Matrix random products with singular harmonic measure |
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Authors: | Vadim A. Kaimanovich Vincent Le Prince |
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Affiliation: | 1.Department of Mathematics and Statistics,University of Ottawa,Ottawa,Canada;2.IRMAR, Université Rennes-1,Rennes,France |
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Abstract: | Any Zariski dense countable subgroup of SL(d, mathbb R){SL(d, mathbb {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 mathbb Rd{mathbb {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 arbitrarily long Lyapunov vector. |
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