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

Let S-i : R-d -> R-d for i = 1,..., N be contracting similarities. Also, let (p(1),..., p(N), p) be a probability vector and let. be a probability measure on Rd with compact support. We show that there exists a unique probability measure mu on R-d such that

mu + Sigma(i)Pi mu oS(i)(-1) + pv.

The measure mu is called an in-homogeneous self-similar measure.

In this paper we study the L-q spectra and the Renyi 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 inhomogeneous self-similar measure, and we provide applications to the study of box-dimensions of in-homogeneous self-similar sets.