## Illinois Journal of Mathematics

### Geometric exponents for hyperbolic Julia sets

#### Abstract

We show that the Hausdorff dimension of the Julia set associated to a hyperbolic rational map is bounded away from $2$, where the bound depends only on certain intrinsic geometric exponents. This result is derived via lower estimates for the iterate-counting function and for the dynamical Poincaré series. We deduce some interesting consequences, such as upper bounds for the decay of the area of parallel-neighbourhoods of the Julia set, and lower bounds for the Lyapunov exponents with respect to the measure of maximal entropy.

#### Article information

Source
Illinois J. Math., Volume 45, Number 3 (2001), 775-785.

Dates
First available in Project Euclid: 13 November 2009

https://projecteuclid.org/euclid.ijm/1258138150

Digital Object Identifier
doi:10.1215/ijm/1258138150

Mathematical Reviews number (MathSciNet)
MR1879234

Zentralblatt MATH identifier
1055.37056

#### Citation

Heinemann, Stefan-M.; Stratmann, Bernd O. Geometric exponents for hyperbolic Julia sets. Illinois J. Math. 45 (2001), no. 3, 775--785. doi:10.1215/ijm/1258138150. https://projecteuclid.org/euclid.ijm/1258138150