On multifractal formalism for self-similar measures with overlaps

Abstract

Let $\mu$ be a self-similar measure generated by an IFS $\Phi =\{{\varphi }_i{\}}_{i=1}^{\ell }$ of similarities on ${\mathbb{R}}^d$ ($d\ge 1$). When $\Phi$ is dimensional regular (see Definition~1.1), we give an explicit formula for the $L^q$-spectrum ${\tau }_{\mu }(q)$ of $\mu$ over $[0,1]$, and show that ${\tau }_{\mu }$ is differentiable over $(0,1]$ and the multifractal formalism holds for $\mu$ at any $\alpha \in [{\tau }_{\mu }^{\prime }(1),{\tau }_{\mu }^{\prime }(0+)]$. We also verify the validity of the multifractal formalism of $\mu$ over $[{\tau }_{\mu }^{\prime }(\infty ),{\tau }_{\mu }^{\prime }(0+)]$ for two new classes of overlapping algebraic IFSs by showing that the asymptotically weak separation condition holds. For one of them, the proof appeals to a recent result of Shmerkin on the $L^q$-spectrum of self-similar measures.

Some minor changes, mainly in the introduction

References in corpus (2)

• Absolute continuity of self-similar measures, their projections and convolutions
• On self-similar sets with overlaps and inverse theorems for entropy in ${\mathbb{R}}^d$