Multifractal spectra of in-homogenous self-similar measures

Abstract

Let Si:Rd?Rd for i=1,...,N be contracting similarities. Also, let (p1,...,pN,p) be a probability vector and let ? be a probability measure on Rd with compact support. Then there exists a unique probability measure µ on Rd such that µ=?ipiµ°Si-1+p?. The measure µ is called an in-homogenous self-similar measure. In previous work we computed the Lq 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.