Hyperbolization Of Hyperspaces And Uniform Cantor Sets

A hyperbolization is a process for converting a geometry object such as metric spaces, fractal sets into a Gromov hyperbolic space. This thesis focuses on the hyperbolization of hyperspaces and a class of fractal sets.Let（X, d） be a metric space, H（X） and F（X） denote the hyperspace of the nondegenerate closed bounded subsets, and the nondegenerate closed subsets of X,respectively. In chapter 3, the hyperbolization of the two hyperspaces（H（X）, d_H）and（F（X）, d_p） is studied. Here d_Hdenotes the Hausdorff metric and d_pdenotes the Busemann-Hausdorff metric. Firstly, we prove that the metric space（H（X）, d_H）is an asymptotically PT₁space if the hyperspace（H（X）, d_H） is a Ptolemy space.Secondly, we construct a family of new metrics d_H,ε, ε ∈(0, 1] and prove that d_H,εcan hyperbolize the hyperspace（H（X）, d_H）. and we obtain that the metric d_H,εis an asymptotically PT₁metric if ε ∈(0, 1/2]. Furthermore, based on BusemannHausdorff metric d_p, we define a new metric dPon F（X） and prove that the metric space（F（X）, dP） is an asymptotically PT₁space if the metric space（F（X）, d_p）is a Ptolemy space. Similarly, a family of new metrics dP,ε, ε ∈(0, 1] is constructed and we prove that the hyperspace（F（X）, d_p） can be hyperbolized by dP,εand we obtain that the metric dP,εis an asymptotically PT₁metric if ε ∈(0, 1/2].In chapter 4, we consider the hyperbolization problem of a special Cantor set—the uniform Cantor set. We construct an asymptotically PT₁space（X, h） and show that the uniform Cantor set is isometric to the Gromov hyperbolic boundary at infinity of the asymptotically PT₁space（X, h）.