IUMJ

Title: Multifractal spectra of in-homogenous self-similar measures

Authors: L. Olsen and N. Snigireva

Issue: Volume 57 (2008), Issue 4, 1789-1844

Abstract:

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.