Multifractal structure of Bernoulli convolutions

Thomas Jordan, Pablo Shmerkin, Boris Solomyak

Research output: Contribution to journalArticlepeer-review

22 Scopus citations


Let vλp be the distribution of the random series, 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 vp for typical?. Namely, we investigate the size of the sets Our main results highlight the fact that for almost all, and in some cases all, ? in an appropriate range, δλ,p (a) is nonempty and, moreover, has positive Hausdorff dimension, for many values of a. This happens even in parameter regions for which vλp is typically absolutely continuous.

Original languageEnglish
Pages (from-to)521-539
Number of pages19
JournalMathematical Proceedings of the Cambridge Philosophical Society
Issue number3
StatePublished - 2011
Externally publishedYes


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