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

• Autorzy: Przemysław Liszka
• Języki publikacji: EN

Abstrakty

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.

Kategorie tematyczne

• 37L40: Invariant measures
• 28A80: Fractals

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Annales Polonici Mathematici
• Rocznik: 2013
• Tom: 109
• Numer: 1
• Strony: 75-92

Daty

wydano

2013

Twórcy

autor

Przemysław Liszka

• Institute of Mathematics, University of Silesia, 40-007 Katowice, Poland

Identyfikatory

DOI

10.4064/ap109-1-6