On inhomogeneous self-similar measures and their $L^{q}$ spectra
Volume 109 / 2013
Annales Polonici Mathematici 109 (2013), 75-92
MSC: Primary 28A80; Secondary 37L40.
DOI: 10.4064/ap109-1-6
Abstract
Let $S_i:\mathbb {R}^d\rightarrow \mathbb {R}^d$ for $i=1,\dots ,N$ be contracting similarities, 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. It is well known that there exists a unique inhomogeneous self-similar probability measure $\mu $ on $\mathbb {R}^d$ such that $\mu =\sum _{i=1}^{N}{p_i\mu \circ S_i^{-1}} + p\nu $.
We give satisfactory estimates for the lower and upper bounds of the $L^q$ spectra of inhomogeneous self-similar measures. The case in which there are a countable number of contracting similarities and probabilities is considered. In particular, we generalise some results obtained by Olsen and Snigireva [Nonlinearity 20 (2007), 151–175] and we give a partial answer to Question 2.7 in that paper.
