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

Itai Benjamini·Ori Gurel-Gurevich·Boris Solomyak
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Abstract

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

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Itai Benjamini, Ori Gurel-Gurevich, Boris Solomyak. 2007-05-30. Branching random walk with exponentially decreasing steps, and stochastically self-similar measures. https://arxiv.org/abs/math/0608271

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