Algebraic dependence number and cardinality of generating iterated function systems
Abstract
For a dust-like self-similar set (generated by IFSs with the strong separation condition), Elekes, Keleti and M\'{a}th\'{e} found an invariant, called `algebraic dependence number', by considering its generating IFSs and isometry invariant self-similar measures. We find an intrinsic quantitative characterisation of this number: it is the dimension over of the vector space generated by the logarithms of all the common ratios of infinite geometric sequences in the gap length set, minus 1. With this concept, we present a lower bound on the cardinality of generating IFS (with or without separation conditions) in terms of the gap lengths of a dust-like set. We also establish analogous result for dust-like graph-directed attractors on complete metric spaces. This is a new application of the ratio analysis method and the gap sequence.
Cite
@article{arxiv.2408.11708,
title = {Algebraic dependence number and cardinality of generating iterated function systems},
author = {Junda Zhang},
journal= {arXiv preprint arXiv:2408.11708},
year = {2026}
}