Matrix random products with singular harmonic measure

Vadim A. Kaimanovich; Vincent Le Prince

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arXiv · 0807.1015

Matrix random products with singular harmonic measure

Vadim A. Kaimanovich·Vincent Le Prince

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Abstract

Any Zariski dense countable subgroup of $SL(d,R)$ is shown to carry a non-degenerate finitely supported symmetric random walk such that its harmonic measure on the flag space is singular. The main ingredients of the proof are: (1) a new upper estimate for the Hausdorff dimension of the projections of the harmonic measure onto Grassmannians in $R^d$ in terms of the associated differential entropies and differences between the Lyapunov exponents; (2) an explicit construction of random walks with uniformly bounded entropy and Lyapunov exponents going to infinity.

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Vadim A. Kaimanovich, Vincent Le Prince. 2008-07-07. Matrix random products with singular harmonic measure. https://arxiv.org/abs/0807.1015

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