2003/10/17 by Itai Benjamini, Itaï Benjamini, Benjamini, Itai +2
Mathematics · #Advanced Operator Algebra Research #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #math.GR #math.PR
paper · pdf · doi:10.48550/arxiv.math/0310258
arxiv created 2003/10/17 · arxiv updated 2009/12/01
Let μ and ν be probability measures on a group Γand let Gμand Gνdenote Green's function with respect to μand ν. The group Γis said to admit instability of Green's function if there are symmetric, finitely supported measures μ and νand a sequence \xn\ such that Gμ(e, xn)/Gν(e,xn) → 0, and Γadmits instability of recurrence if there is a set S that is recurrent with respect to νbut transient with respect to μ. We give a number of examples of groups that have the Liouville property but have both types of instabilities. Previously known groups with these instabilities did not have the Liouville property.