Uniform Cantor sets as hyperbolic boundaries

Abstract

 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.