Multifractal spectra of in-homogenous self-similar measures Lars OlsenN. Snigireva 28A80multifractalsmultifractal spectra$L^q$ spectrain-homogenous self-similar measurein-homogenous self-similar setself similar measure Let $S_{i} : \mathbb{R}^{d} \to \mathbb{R}^{d}$ for $i=1, \dots, N$ be contracting similarities. Also, let $(p_{1}, \dots, p_{N},p)$ be a probability vector and let $\nu$ be a probability measure on $\mathbb{R}^{d}$ with compact support. Then there exists a unique probability measure $\mu$ on $\mathbb{R}^{d}$ such that $$\mu = \sum_{i}p_{i}\mu \circ S_{i}^{-1} + p\nu. $$ The measure $\mu$ is called an in-homogenous self-similar measure. In previous work we computed the $L^{q}$ spectra of in-homogenous self-similar measures. In this paper we study the significantly more difficult problem of computing the multifratal spectra of in-homogenous self-similar measures satisfying the In-homogenous Open Set Condition. We prove that new multifractal phenomena, not exhibited by (ordinary) self-similar measures, appear in the in-homogenous case. In particular, we show that the multifractal spectra of in-homogenous self-similar measures may be non-concave. This is in sharp contrast to the behaviour of the multifractal spectra of (ordinary) self-similar measures satisfying the Open Set Condition. Several applications are presented. Many of our applications are related to the notoriously difficult problem of computing (or simply obtaining non-trivial bounds) for the multifractal spectra of self-similar measures not satisfying the Open Set Condition. We show that our main results can be applied to obtain non-trivial bounds (and in some cases even exact values) for the multifractal spectra of several large and interesting classes of self-similar measures not satisfying the Open Set Condition. Other applications to non-linear self-similar measures introduced by Glickenstein and Strichartz are also presented. Indiana University Mathematics Journal 2008 text pdf 10.1512/iumj.2008.57.3622 10.1512/iumj.2008.57.3622 en Indiana Univ. Math. J. 57 (2008) 1789 - 1844 state-of-the-art mathematics http://iumj.org/access/