A dimension formula for Bernoulli convolutions

Abstract

We present a “dynamical” approach to the study of the singularity of infinitely convolved Bernoulli measuresvβ, for β the golden section. We introducevβ as the transverse measure of the maximum entropy measure μ on the repelling set invariant for contracting maps of the square, the “fat baker's” transformation. Our approach strongly relies on the Markov structure of the underlying dynamical system. Indeed, if β=golden mean, the fat baker's transformation has a very simple Markov coding. The “ambiguity” (of order two) of this coding, which appears when projecting on the line, due to passages for the central, overlapping zone, can be expressed by means of products of matrices (of order two). This product has a Markov distribution inherited by the Markov structure of the map. The dimension of the projected measure is therefore associated to the growth of this product; our dimension formula appears in a natural way as a version of the Furstenberg-Guivarch formula. Our technique provides an explicit dimension formula and, most important, provides a formalism well suited for the multifractal analysis of this measure, as we will show in a forthcoming paper.