Uniform lower bounds on the dimension of Bernoulli convolutions

Smoothness of Projections, Bernoulli Convolutions, and the Dimension of Exceptions

YUVAL PERES and WILHELM SCHLAG

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.

Spectra of Bernoulli Convolutions

A Lower Bound for Garsia’s Entropy for Certain Bernoulli Convolutions

Let β ∈ (1, 2) be a Pisot number and let Hβ denote Garsia’s entropy for the Bernoulli convolution associated with β. Garsia, in 1963 showed that Hβ < 1 for any Pisot β. For the Pisot numbers which satisfy x = xm−1 + xm−2 + · · · + x + 1 (with m ≥ 2) Garsia’s entropy has been evaluated with high precision by Alexander and Zagier for m = 2 and later by Grabner Kirschenhofer and Tichy for m ≥ 3, a...

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...