Branching Random Walk with exponentially decreasing steps, and stochastically self-similar measures

We consider a Branching Random Walk on ℝ whose step size decreases by a fixed factor, 0 < λ < 1, with each turn. This process generates a random probability measure on ℝ; that is, the limit of uniform distribution among the 2ⁿ particles of the n-th step. We present an initial investigation of the limit measure and its support. We show, in particular, that (1) for almost every λ > 1/2 the limit measure is almost surely (a.s.) absolutely continuous with respect to the Lebesgue measure, but for Pisot 1/λ it is a.s. singular; (2) for all λ > (√5-1)/2 the support of the measure is a.s. the closure of its interior; (3) for Pisot 1/λ the support of the measure is "fractured": it is a.s. disconnected, and the components of the complement are not isolated on both sides.