Asian Journal of Mathematics

Volume 25 (2021)

Number 2

The $L^q$-spectrum for a class of self-similar measures with overlap

Pages: 195 – 228

DOI: https://dx.doi.org/10.4310/AJM.2021.v25.n2.a3

Authors

Kathryn E. Hare (Department of Pure Mathematics, University of Waterloo, Ontario, Canada)

Kevin G. Hare (Department of Pure Mathematics, University of Waterloo, Ontario, Canada)

Wanchun Shen (Department of Pure Mathematics, University of Waterloo, Ontario, Canada)

Abstract

It is known that the heuristic principle, referred to as the multifractal formalism, need not hold for self-similar measures with overlap, such as the $3$‑fold convolution of the Cantor measure and certain Bernoulli convolutions. In this paper we study an important function in the multifractal theory, the $L^q$-spectrum, $\tau(q)$, for measures of finite type, a class of self-similar measures that includes these examples. Corresponding to each measure, we introduce finitely many variants on the $L^q$-spectrum which arise naturally from the finite type structure and are often easier to understand than $\tau$. We show that $\tau$ is always bounded by the minimum of these variants and is equal to the minimum variant for $q \geq 0$. This particular variant coincides with the $L^q$-spectrum of the measure $\mu$ restricted to appropriate subsets of its support. If the IFS satisfies particular structural properties, which do hold for the above examples, then $\tau$ is shown to be the minimum of these variants for all $q$. Under certain assumptions on the local dimensions of $\mu$, we prove that the minimum variant for $q \ll 0$ coincides with the straight line having slope equal to the maximum local dimension of $\mu$. Again, this is the case with the examples above. More generally, bounds are given for $\tau$ and its variants in terms of notions closely related to the local dimensions of $\mu$.

Keywords

$L^q$-spectrum, multifractal formalism, self-similar measure, finite type

2010 Mathematics Subject Classification

Primary 28A80. Secondary 28A78.

This research was supported in part by NSERC grants RGPIN 2016-03719 and 2019-03930.

Received 23 September 2019

Accepted 8 July 2020

Published 15 October 2021