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.
@article{bwmeta1.element.bwnjournal-article-fmv155i2p189bwm,
author = {Feliks Przytycki and Steffen Rohde},
title = {Porosity of Collet--Eckmann Julia sets},
journal = {Fundamenta Mathematicae},
volume = {158},
year = {1998},
pages = {189-199},
zbl = {0908.58054},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv155i2p189bwm}
}
Przytycki, Feliks; Rohde, Steffen. Porosity of Collet–Eckmann Julia sets. Fundamenta Mathematicae, Tome 158 (1998) pp. 189-199. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv155i2p189bwm/
[00000] [BC] M. Benedicks and L. Carleson, The dynamics of the Hénon maps, Ann. of Math. 133 (1991), 73-169. | Zbl 0724.58042
[00001] [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. | Zbl 0532.28014
[00002] [CJY] L. Carleson, P. Jones and J.-C. Yoccoz, Julia and John, Bol. Soc. Brasil. Mat. 25 (1994), 1-30.
[00003] [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. | Zbl 0852.46024
[00004] [DU] M. Denker and M. Urbański, On Hausdorff measures on Julia sets of subexpanding rational maps, Israel J. Math. 76 (1992), 193-214. | Zbl 0763.30008
[00005] [G] J. Graczyk, Hyperbolic subsets, conformal measures and Hausdorff dimension of Julia sets, preprint, 1995.
[00006] [GS] J. Graczyk and S. Smirnov, Collet, Eckmann, & Hölder, Invent. Math., to appear.
[00007] [JM] P. Jones and N. Makarov, Density properties of harmonic measure, Ann. of Math. 142 (1995), 427-455. | Zbl 0842.31001
[00008] [KR] P. Koskela and S. Rohde, Hausdorff dimension and mean porosity, Math. Ann., to appear. | Zbl 0890.30013
[00009] [LP] G. Levin and F. Przytycki, When do two rational functions have the same Julia set?, Proc. Amer. Math Soc. 125 (1997), 2179-2190. | Zbl 0870.58085
[00010] [M] R. Mañé, On a theorem of Fatou, Bol. Soc. Brasil. Mat. 24 (1993), 1-12. | Zbl 0781.30023
[00011] [McM] C. McMullen, Self-similarity of Siegel discs and Hausdorff dimension of Julia sets, preprint, 1995.
[00012] [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.
[00013] [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. | Zbl 0664.58014
[00014] [P] C. Pommerenke, Boundary Behaviour of Conformal Maps, Springer, 1992. | Zbl 0762.30001
[00015] [P1] F. Przytycki, Iterations of holomorphic Collet-Eckmann maps: conformal and invariant measures, Trans. Amer. Math. Soc., to appear. | Zbl 0892.58063
[00016] [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.
[00017] [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. | Zbl 0732.58022
[00018] [SS] W. Smith and D. Stegenga, Exponential integrability of the quasihyperbolic metric in Hölder domains, Ann. Acad. Sci. Fenn. 16 (1991), 345-360. | Zbl 0725.46024
[00019] [T] M. Tsujii, Positive Lyapunov exponents in families of one dimensional dynamical systems, Invent. Math. 111 (1993), 113-137. | Zbl 0787.58029
[00020] [U] M. Urbański, Rational functions with no recurrent critical points, Ergodic Theory Dynam. Systems 14 (1994), 391-414. | Zbl 0807.58025