Furstenberg Boundary In Group Dynamics

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Let μ\mu be a likelihood measure on Γ\Gamma with finite entropy, such that its support generates Γ\Gamma as a semigroup. Yet, that is adequate to define a boundary map, therefore it is smart to ask whether or not the ensuing topological boundary is the Poisson boundary. On the other hand, if one drops the second situation, the random walk still converges, however in a weaker sense, i.e. in the Furstenberg-Satake compactification, whose boundary is G/PG/P GR85. In this paper, we drop every moment situation, and establish the Poisson boundary for random walks on discrete subgroups of semisimple Lie groups assuming just that the entropy is finite.

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How Common Is That This Condition?

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