A DIMENSION FORMULA FOR BERNOULLI CONVOLUTIONS

We present a ''dynamical'' approach to the study of the singularity of infinitely convolved Bernoulli measures nu(beta), for beta the golden section. We introduce nu(beta) as the transverse measure of the maxim um entropy measure mu on the repelling set invariant for contracting m aps of the square, the ''fat baker's'' transformation. Our approach st rongly relies on the Markov structure of the underlying dynamical syst em. Indeed, if beta = 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 inh erited by the Markov structure of the map. The dimension of the projec ted measure is therefore associated to the growth of this product; our dimension formula appears in a natural way as a version of the Furste nberg-Guivarch formula. Our technique provides an explicit dimension f ormula and, most important, provides a formalism well suited for the m ultifractal analysis of this measure, as we will show in a forthcoming paper.