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

Przemysław Liszka. "On inhomogeneous self-similar measures and their $L^{q}$ spectra." Annales Polonici Mathematici 109.1 (2013): 75-92. <http://eudml.org/doc/280199>.

@article{PrzemysławLiszka2013,
abstract = {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.},
author = {Przemysław Liszka},
journal = {Annales Polonici Mathematici},
keywords = { spectra; inhomogeneous self-similar set; inhomogeneous selfsimilar measure; invariant measures},
language = {eng},
number = {1},
pages = {75-92},
title = {On inhomogeneous self-similar measures and their $L^\{q\}$ spectra},
url = {http://eudml.org/doc/280199},
volume = {109},
year = {2013},
}

TY - JOUR
AU - Przemysław Liszka
TI - On inhomogeneous self-similar measures and their $L^{q}$ spectra
JO - Annales Polonici Mathematici
PY - 2013
VL - 109
IS - 1
SP - 75
EP - 92
AB - 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.
LA - eng
KW - spectra; inhomogeneous self-similar set; inhomogeneous selfsimilar measure; invariant measures
UR - http://eudml.org/doc/280199
ER -