Multifractal analysis of Bernoulli convolutions associated with Salem numbers

Abstract

We consider the multifractal structure of the Bernoulli convolution ${\nu }_{\lambda }$, where ${\lambda }^{-1}$ is a Salem number in $(1,2)$. Let $\tau (q)$ denote the $L^q$ spectrum of ${\nu }_{\lambda }$. We show that if $\alpha \in [{\tau }^{\prime }(+\infty ),{\tau }^{\prime }(0+)]$, then the level set $$E(\alpha ):=x\in \mathbb{R}:\underset{r\to 0}{\lim }\frac{\log {\nu }_{\lambda }([x-r,x+r])}{\log r}=\alpha$$ is non-empty and ${\dim }_{H}E(\alpha )={\tau }^{\ast }(\alpha )$, where ${\tau }^{\ast }$ denotes the Legendre transform of $\tau$. This result extends to all self-conformal measures satisfying the asymptotically weak separation condition. We point out that the interval $[{\tau }^{\prime }(+\infty ),{\tau }^{\prime }(0+)]$ is not a singleton when ${\lambda }^{-1}$ is the largest real root of the polynomial $x^n-x^{n-1}-...-x+1$, $n\ge 4$. An example is constructed to show that absolutely continuous self-similar measures may also have rich multifractal structures.

arXiv comment26 pages. Accepted by Adv. Math

References in corpus (1)

• Phase transitions for the multifractal analysis of self-similar measures

Cited by in corpus (1)

• Survey article: Seventy years of Salem numbers