On the regularity of the multifractal spectrum of Bernoulli convolutions

In previous work we developed a thermodynamic formalism for the Bernoulli convolution associated with the golden mean, and we obtained by perturbative analysis the existence, regularity, and strict convexity of the pressure F(beta) in a neighborhood of beta = 0. This gives the existence of a multifractal spectrum f(alpha) in a neighborhood of the almost sure Value alpha = f(alpha) = 0, 9957.... In the present paper, by a direct study of the Ruelle-Perron-Frobenius operator associated with the random unbounded matrix product arising in our problem, we can prove the regularity of the pressure F(beta) for (at least) beta is an element of (- 1/2, + infinity). This yields the interval of the singularity spectrum between the minimal value of the dimension of v, alpha(min) = 0.94042.., and the almost sure value, alpha(a.s.) = 0.9957....