2004/08/25 by Chris Connell, Connell, Chris, Roman Muchnik +1 · 1 citation
Mathematics · #20F67 #37A35 #41A65 #60J50 #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds #Group Theory (math.GR) #Probability (math.PR) #math.FA #math.GR #math.PR #msc:20F67 #msc:37A35 #msc:41A65 #msc:60J50
paper · pdf · doi:10.48550/arxiv.math/0408355
56 pages
arxiv created 2004/08/25 · openalex publication_date 2004/08/25 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider a group G of isometries acting on a (not necessarily geodesic) delta-hyperbolic space X and possessing a radial limit set of full measure within its limit set. For any continuous quasiconformal measure w supported on the limit set, we produce a stationary measure m on G. Moreover the limit set together with w forms a m-boundary and w is harmonic with respect to the random walk induced by m. In the case when X is a CAT(-1) space and G acts cocompactly, for instance, we show that m has finite first moment. This implies that the boundary of X with w is the unique Poisson boundary for m. As a bi-product, we establish sufficient conditions for a set of continuous functions to form a positive basis, either in the L1 or sup norm, for the space of uniformly positive lower-semicontinuous functions on a general metric measure space.