Branching random walk with exponentially decreasing steps, and stochastically self-similar measures

We consider a Branching Random Walk on $\mathbb{R}$ whose step size decreases by a fixed factor, $0<\lambda<1$, with each turn. This process generates a random probability measure on $\mathbb{R}$; that is, the limit of uniform distribution among the $2^n$ 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 $\lambda>1/2$ the limit measure is almost surely (a.s.) absolutely continuous with respect to the Lebesgue measure, but for Pisot $1/\lambda$ it is a.s. singular; (2) for all $\lambda> (\sqrt{5}-1)/2$ the support of the measure is a.s. the closure of its interior; (3) for Pisot $1/\lambda$ the support of the measure is ``fractured'': it is a.s. disconnected, and the components of the complement are not isolated on both sides.