Minkowski content and fractal Euler characteristic for conformal graph directed systems

Abstract

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.2010 Mathematics Subject Classification. Primary 28A80; Secondary 28A75, 60K05.

Localization results for Minkowski contents

“…for arbitrary closed sets . We will show that these two measures coincide (whenever one of them exists), generalizing an observation made earlier for self‐similar sets and certain self‐conformal sets . The following statement establishes this phenomenon for compact sets.…”

“…The local S‐content was shown in to exist for for all nonlattice self‐similar sets satisfying OSC and, moreover, to coincide with the local Minkowski content. A similar relation has been observed in for nonlattice self‐conformal sets in and in for nonlattice limit sets of conformal graph directed systems in .…”

“…Completely analogous remarks apply to the (average) local contents of the limit sets of the graph directed conformal iterated function systems studied in . Remark Note that the existence of the local Minkowski content implies the existence of the Minkowski content of a set but not vice versa , and similarly for the S‐contents.…”

Lattice self-similar sets on the real line are not Minkowski measurable

“…Our main results are stated in Section 3 and proved in Section 4. We conclude by showing in Section 5 that for sets in R the above-mentioned results from [KK15,KPW16] are equivalent.…”

“…In the present article we fully remove the assumptions of [LvF00, KK15,KPW16] and in this way provide the last puzzle-piece in proving that under OSC a nontrivial self-similar subset of R is Minkowski measurable iff it arises from a nonlattice IFS. This resolves the conjecture stated in [Lap93, Conjecture 3] and [Gat00, Section 5.2] for self-similar sets in R.…”