Lq spectra and Rényi dimensions of in-homogeneous self-similar measures

Abstract

Let for i = 1, …, N be contracting similarities. Also, let (p1, …, pN, p) be a probability vector and let ν be a probability measure on with compact support. We show that there exists a unique probability measure μ on such that The measure μ is called an in-homogeneous self-similar measure.In this paper we study the Lq spectra and the Rényi dimensions of in-homogeneous self-similar measures. We prove that new multifractal phenomena, not exhibited by (ordinary) self-similar measures, appear in the in-homogeneous case. In particular, we show that in-homogeneous self-similar measures may have phase transitions, i.e. points at which the Lq spectra are non-differentiable. This is in sharp contrast to the behaviour of the Lq spectra of (ordinary) self-similar measures satisfying the open set condition.We also present a number of applications of our results. Namely, we obtain non-trivial upper bounds for the multifractal spectrum of an in-homogeneous self-similar measure, and we provide applications to the study of box-dimensions of in-homogeneous self-similar sets.