Multifractal formalism for almost all self-affine measures

TL;DR

This paper advances the multifractal analysis of self-affine measures by extending Falconer's formula beyond a specific range and verifying the multifractal formalism for most cases.

Contribution

It extends Falconer's $L^q$-spectrum formula outside the range $1< q extless 2$ and partially verifies the multifractal formalism for almost all self-affine measures.

Findings

Extended Falconer's formula outside the range 1<q≤2

Partially verified the multifractal formalism for most self-affine measures

Provided new insights into the multifractal structure of self-affine measures

Abstract

We conduct the multifractal analysis of self-affine measures for "almost all" family of affine maps. Besides partially extending Falconer's formula of $L^q$-spectrum outside the range $1< q\leq 2$, the multifractal formalism is also partially verified.