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

TL;DR

This paper investigates the $L^q$-spectrum of self-similar measures with overlaps, introducing variants that simplify analysis and establishing conditions under which these variants accurately describe the spectrum.

Contribution

It defines and compares variants of the $L^q$-spectrum for measures of finite type, providing new insights into their structure and relation to local dimensions.

Findings

$ au(q)$ is bounded by and equal to the minimum of variants for $q \,\geq 0$

Variants coincide with the $L^q$-spectrum on certain support subsets

Bounds relate $ au$ to local dimensions of the measure

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 $.