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

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.

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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