Multifractal structure of Bernoulli convolutions

Abstract

Let ${\nu }_{\lambda }^p$ be the distribution of the random series ${\sum }_{n=1}^{\infty }{i}_n{\lambda }^n$, where $i_n$ is a sequence of i.i.d. random variables taking the values 0,1 with probabilities $p,1-p$. These measures are the well-known (biased) Bernoulli convolutions. In this paper we study the multifractal spectrum of ${\nu }_{\lambda }^p$ for typical $\lambda$. Namely, we investigate the size of the sets \[ Δ_{λ,p}(α) = \left\{x\in\R: \lim_{r\searrow 0} \frac{\log ν_λ^p(B(x,r))}{\log r} =α\right\}. \] Our main results highlight the fact that for almost all, and in some cases all, $\lambda$ in an appropriate range, ${\Delta }_{\lambda ,p}(\alpha )$ is nonempty and, moreover, has positive Hausdorff dimension, for many values of $\alpha$. This happens even in parameter regions for which ${\nu }_{\lambda }^p$ is typically absolutely continuous.

24 pages, 2 figures

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