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)

 
 

 

Hausdorff dimension and biaccessibility for polynomial Julia sets


Authors: Philipp Meerkamp and Dierk Schleicher
Journal: Proc. Amer. Math. Soc. 141 (2013), 533-542
MSC (2010): Primary 37F10, 37F20, 37F35
DOI: https://doi.org/10.1090/S0002-9939-2012-11323-1
Published electronically: June 4, 2012
MathSciNet review: 2996957
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We investigate the set of biaccessible points for connected polynomial Julia sets of arbitrary degrees $ d\geq 2$. We prove that the Hausdorff dimension of the set of external angles corresponding to biaccessible points is less than $ 1$, unless the Julia set is an interval. This strengthens theorems of Stanislav Smirnov and Anna Zdunik: they proved that the same set of external angles has zero $ 1$-dimensional measure.


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

  • 1. A. Blokh, C. Curry, G. Levin, L. Oversteegen, D. Schleicher, An extended Fatou-Shishikura inequality and wandering branch continua for polynomials. Manuscript, submitted.
  • 2. A. Blokh, G. Levin, An inequality for laminations, Julia sets and ``growing trees''. Ergodic Theory Dynam. Systems 22 1 (2002), 63-97. MR 1889565 (2003i:37045)
  • 3. H. Bruin, A. Kaffl, D. Schleicher, Symbolic dynamics of quadratic polynomials. Monograph, in preparation.
  • 4. H. Bruin, D. Schleicher, Symbolic dynamics of quadratic polynomials, Mittag Leffler Preprint 7 (2001/02). To appear as [3].
  • 5. H. Bruin, D. Schleicher, Hausdorff dimension of biaccessible angles for quadratic polynomials. Manuscript, in preparation.
  • 6. A. Douady, J. H. Hubbard, Étude dynamique des polynômes complexes, I, Publ. Math. Orsay, 1984.MR 0762431 (87f:58072a)
  • 7. K. Falconer, Fractal geometry. Mathematical foundations and applications, 2nd edn., Wiley, 2003. MR 2118797 (2006b:28001)
  • 8. J. Kiwi, Wandering orbit portraits. Trans. Amer. Math. Soc. 354 4 (2002), 1473-1485. MR 1873015 (2002h:37070)
  • 9. J. Milnor, Dynamics in one complex variable, 3rd edn., Princeton University Press, 2006. MR 2193309 (2006g:37070)
  • 10. C. Pommerenke, Boundary behaviour of conformal maps, Springer, 1992. MR 1217706 (95b:30008)
  • 11. R. Radu, Hausdorff dimension and biaccessibility for polynomial Julia sets. Bachelor's thesis, Jacobs University, 2007.
  • 12. D. Schleicher, S. Zakeri, On biaccessible points in the Julia set of a Cremer quadratic polynomial. Proc. AMS 128 3 (2000), 933-937. MR 1637424 (2000e:37057)
  • 13. S. Smirnov, On supports of dynamical laminations and biaccessible points in polynomial Julia sets, Colloq. Math. 87 (2001) 2, 287-295. MR 1814670 (2001m:37092)
  • 14. W. Thurston, On the geometry and dynamics of iterated rational maps. In: D. Schleicher (ed.), Complex dynamics: families and friends, A K Peters, Wellesley, MA, 2009, 1-137. MR 2508255 (2010m:37076)
  • 15. S. Zakeri, Biaccessibility in quadratic Julia sets, Ergod. Th. & Dynam. Sys. 20 (2000), 1859-1883. MR 1804961 (2001k:37068)
  • 16. S. Zakeri, External rays and the real slice of the Mandelbrot set, Ergod. Th. & Dynam. Sys. 23 (2003), 637-660. MR 1972243 (2004a:37057)
  • 17. A. Zdunik, On biaccessible points in Julia sets of polynomials, Fund. Math. 163 (2000), 277-286. MR 1758329 (2001f:37058)
  • 18. A. Zdunik, Parabolic orbifolds and the dimension of the maximal measure for rational maps, Invent. Math. 99 (1990), 627-649. MR 1032883 (90m:58120)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 37F10, 37F20, 37F35

Retrieve articles in all journals with MSC (2010): 37F10, 37F20, 37F35


Additional Information

Philipp Meerkamp
Affiliation: Department of Mathematics, Malott Hall, Cornell University, Ithaca, New York 14853-4201
Email: pmeerkamp@math.cornell.edu

Dierk Schleicher
Affiliation: School of Engineering and Science, Jacobs University, Postfach 750 561, D-28725 Bremen, Germany
Email: dierk@jacobs-university.de

DOI: https://doi.org/10.1090/S0002-9939-2012-11323-1
Keywords: Julia set, polynomial, biaccessible, Hausdorff dimension
Received by editor(s): April 14, 2011
Received by editor(s) in revised form: June 28, 2011
Published electronically: June 4, 2012
Communicated by: Bryna Kra
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society