Spherical functions and conformal densities on spherically symmetric $CAT(-1)$-spaces
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- by Michel Coornaert and Athanase Papadopoulos PDF
- Trans. Amer. Math. Soc. 351 (1999), 2745-2762 Request permission
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.References
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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
- MR Author ID: 135835
- Email: papadopoulos@math.u-strasbg.fr
- 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)
- © Copyright 1999 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 351 (1999), 2745-2762
- MSC (1991): Primary 53C35
- DOI: https://doi.org/10.1090/S0002-9947-99-02155-8
- MathSciNet review: 1466945