Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

Multifractal structure of Bernoulli convolutions

THOMAS JORDANa1, PABLO SHMERKINa2 and BORIS SOLOMYAKa3

a1 School of Mathematics, University of Bristol, University Walk, Bristol, BS8 1TW. e-mail: Thomas.Jordan@bristol.ac.uk

a2 Department of Mathematics, Faculty of Engineering and Physical Sciences, University of Surrey, Guildford, GU2 7XH, UK. e-mail: pablo.shmerkin@surrey.ac.uk

a3 Department of Mathematics, University of Washington, Box 354 350, Seattle WA 98195-5350, U.S.A. e-mail: solomyak@math.washington.edu

Abstract

Let νpλ be the distribution of the random series $\sum_{n=1}^\infty i_n \lam^n$, where in 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 νpλ for typical λ. Namely, we investigate the size of the sets

\begin{linenomath}
\[
\Delta_{\lam, p}(\alpha) = \left\{x\in\R: \lim_{r\searrow 0} \frac{\log \nula^p(B(x, r))}{\log r} =\alpha\right\}\!.
\]\end{linenomath}

Our main results highlight the fact that for almost all, and in some cases all, λ in an appropriate range, Δλ, p(α) is nonempty and, moreover, has positive Hausdorff dimension, for many values of α. This happens even in parameter regions for which νpλ is typically absolutely continuous.

(Received November 09 2010)

(Online publication August 19 2011)

Footnotes

† Supported by EPSRC grant EP/E050441/1 and the University of Manchester.

‡ Supported in part by NSF grants DMS-0654408 and DMS-0968879.