On the minimal generating weighted IFS of self-similar measure

We concern the structure 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 results of exponential polynomials (factorization theory and the distribution of zeros), logarithmic commensurability (with a dynamical system argument), and results on the structure of generating IFSs of a self-similar sets.