Iterates of Fractional Order

This problem typifies the general one of iteration. Let g(x) be the &th; order iterate of g [i.e. g°(x) = x, g(x) = g(g(x))]. The iteration problem is that of attaching a consistent meaning to this expression for fractional k (in the sense of preserving the additive law of exponents). A n / satisfying (1) is thus g(x). By ideas similar to those discussed herein, we can find the most general g and then by iterating it, the most general iterate of any rational order. Without introducing continuity, this is as far as it is possible to go. We confine ourselves to the case of k = 1/2 to avoid oppressive detail; the generalization to k = 1/m is indicated later. The iteration problem has received attention for many years, alone or as part of another topic (functional equations, fractional derivatives, the trioperational algebra of Menger [1], etc.). Some of these applications require subsidiary conditions on the functions (continuity, differentiability, etc.). We deal with the general problem without such side conditions; thus our work might be called combinatorial. The problem with a side condition such as continuity appears highly interesting. In all the literature we have encountered, the general problem is approached in but one way—through the Abel function. The idea here is to ascertain a numerically valued function on E satisfying