University of Oulu

Troscheit, S. (2023). On the Minkowski content of self-similar random homogeneous iterated function systems. Annales Fennici Mathematici, 48(1), 345–359.

On the Minkowski content of self-similar random homogeneous iterated function systems

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Author: Troscheit, Sascha1
Organizations: 1University of Oulu, Research Unit of Mathematical Sciences
Format: article
Version: published version
Access: open
Online Access: PDF Full Text (PDF, 0.2 MB)
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Language: English
Published: Finnish Mathematical Society, 2023
Publish Date: 2023-09-21


The Minkowski content of a compact set is a fine measure of its geometric scaling. For Lebesgue null sets it measures the decay of the Lebesgue measure of epsilon neighbourhoods of the set. It is well known that self-similar sets, satisfying reasonable separation conditions and non-log commensurable contraction ratios, have a well-defined Minkowski content. When dropping the contraction conditions, the more general notion of average Minkowski content still exists. For random recursive self-similar sets the Minkowski content also exists almost surely, whereas for random homogeneous self-similar sets it was recently shown by Zähle that the Minkowski content exists in expectation.

In this short note we show that the upper Minkowski content, as well as the upper average Minkowski content of random homogeneous self-similar sets is infinite, almost surely, answering a conjecture posed by Zähle. Additionally, we show that in the random homogeneous equicontractive self-similar setting the lower Minkowski content is zero and the lower average Minkowski content is also infinite. These results are in stark contrast to the random recursive model or the mean behaviour of random homogeneous attractors.

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Series: Annales Fennici mathematici
ISSN: 2737-0690
ISSN-E: 2737-114X
ISSN-L: 2737-0690
Volume: 48
Issue: 1
Pages: 345 - 359
DOI: 10.54330/afm.129638
Type of Publication: A1 Journal article – refereed
Field of Science: 111 Mathematics
Funding: The author was financially supported by Austrian Science Fund (FWF) Lise Meitner Senior Fellowship M-2813.
Copyright information: © 2023 The Finnish Mathematical Society. This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.