A Fatou-type theorem for functions associated with conformal densities on the boundary of a metric tree (Q1356283)

The authors prove a Fatou type theorem, that is a theorem about the (radial) boundary behavior of a certain class of functions. Here the setting is on a metric tree and the class of functions are related to conformal densities. Let \(X\) be a complete and locally compact space and a metric tree, which means that between any two points in \(X\) there is a unique arc, and that this arc is isometric to an interval of \(\mathbb{R}\). The distance between two points in \(X\) is \(| x-y| \). A geodesic ray is a map \(f\) from \([0,\infty[\), or a closed interval, into \(X\) such that \(| f(t)-f(s)| =| t-s| \). The boundary \(\partial X\) of \(X\) is the space of ends. A conformal density of dimension \(d\) is a family of pairwise absolutely continuous measures \((\mu_{x})_{x\in X}\) on \(\partial X\), such that \((d\mu_{y}/d\mu_{x})(\xi)=e^{d(| x-p| -| p-y|)}\), where \(p\) is the projection of \(\xi\) on the geodesic from \(x\) to \(y\), this point \(p\) also satisfies that \([p,\xi[=[x,\xi[ \cap [y,\xi[\). For a conformal density define \(\phi_\mu (x)=\mu_{x}(\partial X)\). The Fatou type theorem is as follows. Assuming that \(\mu\) and \(\nu\) are two conformal densities of the same dimension, and that \(\nu\) is absolutely continuous with respect to \(\mu\) then for a fixed point \(x_{0}\) and for \(\mu\)-almost all \(\xi_{0}\) does \(\phi_{\nu}/\phi_{\mu}(x)\) converge to \(d\nu_{x_{0}}/d\mu_{x_{0}}(\xi_{0})\) as \(x\) tends to \(\xi_{0}\) along the geodesic ray \([x_{0},\xi_{0}[\).