Matrix representations for some self-similar measures on $\mathbb{R}^{d}$

TL;DR

This paper develops matrix representations for certain self-similar measures in higher dimensions and proves their $L^q$-spectra are differentiable, extending previous one-dimensional results.

Contribution

It introduces matrix representations for self-similar measures satisfying the finite type condition in higher dimensions and proves the differentiability of their $L^q$-spectra.

Findings

Matrix representations for self-similar measures in $\,\mathbb{R}^d$

Differentiability of the $L^q$-spectrum for these measures

Extension of Feng's 2003 result to higher dimensions

Abstract

We establish matrix representations for self-similar measures on $\mathbb{R}^d$ generated by equicontractive IFSs satisfying the finite type condition. As an application, we prove that the $L^q$-spectrum of every such self-similar measure is differentiable on $(0,\infty)$. This extends an earlier result of Feng (J. Lond. Math. Soc.(2) 68(1):102--118, 2003) to higher dimensions.