A Fatou-type theorem for functions associated with conformal densities on the boundary of a metric tree

Abstract

A metric tree is a metric space such that between any two points there is a unique topological segment (i.e., a subset homeomorphic to an interval of the real line), and such that this segment is a geodesic (i.e., the isometric image of an interval). The theory of group actions on metric trees and of the structure of these trees on the boundary at infinity has many different aspects, and it has been made clear that this theory captures a lot of the interesting phenomena of hyperbolic geometry, or, to be more precise, of the theory of groups acting by isometrics on the hyperbolic space I/~ and on its sphere at infinity. One of the particular features, in the case of metric trees, is that the ideas and methods which are used are often of a combinatorial nature, and, probably, simpler and more intuitive than those of classical hyperbolic geometry. In particular, in [2] and [3], we give a series of criteria for the ergodicity of the geodesic flow associated with groups acting properly and isometrically on metric trees, which are discrete analogs of theorems due to Hopf and Sullivan in the case of hyperbolic manifolds. The invariant measures on these flows come from conformal densities on the boundary. This theory of conformal densities on the boundary was developed, in the case of hyperbolic manifolds, by Patterson and Sullivan (see [5] and [6]), and later, in the case of metric trees, in [11 . The result that we prove here concerns a Fatou-type behavior of certain natural functions associated with conformal densities on the boundary of a metric tree. A somehow more special result of this kind, in the case of conformal densities on the boundary of the hyperbolic n-space, is used by Sullivan in [6], in his proof for the probabilistic criterion of the ergodicity of the geodesic flow on a hyperbolic manifold (see also [7, Theorem 2.14]). T h e p r o o f of the corresponding result for metric trees, which we give in [31, involves a different kind of argument, although the Fatou-type behavior is also valid in that case, and, indeed, the aim of the present paper is to prove this result in the case of metric trees. Let us begin by describing the background. Let X be a metric tree which is complete and locally compact. We denote the distance between two points in X by Ix YlA geodesic ray (resp. a geodesic segment) is a map f from [0, oo[ (resp. from a closed interval) into X such that I f(r t ) f(r2)l = Irt r~l for every r, and r2. The boundary OX is the space of the ends of X. Every geodesic ray r defines a point in OX, which is denoted by r(oo). Let us also recall that between any two points x E X and r /E OX, there is a unique image of a geodesic ray, which is denoted by Ix, r/[. There is a natural topology on X U OX, which can easily be defined with the use of the ray structure from any given point in X, and for this topology, the space X U OX is compact and X lies in i t as an open dense subset. Better than that, X tO OX is, in fact, equipped with a natural class of metrics, and the induced class of metrics on OX, called visual metrics, is most useful. Let us recall its definition: For every point z E X, the visual metric I I~ on OX is defined by