Multifractal analysis of Bernoulli convolutions associated with Salem numbers

We consider the multifractal structure of the Bernoulli convolution $ν_λ$, where $λ^{-1}$ is a Salem number in $(1,2)$. Let $τ(q)$ denote the $L^q$ spectrum of $ν_λ$. We show that if $α\in [τ'(+\infty), τ'(0+)]$, then the level set $E(α):={x\in \R:\; \lim_{r\to 0}\frac{\log ν_λ([x-r, x+r])}{\log r}=α}$ is non-empty and $\dim_HE(α)=τ^*(α)$, where $τ^*$ denotes the Legendre transform of $τ$. This result extends to all self-conformal measures satisfying the asymptotically weak separation condition. We point out that the interval $[τ'(+\infty), τ'(0+)]$ is not a singleton when $λ^{-1}$ 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.

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