- journal article metadata
European Mathematical Society Publishing House
2016-09-19 17:05:11
Journal of Fractal Geometry
J. Fractal Geom.
JFG
2308-1309
2308-1317
Measure and integration
General
10.4171/JFG
http://www.ems-ph.org/doi/10.4171/JFG
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
2
2015
2
Minkowski content and fractal Euler characteristic for conformal graph directed systems
Marc
Kesseböhmer
Universität Bremen, BREMEN, GERMANY
Sabrina
Kombrink
Universität zu Lübeck, LÜBECK, GERMANY
Minkowski content, fractal Euler characteristic, conformal graph directed system, fractal curvature measures, renewal theory
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.
Measure and integration
Probability theory and stochastic processes
171
227
10.4171/JFG/19
http://www.ems-ph.org/doi/10.4171/JFG/19