CONWAY NUMBERS AND ITERATION THEORY

Conway (''On Numbers and Games,'' Academic Press, New York, 1976) has given an inductive procedure for generating the real numbers that exte nds in a natural way to a new class of numbers called the surreals. Th e number 0 is defined at the first step in terms of a pair of empty se ts. At step 1, the number 1 and its negative are generated, giving the set {1, 0, -1}; at step 2, the numbers 2, 1/2 and their negatives are generated, giving the set {2, 1, 1/2, 0, - 1/2, - 1, - 2}; at step 3, the numbers 3,3/2,3/4,1/4 and their negatives are generated, giving t he set {3,2,3/2,1,3/4,1/2,1/4,0, - 1/4, - 1/2, -3/4, -1, -3/2, -2, -3} , etc. It is shown that these numbers are generated in one-to-one corr espondence with certain sequences of positive and negative sequences o f integers: At step 0, the sequence (0) is introduced; at step 1, the sequence (1) and its negative (-1) are generated, giving the set {(1), (0), (-1)}; at step 2, the sequences (2), (1, 1) and their negatives (-2),(-1, -1) are generated, giving the set {(2),(1),(1, 1),(0),(-1, - 1), (-1), (-2)}; at step 3, the sequences (3), (2, 1), (1,1,1), (1, 2) and their negatives (-3), (-2, -1), (-1, -1, -1),(-1, -2) are generat ed, giving the set [(3), (2), (2,1), (1), (1,1,1), (1,1), (1,2), (0), (-1, -2), (-1, -1), (-1, -1, -1), (-1), (-2, -1), (-2), (-3}, etc. Thi s generation of sequences in not ad hoc. The positive and negative seq uences given here, and their generalizations, arise in iteration theor y and in the theory of words associated with that theory. There is a n atural order relation on these sequences that is one-to-one with the C onway numbers and which is rooted in the ordering of the inverse funct ions that arise in the description of the graph inverse to the nth ite rate of certain classes of maps of an interval. A simple transformatio n of the points occurring in the cycles of the nth iterate of the trap ezodal map of the interval [0,2] for n = 1,2,... give all the dyadic C onway numbers, in one-to-one correspondence with sequences, and the in clusion of infinite iterates yields all the reals. The implication of this result is that all such Conway numbers arise as fixed points of t he trapezodal map, one-to-one with the sequences that label uniquely t hese fixed points. However, because of certain ''universality'' proper ties, the sequences themselves have application to and significance fo r other maps of the interval. The various aspects of iteration theory and its relation to the subset of real Conway numbers are discussed in some detail. The question of whether the surreal Conway numbers have application to iteration theory, hence, possibly to chaos, is left ope n. (C) 1997 Academic Press.