JournalsjfgVol. 2, No. 2pp. 171–227

Minkowski content and fractal Euler characteristic for conformal graph directed systems

  • Marc Kesseböhmer

    Universität Bremen, Germany
  • Sabrina Kombrink

    Universität zu Lübeck, Germany
Minkowski content and fractal Euler characteristic for conformal graph directed systems cover
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Abstract

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