Porosity of Collet-Eckmann Julia sets.

*(English)*Zbl 0908.58054The Collet-Eckmann condition came first to be studied for unimodal maps of the interval. The reason it is interesting is because it implies fairly strong stochastic properties (an absolutely continuous invariant measure with exponential decay of correlations) and still occurs on sets of positive measure in families of maps [J. Graczyk and S. Smirnov, Invent. Math. 133, No. 1, 69-96 (1998)]. More recently, the Collet-Eckmann condition was considered for rational maps.

The present paper shows that the Julia set of a rational map which satisfies the Collet-Eckmann condition has box dimension less than 2. The same result was announced earlier for polynomials and has meanwhile appeared in [P. Thieullen, C. Tresser and L. S. Young, C. R. Acad. Sci., Paris, Sér. I 315, No. 1, 69-72 (1992; Zbl 0786.58026)]. The method of proof in the current paper is based on the notion of porosity, introduced a few years ago by the second author and P. Koskela.

The present paper shows that the Julia set of a rational map which satisfies the Collet-Eckmann condition has box dimension less than 2. The same result was announced earlier for polynomials and has meanwhile appeared in [P. Thieullen, C. Tresser and L. S. Young, C. R. Acad. Sci., Paris, Sér. I 315, No. 1, 69-72 (1992; Zbl 0786.58026)]. The method of proof in the current paper is based on the notion of porosity, introduced a few years ago by the second author and P. Koskela.

Reviewer: G.Swiatek (University Park)

##### MSC:

37F99 | Dynamical systems over complex numbers |

30D05 | Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable |

28A80 | Fractals |

28A78 | Hausdorff and packing measures |