Uniform Cantor sets as hyperbolic boundaries

Xiao Yingqing


 In this paper, we prove that the Gromov hyperbolic space $(\mathcal{X}, h)$ which was introduced by Z. Ibragimov and J. Simanyi in \cite{ZIJS} is an asymptotically $\mathrm{PT}_{-1}$ space and extend the methods of \cite{ZIJS} to the case of uniform Cantor sets, show that the uniform Cantor set is isometric to the hyperbolic boundary of some asymptotically $\mathrm{PT}_{-1}$ space.

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