Algebraic dependence number and cardinality of generating iterated function systems

TL;DR

This paper characterizes the algebraic dependence number of self-similar sets through the dimension over rationals of logarithms of common ratios, providing bounds on the number of generating IFSs based on gap lengths.

Contribution

It offers an intrinsic, quantitative characterization of the algebraic dependence number and establishes bounds on the number of generating IFSs using gap length analysis.

Findings

Provides a new intrinsic characterization of the algebraic dependence number.

Establishes lower bounds on the number of generating IFSs based on gap lengths.

Extends results to dust-like graph-directed attractors.

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 $\mathbb{Q}$ 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.