Shannon's theorem for locally compact groups

2018/12/18 by Behrang Forghani, Forghani, Behrang, Giulio Tiozzo +1
Mathematics · #22D05 #60G50 #60J50 #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR) #Probability (math.PR) #math.DS #math.GR #math.GT #math.PR #msc:22D05 #msc:60G50 #msc:60J50

paper · pdf · doi:10.48550/arxiv.1812.07292

34 pages, no figures

arxiv created 2020/03/09 · arxiv updated 2020/03/10

Abstract

We consider random walks on locally compact groups, extending the geometric criteria for the identification of their Poisson boundary previously known for discrete groups. First, we prove a version of the Shannon-McMillan-Breiman theorem, which we then use to generalize Kaimanovich's ray approximation and strip approximation criteria. We give several applications to identify the Poisson boundary of locally compact groups which act by isometries on nonpositively curved spaces, as well as on Diestel-Leader graphs and horocylic products.

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