Tangential Limits of Potentials on Homogeneous Trees

Abstract

Let T be a homogeneous tree of homogeneity q+1. Let Δ denote the boundary of T, consisting of all infinite geodesics b=[b0,b1,b2,] beginning at the root, 0. For each beΔ, τ≥1, and a≥0 we define the approach region Ωτ,a(b) to be the set of all vertices t such that, for some j, t is a descendant of bj and the geodesic distance of t to bj is at most (τ−1)j+a. If τ>1, we view these as tangential approach regions to b with degree of tangency τ. We consider potentials Gf on T for which the Riesz mass f satisfies the growth condition ∑Tfp(t)q−γ|t| 1 and 0 0inf ∑iq−τγ|ti|:E a subset of the boundary points passing through ti for some i,|ti|>log q(1/δ).