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Transfinite Function Interaction and Surreal Numbers

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Transfinite Function Interaction and Surreal Numbers ADVANCES IN APPLIED MATHEMATICS 18, 333]350Ž. 1997 ARTICLE NO. AM960513 Transfinite Function Iteration and Surreal Numbers W. A. Beyer and J. D. Louck Theoretical Di¨ision, Los Alamos National Laboratory, Mail Stop B284, Los Alamos, New Mexico 87545 Received August 25, 1996 Louck has developed a relation between surreal numbers up to the first transfi- nite ordinal v 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 addi- tion, 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. Q 1997 Academic Press 1. INTRODUCTION In this paper, we assume the reader is familiar with the interesting topic of surreal numbers, invented by Conway and first presented in book form in Conwaywx 3 and Knuth wx 6 . In this paper we follow the development given in Gonshor's bookwx 4 which is quite different from that of Conway and Knuth. Inwx 10 , a relation was shown between surreal numbers up to the first transfinite ordinal v and aspects of iterated maps of the intervalwx 0, 2 . In this paper, we specialize the results inwx 10 to the graph of the nth iterate of the inverse tent map and extend the results ofwx 10 to all the surreal numbers. The extension is made by transfinite induction. One can define the arithmetic operations of addition, multiplication, division, square roots, etc., of transfinite iterates by conversion of them to surreal numbers. Gonshor's theory of surreals uses sequences of ordinal length of pluses and minuses rather than left and right sets as did Conwaywx 3 . The Gonshor method was expounded by Kruskal in two popular articles, one with Matthewswx 7 and a second one with Shulman wx 8 . In those two articles 333 0196-8858r97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved. 334 BEYER AND LOUCK up arrows ­ and down arrows x are used in place of Gonshor's pluses and minuses. In this work, we give in Section 4 the explicit map between a-sequences and Gonshor's notation. It is this result that allows us to extend the a-sequence notation ofwx 10 to all the surreals. But much of the theory of a-sequences for the tent map remains to be extended to the set of surreal numbers. It is hoped that the theory of a-sequences can be extended to much of the class of surreal numbers. It is for this reason that we present some of the a-sequence theory here. 2. THE TENT MAP The well-known tent map originated with Ulam and first appeared on page 497 of Rechardwx 12 with attribution to Ulam.Ž For this reason, the tent map could well be called the Ulam map.. We follow the notation of wx10 . The direct tent map has the form onw 0, 2 x : L: x ª 2 x ' txŽ.,0FxF1, Ž.2.1 R:xª22Ž.Ž.yx 'tx,1FxF2.Ž. 2.2 The corresponding two inverse maps onwx 0, 2 have the form: x y1 L : x ' tx1Ž.,0FxF22.3Ž. ª 2 x y1 R :x 2y 'tx1Ž.,0FxF2.Ž. 2.4 ª 2 y The tent map is the simplest map of the intervalwx 0, 2 realizing certain properties, but not all, of the more general theory given inwx 10 . The tent map or tent function is well-known, there being at least 67 mentions of it in Mathematical Re¨iews. A complete description of the graph of the nth iterate and its inverse may be given, including all of its fixed points and the decomposition of this set of fixed points into cycles. This description is obtained from results already proved inwx 10 and wx 11 , and these proofs will not be repeated here. Rather, we give a synthesis of that theory in the context of the inverse tent map. Moreover, because of the Ulam topologi- cal equivalence between the direct tent map txŽ.and the quadratic map qxŽ.s2x Ž2yx .onwx 0, 2 , we also obtain the same detailed description of this latter map. This topological equivalence is expressed by 1 q s hy (t( h,2.5Ž. FUNCTION ITERATION AND SURREAL NUMBERS 335 where 4 x hxŽ.s arcsin , 0 F x F 2.