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2011.8: Multifractal Structure of Bernoulli Convolutions

2011.8: Thomas Jordan, Pablo Shmerkin and Boris Solomyak (2011) Multifractal Structure of Bernoulli Convolutions.

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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 \[ \Delta_{\lambda,p}(\alpha) = \left\{x\in\R: \lim_{r\searrow 0} \frac{\log \nu_\lambda^p(B(x,r))}{\log r} =\alpha\right\}. \] 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.

Item Type:MIMS Preprint
Uncontrolled Keywords:Bernoulli convolutions, multifractal analysis, CICADA
Subjects:MSC 2000 > 28 Measure and integration
MSC 2000 > 37 Dynamical systems and ergodic theory
MIMS number:2011.8
Deposited By:Mr Pablo Shmerkin
Deposited On:17 January 2011

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