Minkowski content and fractal Euler characteristic for conformal graph directed systems

TL;DR

This paper investigates the Minkowski content and fractal Euler characteristic of limit sets from conformal graph directed systems, establishing conditions for their existence and measurability, and confirming a conjecture for Fuchsian groups.

Contribution

It provides new results on the existence and properties of Minkowski content and fractal Euler characteristic for conformal graph directed systems, including non-lattice and lattice cases, and confirms a longstanding conjecture.

Findings

Limit sets of Fuchsian groups of Schottky type are Minkowski measurable.

Local quantities exist and are proportional to the conformal measure in non-lattice cases.

Existence of Minkowski content and Euler characteristic is characterized by the lattice property of the system.

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 Ces\`aro 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.