2013/12/30 by C. R. E. Raja, Raja, C. R. E.
Mathematics · #14E20 #20C20 #46E25 #54C40 #Advanced Operator Algebra Research #Advanced Topology and Set Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1312.7654
openalex publication_date 2013/12/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let G be a locally compact group and E be a G-space. An irreducible probability measure μ on G is said to have Liouville property on E if G-invariant functions on E are the only continuous bounded functions on E that satisfy the mean value property with respect to μ. We first prove that the random walk induced by μ on E is transient outside a closed set and on the closed set μ has Liouville. We mainly consider actions on vector spaces and projective spaces. We show that measures on GL(V) that are supported inside a ball of radius less than a<1 have Liouville property on V. We also prove that measures on GL(\R 2) have Liouville property on the projective line. We next exhibit subgroups of GL(V) so that irreducible measures on such subgroups have Liouville on the projective space \mP (V) of V. We also prove irreducible measures on SL(V) have Liouville property on \mP (\cSL (V)) where \cSL (V) is the Lie algebra of SL(V)