Stationary Measures for Projective Transformations: The Blackwell and Furstenberg Measures

In this paper we study the Blackwell and Furstenberg measures, which play an important role in information theory and the study of Lyapunov exponents. For the Blackwell measure we determine parameter domains of singularity and give upper bounds for the Hausdorff dimension. For the Furstenberg measure, we establish absolute continuity for some parameter values. Our method is to analyze linear fractional iterated function schemes which are contracting on average, have no separation properties (that is, we do not assume that the open set condition holds, see Hutchinson in Indiana Univ. Math. J. 30:713–747, 1981) and, in the case of the Blackwell measure, have place dependent probabilities. In such a general setting, even an effective upper bound on the dimension of the measure is difficult to achieve.