## Hausdorff dimension and biaccessibility for polynomial Julia sets

HTML articles powered by AMS MathViewer

- by Philipp Meerkamp and Dierk Schleicher PDF
- Proc. Amer. Math. Soc.
**141**(2013), 533-542 Request permission

## 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

- A. Blokh, C. Curry, G. Levin, L. Oversteegen, D. Schleicher,
*An extended Fatou-Shishikura inequality and wandering branch continua for polynomials*. Manuscript, submitted. - A. Blokh and G. Levin,
*An inequality for laminations, Julia sets and “growing trees”*, Ergodic Theory Dynam. Systems**22**(2002), no. 1, 63–97. MR**1889565**, DOI 10.1017/S0143385702000032 - H. Bruin, A. Kaffl, D. Schleicher,
*Symbolic dynamics of quadratic polynomials*. Monograph, in preparation. - H. Bruin, D. Schleicher,
*Symbolic dynamics of quadratic polynomials*, Mittag Leffler Preprint**7**(2001/02). To appear as [3]. - H. Bruin, D. Schleicher,
*Hausdorff dimension of biaccessible angles for quadratic polynomials*. Manuscript, in preparation. - A. Douady and J. H. Hubbard,
*Étude dynamique des polynômes complexes. Partie I*, Publications Mathématiques d’Orsay [Mathematical Publications of Orsay], vol. 84, Université de Paris-Sud, Département de Mathématiques, Orsay, 1984 (French). MR**762431** - Kenneth Falconer,
*Fractal geometry*, 2nd ed., John Wiley & Sons, Inc., Hoboken, NJ, 2003. Mathematical foundations and applications. MR**2118797**, DOI 10.1002/0470013850 - Jan Kiwi,
*Wandering orbit portraits*, Trans. Amer. Math. Soc.**354**(2002), no. 4, 1473–1485. MR**1873015**, DOI 10.1090/S0002-9947-01-02896-3 - John Milnor,
*Dynamics in one complex variable*, 3rd ed., Annals of Mathematics Studies, vol. 160, Princeton University Press, Princeton, NJ, 2006. MR**2193309** - Ch. Pommerenke,
*Boundary behaviour of conformal maps*, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 299, Springer-Verlag, Berlin, 1992. MR**1217706**, DOI 10.1007/978-3-662-02770-7 - R. Radu,
*Hausdorff dimension and biaccessibility for polynomial Julia sets*. Bachelor’s thesis, Jacobs University, 2007. - Dierk Schleicher and Saeed Zakeri,
*On biaccessible points in the Julia set of a Cremer quadratic polynomial*, Proc. Amer. Math. Soc.**128**(2000), no. 3, 933–937. MR**1637424**, DOI 10.1090/S0002-9939-99-05111-4 - Stanislav K. Smirnov,
*On supports of dynamical laminations and biaccessible points in polynomial Julia sets*, Colloq. Math.**87**(2001), no. 2, 287–295. MR**1814670**, DOI 10.4064/cm87-2-11 - William P. Thurston,
*On the geometry and dynamics of iterated rational maps*, Complex dynamics, A K Peters, Wellesley, MA, 2009, pp. 3–137. Edited by Dierk Schleicher and Nikita Selinger and with an appendix by Schleicher. MR**2508255**, DOI 10.1201/b10617-3 - Saeed Zakeri,
*Biaccessibility in quadratic Julia sets*, Ergodic Theory Dynam. Systems**20**(2000), no. 6, 1859–1883. MR**1804961**, DOI 10.1017/S0143385700001024 - Saeed Zakeri,
*External rays and the real slice of the Mandelbrot set*, Ergodic Theory Dynam. Systems**23**(2003), no. 2, 637–660. MR**1972243**, DOI 10.1017/S0143385702001335 - Anna Zdunik,
*On biaccessible points in Julia sets of polynomials*, Fund. Math.**163**(2000), no. 3, 277–286. MR**1758329**, DOI 10.4064/fm-163-3-277-286 - Anna Zdunik,
*Parabolic orbifolds and the dimension of the maximal measure for rational maps*, Invent. Math.**99**(1990), no. 3, 627–649. MR**1032883**, DOI 10.1007/BF01234434

## 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
- MR Author ID: 359328
- Email: dierk@jacobs-university.de
- 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
- © Copyright 2012
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2996957