# Minkowski content and fractal Euler characteristic for conformal graph directed systems

### Marc Kesseböhmer

Universität Bremen, Germany### Sabrina Kombrink

Universität zu Lübeck, Germany

## Abstract

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.

## Cite this article

Marc Kesseböhmer, Sabrina Kombrink, Minkowski content and fractal Euler characteristic for conformal graph directed systems. J. Fractal Geom. 2 (2015), no. 2, pp. 171–227

DOI 10.4171/JFG/19