# Porosity of Collet–Eckmann Julia sets

Feliks Przytycki; Steffen Rohde

Fundamenta Mathematicae (1998)

- Volume: 155, Issue: 2, page 189-199
- ISSN: 0016-2736

## Access Full Article

top## Abstract

top## How to cite

topPrzytycki, Feliks, and Rohde, Steffen. "Porosity of Collet–Eckmann Julia sets." Fundamenta Mathematicae 155.2 (1998): 189-199. <http://eudml.org/doc/212251>.

@article{Przytycki1998,

abstract = {We prove that the Julia set of a rational map of the Riemann sphere satisfying the Collet-Eckmann condition and having no parabolic periodic point is mean porous, if it is not the whole sphere. It follows that the Minkowski dimension of the Julia set is less than 2.},

author = {Przytycki, Feliks, Rohde, Steffen},

journal = {Fundamenta Mathematicae},

keywords = {Hölder domain; shrinking neighborhoods; Collet-Eckmann condition; Julia set; rational map; box dimension; porosity},

language = {eng},

number = {2},

pages = {189-199},

title = {Porosity of Collet–Eckmann Julia sets},

url = {http://eudml.org/doc/212251},

volume = {155},

year = {1998},

}

TY - JOUR

AU - Przytycki, Feliks

AU - Rohde, Steffen

TI - Porosity of Collet–Eckmann Julia sets

JO - Fundamenta Mathematicae

PY - 1998

VL - 155

IS - 2

SP - 189

EP - 199

AB - We prove that the Julia set of a rational map of the Riemann sphere satisfying the Collet-Eckmann condition and having no parabolic periodic point is mean porous, if it is not the whole sphere. It follows that the Minkowski dimension of the Julia set is less than 2.

LA - eng

KW - Hölder domain; shrinking neighborhoods; Collet-Eckmann condition; Julia set; rational map; box dimension; porosity

UR - http://eudml.org/doc/212251

ER -

## References

top- [BC] M. Benedicks and L. Carleson, The dynamics of the Hénon maps, Ann. of Math. 133 (1991), 73-169. Zbl0724.58042
- [CE] P. Collet and J.-P. Eckmann, Positive Lyapunov exponents and absolute continuity for maps of the interval, Ergodic Theory Dynam. Systems 3 (1983), 13-46. Zbl0532.28014
- [CJY] L. Carleson, P. Jones and J.-C. Yoccoz, Julia and John, Bol. Soc. Brasil. Mat. 25 (1994), 1-30.
- [DPU] M. Denker, F. Przytycki and M. Urbański, On the transfer operator for rational functions on the Riemann sphere, Ergodic Theory Dynam. Systems 16 (1996), 255-266. Zbl0852.46024
- [DU] M. Denker and M. Urbański, On Hausdorff measures on Julia sets of subexpanding rational maps, Israel J. Math. 76 (1992), 193-214. Zbl0763.30008
- [G] J. Graczyk, Hyperbolic subsets, conformal measures and Hausdorff dimension of Julia sets, preprint, 1995.
- [GS] J. Graczyk and S. Smirnov, Collet, Eckmann, & Hölder, Invent. Math., to appear.
- [JM] P. Jones and N. Makarov, Density properties of harmonic measure, Ann. of Math. 142 (1995), 427-455. Zbl0842.31001
- [KR] P. Koskela and S. Rohde, Hausdorff dimension and mean porosity, Math. Ann., to appear. Zbl0890.30013
- [LP] G. Levin and F. Przytycki, When do two rational functions have the same Julia set?, Proc. Amer. Math Soc. 125 (1997), 2179-2190. Zbl0870.58085
- [M] R. Mañé, On a theorem of Fatou, Bol. Soc. Brasil. Mat. 24 (1993), 1-12. Zbl0781.30023
- [McM] C. McMullen, Self-similarity of Siegel discs and Hausdorff dimension of Julia sets, preprint, 1995.
- [NPV] F. Nazarov, I. Popovici and A. Volberg, Domains with quasi-hyperbolic boundary condition, domains with Greens boundary condition, and estimates of Hausdorff dimension of their boundary, preprint, 1995.
- [NP] T. Nowicki and F. Przytycki, The conjugacy of Collet-Eckmann's map of the interval with the tent map is Hölder continuous, Ergodic Theory Dynam. Systems 9 (1989), 379-388. Zbl0664.58014
- [P] C. Pommerenke, Boundary Behaviour of Conformal Maps, Springer, 1992. Zbl0762.30001
- [P1] F. Przytycki, Iterations of holomorphic Collet-Eckmann maps: conformal and invariant measures, Trans. Amer. Math. Soc., to appear. Zbl0892.58063
- [P2] F. Przytycki, On measure and Hausdorff dimension of Julia sets for holomorphic Collet-Eckmann maps, in: Internat. Conf. on Dynamical Systems, Montevideo 1995 - a Tribute to Ricardo Mañ (F. Ledrappier, J. Lewowicz and S. Newhouse, eds.), Pitman Res. Notes Math. 362, Longman, 1996, 167-181.
- [PUZ] F. Przytycki, M. Urbański and A. Zdunik, Harmonic, Gibbs and Hausdorff measures on repellers for holomorphic maps, II, Studia Math. 97 (1991), 189-225. Zbl0732.58022
- [SS] W. Smith and D. Stegenga, Exponential integrability of the quasihyperbolic metric in Hölder domains, Ann. Acad. Sci. Fenn. 16 (1991), 345-360. Zbl0725.46024
- [T] M. Tsujii, Positive Lyapunov exponents in families of one dimensional dynamical systems, Invent. Math. 111 (1993), 113-137. Zbl0787.58029
- [U] M. Urbański, Rational functions with no recurrent critical points, Ergodic Theory Dynam. Systems 14 (1994), 391-414. Zbl0807.58025

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.