Abstract
For Damek-Ricci spaces $(X,g)$ we compute the exact form of the Busemann function which is needed to represent the Poisson kernel of $(X,g)$ in exponential form in terms of the Busemann function and the volume entropy. From this fact, we show that the Poisson kernel map $\varphi: (X,g) \rightarrow (\mathcal{P}(\partial X),G)$ is a homothetic embedding. Here $\mathcal{P}(\partial X)$ is the space of probability measures having positive density function on the ideal boundary $\partial X$ of $X$, and $G$ is the Fisher information metric on $\mathcal{P}(\partial X)$.
Citation
Mitsuhiro ITOH. Hiroyasu SATOH. "Information Geometry of Poisson Kernels on Damek-Ricci Spaces." Tokyo J. Math. 33 (1) 129 - 144, June 2010. https://doi.org/10.3836/tjm/1279719582
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