Hyperbolic construction of Cantor sets. (English) Zbl 1408.30026

Ibragimov, Zair; Simanyi, John

Involve 6, No. 3, 333-343 (2013).

Summary: In this paper we present a new construction of the ternary Cantor set within the context of Gromov hyperbolic geometry. Unlike the standard construction, where one proceeds by removing middle-third intervals, our construction uses the collection of the removed intervals. More precisely, we first hyperbolize (in the sense of Gromov) the collection of the removed middle-third open intervals, then we define a visual metric on its boundary at infinity and then we show that the resulting metric space is isometric to the Cantor set.

MSC:

30C65	Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
05C25	Graphs and abstract algebra (groups, rings, fields, etc.)

Keywords:

Cantor set; Gromov hyperbolic spaces