Bernoulli convolutions and differentiable functions

Spectra of Bernoulli Convolutions

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...

Ergodic-theoretic Properties of Certain Bernoulli Convolutions

In [17] the author and A. Vershik have shown that for β = 1 2 (1 + √ 5) and the alphabet {0, 1} the infinite Bernoulli convolution (= the Erdös measure) has a property similar to the Lebesgue measure. Namely, it is quasi-invariant of type II1 under the β-shift, and the natural extension of the β-shift provided with the measure equivalent to the Erdös measure, is Bernoulli. In this note we exten...