1985) (deuxieme partie). MR 0762431 (87f:58072a) MR 0812271 (87f:58072b)
  • [DH2] A. Douady and J. Hubbard, On the dynamics of polynomial-like mappings, Ann. Sci. Ec. Norm. Sup., 18 (1985) 287-343. MR 0816367 (87f:58083)
  • [GM] L. Goldberg and J. Milnor, Fixed points of polynomial maps II, Ann. Sci. Ec. Norm. Sup., 26 (1993) 51-98. MR 1209913 (95d:58107)
  • [GS] J. Graczyk and G. Swiatek, Harmonic measure and expansion on the boundary of the connectedness locus, Invent. Math., 142 (2000) 605-629. MR 1804163 (2001k:37066)
  • [H] J. Hubbard, Local connectivity of Julia sets and bifurcation loci, in ``Topological Methods in Modern Mathematics,'' Publish or Perish, 1993, 457-511. MR 1215974 (94c:58172)
  • [L] M. Lyubich, On the Lebesgue measure of the Julia set of a quadratic polynomial, Stony Brook IMS preprint # 1991/10.
  • [Ma] A. Manning, Logarithmic capacity and renormalizability for landing on the Mandelbrot set, Bull. London Math. Soc., 28 (1996) 521-526. MR 1396155 (97g:58141)
  • [Mc] C. McMullen, Complex Dynamics and Renormalization, Annals of Math Studies 135, Princeton University Press, 1994. MR 1312365 (96b:58097)
  • [M1] J. Milnor, Dynamics in one complex variable: Introductory lectures, Vieweg, 1999. MR 1721240 (2002i:37057)
  • [M2] J. Milnor, Local connectivity of Julia sets: expository lectures, in ``The Mandelbrot set, Theme and Variations,'' LMS Lecture Note Series 274, Cambr. U. Press 2000, 67-116. MR 1765085 (2001b:37073)
  • [M3] J. Milnor, Periodic orbits, external rays and the Mandelbrot set: An expository account, Astérisque, 261 (2000) 277-333. MR 1755445 (2002e:37067)
  • [M4] J. Milnor, Pasting together Julia sets: a worked out example of mating, Experiment. Math., 13 (2004) 55-92. MR 2065568 (2005c:37087)
  • [P] C. Pommerenke, Boundary behaviour of conformal maps, Springer-Verlag, 1992. MR 1217706 (95b:30008)
  • [R] P. R\oesch, Holomorphic motions and puzzles, in ``The Mandelbrot set, Theme and Variations,'' LMS Lecture Note Series 274, Cambr. U. Press 2000, 117-131. MR 1765086 (2001c:37046)
  • [S1] S. Smirnov, Symbolic dynamics and the Collet-Eckmann condition, Internat. Math. Res. Notices, 7 (2000) 333-351. MR 1749741 (2001i:37071)
  • [S2] S. Smirnov, On support of dynamical laminations and biaccessible points in Julia set, Colloq. Math., 87 (2001) 287-295. MR 1814670 (2001m:37092)
  • [T1] Tan Lei, Voisinages connexes des points de Misiurewicz, Ann. Inst. Fourier Grenoble, 42 (1992) 707-735. MR 1196091 (94a:58165)
  • [T2] Tan Lei, Local properties of the Mandelbrot set at parabolic points, in ``The Mandelbrot set, Theme and Variations,'' LMS Lecture Note Series 274, Cambr. U. Press 2000, 133-160. MR 1765087 (2001g:37065)
  • [Th] W. Thurston, The geometry and topology of $ 3$-manifolds, Lecture notes, Princeton University, 1976-1979.
  • [Z1] S. Zakeri, Biaccessibility in quadratic Julia sets, Ergod. Th. & Dyn. Sys., 20 (2000) 1859-1883. MR 1804961 (2001k:37068)
  • [Z2] S. Zakeri, External rays and the real slice of the Mandelbrot set, Ergod. Th. & Dyn. Sys., 23 (2003) 637-660. MR 1972243 (2004a:37057)
  • [Zd] A. Zdunik, On biaccessible points in Julia sets of polynomials, Fund. Math., 163 (2000) 277-286. MR 1758329 (2001f:37058)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 37F10, 37F20, 37F35, 35F45

Retrieve articles in all journals with MSC (2000): 37F10, 37F20, 37F35, 35F45

Additional Information

Saeed Zakeri
Affiliation: Institute for Mathematical Sciences, Stony Brook University, Stony Brook, New York 11794
Address at time of publication: Department of Mathematics, Queens College of CUNY, Flushing, New York 11367

Received by editor(s): February 5, 2004
Received by editor(s) in revised form: January 24, 2005
Published electronically: March 14, 2006
Communicated by: Linda Keen
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society