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Multifractal structure of Bernoulli convolutions

Published online by Cambridge University Press:  19 August 2011

THOMAS JORDAN ,
PABLO SHMERKIN  and
BORIS SOLOMYAK
THOMAS JORDAN
Affiliation:
School of Mathematics, University of Bristol, University Walk, Bristol, BS8 1TW. e-mail: Thomas.Jordan@bristol.ac.uk
PABLO SHMERKIN
Affiliation:
Department of Mathematics, Faculty of Engineering and Physical Sciences, University of Surrey, Guildford, GU2 7XH, UK. e-mail: pablo.shmerkin@surrey.ac.uk
BORIS SOLOMYAK
Affiliation:
Department of Mathematics, University of Washington, Box 354 350, Seattle WA 98195-5350, U.S.A. e-mail: solomyak@math.washington.edu
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Abstract

Let νpλ 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 νpλ for typical λ. Namely, we investigate the size of the sets 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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 151 , Issue 3 , November 2011 , pp. 521 - 539
Copyright
Copyright © Cambridge Philosophical Society 2011

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