On inhomogeneous self-similar measures and their $L^{q}$ spectra

Przemysław Liszka

On inhomogeneous self-similar measures and their

L^{q}

spectra

Przemysław Liszka

Annales Polonici Mathematici, Tome 107 (2013), p. 75-92 / Harvested from The Polish Digital Mathematics Library

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Résumé

Let $S_{i} : ℝ^{d} \to ℝ^{d}$ for i = 1,..., N be contracting similarities, let $(p ₁, . . ., p_{N}, p)$ 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 $μ = \sum_{i = 1}^{N} p_{i} μ \circ S_{i}^{- 1} + p ν$ . 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.

Publié le : 2013-01-01

Zbl 06176235

EUDML-ID : urn:eudml:doc:280199

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     title = {On inhomogeneous self-similar measures and their $L^{q}$ spectra},
     journal = {Annales Polonici Mathematici},
     volume = {107},
     year = {2013},
     pages = {75-92},
     zbl = {06176235},
     language = {en},
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Przemysław Liszka. On inhomogeneous self-similar measures and their $L^{q}$ spectra. Annales Polonici Mathematici, Tome 107 (2013) pp. 75-92. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap109-1-6/