A Multifractal Decomposition for Self-similar Measures with Exact Overlaps

TL;DR

This paper introduces a multifractal decomposition for self-similar measures with overlaps, characterizing their spectra via a finite set of concave functions, and establishes conditions under which the multifractal formalism holds.

Contribution

It provides a novel multifractal decomposition framework for self-similar measures with overlaps under weak separation, linking spectra to concave functions and formalism conditions.

Findings

The $L^q$-spectrum is the minimum of finitely many concave functions.

The multifractal spectrum is the maximum of the conjugates of these functions.

The measure satisfies the multifractal formalism at points of spectrum differentiability.

Abstract

We study self-similar measures in $\mathbb{R}$ satisfying the weak separation condition along with weak technical assumptions which are satisfied in all known examples. For such a measure $\mu$, we show that there is a finite set of concave functions $\{\tau_1,\ldots,\tau_m\}$ such that the $L^q$-spectrum of $\mu$ is given by $\min\{\tau_1,\ldots,\tau_m\}$ and the multifractal spectrum of $\mu$ is given by $\max\{\tau_1^*,\ldots,\tau_m^*\}$, where $\tau_i^*$ denotes the concave conjugate of $\tau_i$. In particular, the measure $\mu$ satisfies the multifractal formalism if and only if its multifractal spectrum is a concave function. This implies that $\mu$ satisfies the multifractal formalism at values corresponding to points of differentiability of the $L^q$-spectrum. We also verify existence of the limit for the $L^q$-spectra of such measures for every $q\in\mathbb{R}$. As a direct application, we obtain many new results and simple proofs of well-known results in the multifractal analysis of self-similar measures satisfying the weak separation condition.