J Austral Math Soc Ser A 59 pp148--172, 1995.
(Received 4 January 1993; revised 5 October 1994)
Szekeres defined a continuous analogue of the addictive ordinary continued fraction expansion, which iteratees a map T on a domain which can be identified with the unit square [0, 1]2. Associated to it are continuous analogues of the Lagrange and Markoff spectrum. Our main result is that these are identical with the usual Lagrange and Markoff spectra, respectively; thus providing a alternative characterization of them. Szekeres also described a multi-dimensional analogue of T, which iterates a map Td on a higher-dimensional domain; he proposed using it to bound d-dimensional Diophantine approximation constants. We formulate several open problems concerning the Diophantine approximation properties of the map Td.
1991 AMS Subject Classification: primary 11J06; secondary 58F03, 58F08
Last Modified: Thu Jan 9 9:04:24 2003