On the minimal generating weighted IFS of self-similar measure

TL;DR

This paper investigates the structure of minimal generating weighted IFSs for self-similar measures on the real line, providing conditions for their existence and showing that most measures have such IFSs without separation assumptions.

Contribution

It offers new sufficient conditions for the existence of minimal generating weighted IFSs and demonstrates their prevalence among most self-similar measures on the real line.

Findings

Most self-similar measures on the real line have a minimal generating weighted IFS.

The paper establishes sufficient conditions for the existence of such IFSs.

Uses zero distribution, factorization of exponential polynomials, and dynamical systems in proofs.

Abstract

We concern the structrue of generating weighted IFSs of a self-similar measure on the real line. We provide various sufficient conditions for the existence of a minimal generating weighted IFS of a self-similar measure on the real line. Under the homogeneity, we show that `most' self-similar measures on the real line have a minimal generating weighted IFS, without separation conditions. The ingredients of our proofs are based on the zero distribution and factorization theory of exponential polynomials, logarithmic commensurability (with a dynamical system argument), and results on the structure of generating IFSs of a self-similar sets.