Multifractal analysis of Bernoulli convolutions associated with Salem numbers

Abstract

We consider the multifractal structure of the Bernoulli convolution νλ, where λ−1 is a Salem number in (1,2). Let τ(q) denote the Lq-spectrum of νλ. We show that if α∈[τ′(+∞),τ′(0+)], then the level setE(α):={x∈R:limr→0logνλ([x−r,x+r])logr=α} is non-empty and dimHE(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′(+∞),τ′(0+)] is not a singleton when λ−1 is the largest real root of the polynomial xn−xn−1−⋯−x+1, n⩾4. An example is constructed to show that absolutely continuous self-similar measures may also have rich multifractal structures.