On the L q spectra of in-homogeneous self-similar measures

Abstract

Abstract The in-homogeneous self-similar measure μ is defined by the relation μ = ∑ i = 1 N p i ⁢ μ ∘ S i - 1 + p ⁢ ν , \mu=\sum_{i=1}^{N}p_{i}\mu\circ S_{i}^{-1}+p\nu, where ( p 1 , … , p N , p ) {(p_{1},\ldots,p_{N},p)} is a probability vector, each S i : ℝ d → ℝ d {S_{i}:\mathbb{R}^{d}\to\mathbb{R}^{d}} , i = 1 , … , N {i=1,\ldots,N} , is a contraction similarity, and ν is a compactly supported Borel probability measure on ℝ d {\mathbb{R}^{d}} . In this paper, we study the L q {L^{q}} -spectra of in-homogeneous self-similar measures. We obtain non-trivial lower and upper bounds for the L q {L^{q}} -spectra of an arbitrary in-homogeneous self-similar measure. Moreover, if the IFS satisfies some separation conditions, the bounds for the L q {L^{q}} -spectra can be improved.