Vol. 6, No. 3, 2013

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Hyperbolic construction of Cantor sets

Zair Ibragimov and John Simanyi

Vol. 6 (2013), No. 3, 333–343
Abstract

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.

Keywords
Cantor set, Gromov hyperbolic spaces
Mathematical Subject Classification 2010
Primary: 30C65
Secondary: 05C25
Milestones
Received: 6 July 2012
Revised: 21 December 2012
Accepted: 6 January 2013
Published: 8 September 2013

Communicated by Kenneth S. Berenhaut
Authors
Zair Ibragimov
Department of Mathematics
California State University, Fullerton
McCarthy Hall 154
Fullerton, CA 92831
United States
John Simanyi
Department of Mathematics
California State University, Fullerton
McCarthy Hall 154
Fullerton, CA 92831
United States