TRANSFINITE FUNCTION ITERATION AND SURREAL NUMBERS

Louck has developed a relation between surreal numbers up to the first transfinite ordinal omega and aspects of iterated trapezoid maps. In this paper, we present a simple connection between transfinite iterate s of the inverse of the tent map and the class of all the surreal numb ers. This connection extends Louck's work to all surreal numbers. In p articular, one can define the arithmetic operations of addition, multi plication, division, square roots, etc., of transfinite iterates by co nversion of them to surreal numbers. The extension is done by transfin ite induction. Inverses of other unimodal onto maps of a real interval could be considered and then the possibility exists of obtaining diff erent structures for surreal numbers. (C) 1997 Academic Press.