Journal of Fractal Geometry


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Volume 2, Issue 2, 2015, pp. 171–227
DOI: 10.4171/JFG/19

Published online: 2015-05-27

Minkowski content and fractal Euler characteristic for conformal graph directed systems

Marc Kesseböhmer[1] and Sabrina Kombrink[2]

(1) Universität Bremen, Germany
(2) Universität zu Lübeck, Germany

We study the (local) Minkowski content and the (local) fractal Euler characteristic of limit sets $F \subset \mathbb R$ of conformal graph directed systems (cGDS) $\Phi$. For the local quantities we prove that the logarithmic Cesàro averages always exist and are constant multiples of the $\delta$-conformal measure. If $\Phi$ is non-lattice, then also the non-average local quantities exist and coincide with their respective average versions. When the conformal contractions of $\Phi$ are analytic, the local versions exist if and only if $\Phi$ is non-lattice. For the non-local quantities the above results in particular imply that limit sets of Fuchsian groups of Schottky type are Minkowski measurable, proving a conjecture of Lapidus from 1993. Further, when the contractions of the cGDS are similarities, we obtain that the Minkowski content and the fractal Euler characteristic of $F$ exist if and only if $\Phi$ is non-lattice, generalising earlier results by Falconer, Gatzouras, Lapidus and van Frankenhuijsen for non-degenerate self-similar subsets of $\mathbb R$ that satisfy the open set condition.

Keywords: Minkowski content, fractal Euler characteristic, conformal graph directed system, fractal curvature measures, renewal theory

Kesseböhmer Marc, Kombrink Sabrina: Minkowski content and fractal Euler characteristic for conformal graph directed systems. J. Fractal Geom. 2 (2015), 171-227. doi: 10.4171/JFG/19