Journal of Fractal Geometry


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

Minkowski content and fractal Euler characteristic for conformal graph directed systems

Marc Kesseböhmer[1] and Sabrina Kombrink[2]

(1) FB 3 - Mathematik, Universität Bremen, Bibliothekstr. 1, 28359, Bremen, Germany
(2) Institut für Mathematik, Universität zu Lübeck, Ratzeburger Allee 160, 23562, 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