Almost sure absolute continuity of Bernoulli convolutions

Absolute Continuity of Bernoulli Convolutions, a Simple Proof

A bstract . The distribution νλ of the random series ∑ ±λn has been studied by many authors since the two seminal papers by Erdős in 1939 and 1940. Works of Alexander and Yorke, Przytycki and Urbański, and Ledrappier showed the importance of these distributions in several problems in dynamical systems and Hausdorff dimension estimation. Recently the second author proved a conjecture made by Gar...

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...