On symmetric Bernoulli convolutions

Spectra of Bernoulli Convolutions

On the Gibbs properties of Bernoulli convolutions

We consider infinitely convolved Bernoulli measures (or simply Bernoulli convolutions) related to the β-numeration. A matrix decomposition of these measures is obtained in the case when β is a PV number. We also determine their Gibbs properties for β being a multinacci number, which makes the multifractal analysis of the corresponding Bernoulli convolution possible.

Sixty Years of Bernoulli Convolutions

The distribution νλ of the random series ∑ ±λ is the infinite convolution product of 1 2 (δ−λn + δλn). These measures have been studied since the 1930’s, revealing connections with harmonic analysis, the theory of algebraic numbers, dynamical systems, and Hausdorff dimension estimation. In this survey we describe some of these connections, and the progress that has been made so far on the funda...

Multifractal Structure of Bernoulli Convolutions

Let ν λ be the distribution of the random series ∑∞ n=1 inλ , 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 ν λ for typical λ. Namely, we investigate the size of the sets ∆λ,p(α) = { x ∈ R : lim r↘0 log ν λ(B(x, r)) log r =...

Spectral property of the Bernoulli convolutions ✩

For 0 < ρ < 1, let μρ be the Bernoulli convolution associated with ρ. Jorgensen and Pedersen [P. Jorgensen, S. Pedersen, Dense analytic subspaces in fractal L2-spaces, J. Anal. Math. 75 (1998) 185–228] proved that if ρ = 1/q where q is an even integer, then L(μρ) has an exponential orthonormal basis. We show that for any 0 < ρ < 1, L(μρ) contains an infinite orthonormal set of exponential funct...