L^q spectra and Rényi dimensions of in-homogeneous self-similar measures

Let for i ≤ 1, ..., N be contracting similarities. Also, let (p ₁, ..., p_N, 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 L^q 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 L^q spectra are non-differentiable. This is in sharp contrast to the behaviour of the L^q 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.