Transfinite Function Iteration and Surreal Numbers

AbstractLouck has developed a relation between surreal numbers up to the first transfinite ordinal ω and aspects of iterated trapezoid maps. In this paper, we present a simple connection between transfinite iterates of the inverse of the tent map and the class of all the surreal numbers. This connection extends Louck's work to all surreal numbers. In particular, one can define the arithmetic operations of addition, multiplication, division, square roots, etc., of transfinite iterates by conversion of them to surreal numbers. The extension is done by transfinite induction. Inverses of other unimodal onto maps of a real interval could be considered and then the possibility exists of obtaining different structures for surreal numbers