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The L q spectra and Rényi dimension of generalized inhomogeneous self-similar measures

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Liszka P.
Open Mathematics
|
2014
|
tom 12
|
nr 9
1305-1319
EN
Very recently bounds for the L q spectra of inhomogeneous self-similar measures satisfying the Inhomogeneous Open Set Condition (IOSC), being the appropriate version of the standard Open Set Condition (OSC), were obtained. However, if the IOSC is not satisfied, then almost nothing is known for such measures. In the paper we study the L q spectra and Rényi dimension of generalized inhomogeneous self-similar measures, for which we allow an infinite number of contracting similarities and probabilities depending on positions. As an application of the results, we provide a systematic approach to obtaining non-trivial bounds for the L q spectra and Rényi dimension of inhomogeneous self-similar measures not satisfying the IOSC and of homogeneous ones not satisfying the OSC. We also provide some non-trivial bounds without any separation conditions.
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On inhomogeneous self-similar measures and their $L^{q}$ spectra

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Liszka P.
Annales Polonici Mathematici
|
2013
|
tom 109
|
nr 1
75-92
EN
Let $S_i:ℝ^d → ℝ^d$ for i = 1,..., N be contracting similarities, let $(p₁,..., p_N,p)$ be a probability vector and let ν be a probability measure on $ℝ^d$ with compact support. It is well known that there exists a unique inhomogeneous self-similar probability measure μ on $ℝ^d$ such that $μ = ∑_{i=1}^{N}{p_iμ ∘ S_i^{-1}} + pν$. 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.
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