Hyperbolic construction of Cantor sets. (English) Zbl 1408.30026

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.


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