University of Surrey

Test tubes in the lab Research in the ATI Dance Research

Multifractal structure of Bernoulli convolutions

Jordan, T, Shmerkin, P and Solomyak, B (2011) Multifractal structure of Bernoulli convolutions Cambridge Philosophical Society. Mathematical Proceedings, 151 (3). pp. 121-139.

Text (licence)

Download (33kB)

Download (423kB) | Preview


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: Article
Divisions : Faculty of Engineering and Physical Sciences > Mathematics
Authors :
Jordan, T
Shmerkin, P
Solomyak, B
Date : August 2011
DOI : 10.1017/S0305004111000466
Related URLs :
Additional Information : Copyright 2011 Cambridge University Press.
Depositing User : Symplectic Elements
Date Deposited : 27 Jan 2012 09:35
Last Modified : 25 Nov 2013 15:36

Actions (login required)

View Item View Item


Downloads per month over past year

Information about this web site

© The University of Surrey, Guildford, Surrey, GU2 7XH, United Kingdom.
+44 (0)1483 300800