Asian Journal of Mathematics

Volume 27 (2023)

Number 6

$L^q$-spectrum of a class of self-similar measures

Pages: 867 – 892

DOI: https://dx.doi.org/10.4310/AJM.2023.v27.n6.a3

Authors

Sze-Man Ngai (Key Laboratory of High Performance Computing and Stochastic Information Processing (HPCSIP), Hunan Normal University, Changsha, China; and Department of Mathematical Sciences, Georgia Southern University, Statesboro, GA, USA)

Wen-Quan Zhao (Key Laboratory of Computing and Stochastic Mathematics (Ministry of Education), School of Mathematics and Statistics, Hunan Normal University, Changsha, Hunan, China)

Abstract

We compute the $L^q$-spectrum of self-similar measures defined by an iterated function system of the form $S_i(x) = (x + i)/2, i = 0, 1, \dotsc , m, m \geq 2$. For an iterated function system of the form $S_i(x) = (x + (N − 1)i)/N, i = 0, 1, \dotsc , N, N \geq 3$, the $L^q$-spectrum of a corresponding self-similar measure was computed by Lau and Ngai $\href{https://www.jstor.org/stable/24901125 }{\textrm{[Indiana Univ. Math. J. 49 (2000), 925–972]}}$ for $q \geq 0$ and by Feng, Lau and Wang $\href{https://dx.doi.org/10.4310/AJM.2005.v9.n4.a2 }{\textrm{[Asian J. Math. 9 (2005), 473–488]}}$ for the case $N = 3$ and $q \lt 0$. The method used by Lau and Ngai fails if the contraction ratio is $1/2$. We use some techniques by Feng, Lau and Wang to express the $L^q$-spectrum of $\mu$ as a limit of matrix products. By defining a sub-multiplicative sequence and using its properties, we obtain a formula for the $L^q$-spectrum, $q \in \mathbb{R}$, under the assumption that some limit function $r(q)$ exists when $q \lt 0$. We study systems for which the limit defining $r(q)$ exists.

Keywords

$L^q$-spectrum, multifractal formalism, iterated function systems with overlaps, self-similar measure, sub-multiplicative sequence

2010 Mathematics Subject Classification

Primary 28A80. Secondary 28A78.

Received 6 September 2022

Accepted 26 September 2023

Published 7 August 2024