Conway Numbers and Iteration Theory

Abstract

Conway (“On Numbers and Games,” Academic Press, New York, 1976) has given an inductive procedure for generating the real numbers that extends in a natural way to a new class of numbers called the surreals. The number 0 is defined at the first step in terms of a pair of empty sets. 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 the 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 correspondence with certain sequences of positive and negative sequences of 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 generated, 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. This generation of sequences in not ad hoc. The positive and negative sequences given here, and their generalizations, arise in iteration theory and in the theory of words associated with that theory. There is a natural order relation on these sequences that is one-to-one with the Conway numbers and which is rooted in the ordering of the inverse functions that arise in the description of the graph inverse to thenth iterate of certain classes of maps of an interval. A simple transformation of the points occurring in the cycles of thenth iterate of the trapezodal map of the interval [0,2] forn=1,2,… give all the dyadic Conway numbers, in one-to-one correspondence with sequences, and the inclusion of infinite iterates yields all the reals. The implication of this result is that all such Conway numbers arise as fixed points of the trapezodal map, one-to-one with the sequences that label uniquely these fixed points. However, because of certain “universality” properties, the sequences themselves have application to and significance for 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 open.