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 $\alpha \in [\tau'(+\infty), \tau'(0+)]$, then the level set $$E(\alpha):={x\in \R:\; \lim_{r\to 0}\frac{\log \nu_\lambda([x-r, x+r])}{\log r}=\alpha}$$ is non-empty and $\dim_HE(\alpha)=\tau^*(\alpha)$, where $\tau^*$ 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 $[\tau'(+\infty), \tau'(0+)]$ is not a singleton when $\lambda^{-1}$ is the largest real root of the polynomial $x^{n}-x^{n-1}-... -x+1$, $n\geq 4$. An example is constructed to show that absolutely continuous self-similar measures may also have rich multifractal structures.