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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Spherical functions and conformal densities
on spherically symmetric $CAT(-1)$-spaces


Authors: Michel Coornaert and Athanase Papadopoulos
Journal: Trans. Amer. Math. Soc. 351 (1999), 2745-2762
MSC (1991): Primary 53C35
Published electronically: February 5, 1999
MathSciNet review: 1466945
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Abstract: Let $X$ be a $CAT(-1)$-space which is spherically symmetric around some point $x_{0}\in X$ and whose boundary has finite positive $s-$dimensional Hausdorff measure. Let $\mu =(\mu _{x})_{x\in X}$ be a conformal density of dimension $d>s/2$ on $\partial X$. We prove that $\mu _{x_{0}}$ 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 $\mu $ and of the $d-$dimensional spherical function on $X$. In particular, this result proves that $\mu $ 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.


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Additional Information

Michel Coornaert
Affiliation: Institut de Recherche Mathématique Avancée, Université Louis Pasteur et CNRS, 7, rue René Descartes, 67084 Strasbourg Cedex France
Email: coornaert@math.u-strasbg.fr

Athanase Papadopoulos
Affiliation: Institut de Recherche Mathématique Avancée, Université Louis Pasteur et CNRS, 7, rue René Descartes, 67084 Strasbourg Cedex France
Email: papadopoulos@math.u-strasbg.fr

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02155-8
PII: S 0002-9947(99)02155-8
Received by editor(s): January 30, 1996
Received by editor(s) in revised form: June 12, 1997
Published electronically: February 5, 1999
Additional Notes: The second author was also supported by the Max-Planck-Institut für Mathematik (Bonn)
Article copyright: © Copyright 1999 American Mathematical Society