Illinois Journal of Mathematics

Geometric exponents for hyperbolic Julia sets

Stefan-M. Heinemann and Bernd O. Stratmann

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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

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

First available in Project Euclid: 13 November 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37F50: Small divisors, rotation domains and linearization; Fatou and Julia sets
Secondary: 28A80: Fractals [See also 37Fxx] 37F15: Expanding maps; hyperbolicity; structural stability 37F35: Conformal densities and Hausdorff dimension


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.

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