Minkowski content and fractal Euler characteristic for conformal graph directed systems

We study the (local) Minkowski content and the (local) fractal Euler characteristic of limit sets F⊂R of conformal graph directed systems (cGDS) Φ. For the local quantities we prove that the logarithmic Cesàro averages always exist and are constant multiples of the δ-conformal measure. If Φ is non-lattice, then also the non-average local quantities exist and coincide with their respective average versions. When the conformal contractions of Φ are analytic, the local versions exist if and only if Φ 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 Φ is non-lattice, generalising earlier results by Falconer, Gatzouras, Lapidus and van Frankenhuijsen for non-degenerate self-similar subsets of R that satisfy the open set condition.