SPHERICAL FUNCTIONS AND CONFORMAL DENSITIES ON SPHERICALLY SYMMETRIC CAT( 1)-SPACES (original) (raw)

| Let X be a CAT(?1)? space which is spherically symmetric around some point x 0 2 X and whose boundary has nite positive s?dimensional Hausdor measure. Let = ( x ) x2X be a conformal density of dimension d > s=2 on @X. We prove that x 0 is a weak limit of measures supported on spheres centered at x 0 . These measures are expressed in terms of the total mass function of and of the d?dimensional spherical function on X. In particular, this result proves that is entirely determined by its dimension and its total mass function. The results of this paper apply in particular for symmetric spaces of rank one and semi-homogeneous trees.