Ž. 2.6 p (2 The homeomorphism hxŽ.onwx 0, 2 preserves fixed points and order of points on the line. What is missing from these special unimodal maps of the intervalwx 0, 2 in the general theory for the direct map is the cycle containing x s 1, which degenerates to the three pointsÄ4 1, 2, 0 . The inverse tent map does, however, give the full description of the set of dyadic Conway numbers, their extension to the reals, and finally to all the surreal numbers. For these reasons, it is useful to present this complete description of the inverse tent map and its associated objects. a B A The sets of -sequences, denoted by ny1 and n inwx 10 , have exactly the same role in the construction and labeling of the graph inverse to the graph of the nth iterate of the tent map defined byŽ. 2.1 and Ž. 2.2 . We denote the inverse graph by Gn, Ž1. Ž2. Žn. Gn: Ž.x, P ( P ( ??? ( P Ž.x ,0FxF2, where P Ži. is either Ly1 or Ry1. The graphs analogous to those given in Figs. 2, 3 inwx 10 are given in the present paper as Figs. 1, 2 Ž.n s 1, 2 and we have included here also the graph for n s 3 as Fig. 3. Ž. Ž . Ž. FIG. 1. The inverse graph G1. The a-sequences 1 and y1 label the lines y s tx1 s FŽ. 11Ž.ŽF . Ž. 1; x s 2 y 22x and y s txy1 sy1; x s x. The sequence 0 labels the central boundary abscissa y s FŽ.0; 1 s 1. 336 BEYER AND LOUCK Ž. Ž . Ž . Ž . FIG. 2. The inverse graph G2 . The a-sequences 2 ) 1, 1 ) y1, y1 ) y2 label the 2 2 lines y s FŽ.2; x s 2 y Ž1r2 .Ž.Žx, y s F 1, 1; x s 1 q 1r2 .Ž.x, y s F y1 y1; x s 1 y 2 2 Ž1r2.Ž.Žx, and y s F y2; x s 1r2 .Žx. The a-sequences 1.) Ž0.) Žy1. label the bound- ary abscissae y s FŽ.1; 0 s 3r2, y s F Ž.0, 1 s 1, and y s F Žy1; 1 .s 1r2. An a-sequence is a finite sequence of positive integers: a a a a a N s Ž.01, ,..., ki, g q, ig Ž.0,1,2,..., k , or an infinite sequence of positive integers: a a a a N s Ž.01, ,... , ig q, ig Ž.0,1,2,... Corresponding negative a-sequences are defined as the negative of the above: a a a a a N syŽ.Ž.01,y ,...,y ki, g q, ig 0,1,2,..., k , aa a a N syŽ.01,y ,... , ig q, ig Ž.0,1,2,... An order relation among the a-sequences is given in the third paragraph of the introduction inwx 10 . Here we paraphrase this relation. Consider two a-sequences. First, adjoin to the right side of each sequence an infinite sequence of zeros. Then change the sign of each odd indexed place, counting the first place as zero. Order the two resulting sequences by the order of integers in the first place where the sequences differ and then let this order be reflected back to the original sequences. An equivalent rule in terms of the Gonshor notation given in Section 5 is the following. Write the two sequences in Gonshor's notation and then order the resulting two sequences lexicographically using q) blank ) y. FUNCTION ITERATION AND SURREAL NUMBERS 337 The degree of a finite a-sequence or its negative is defined by: DŽ.a s D Ž.a s Ýai. i We also define the sets of sequences: A ns Ä4aa< is positive, DŽ.a s n , Bn sA01jAj??? j A n. The inverse graph Gn will now be described. First define the following function for x g wx0, 2 and a g A n: 11a 01y1 11a y1 FŽ.a;x'žRy( Ž.Ly(Ry( Ž.Ly(??? 1 1aky1 (Ry(Ž.Lxy / Ž.Ž..2.7 We also need the value of the function FŽ.a; x for a negative a sequence: a 1 0 1 1 a 1y1 1 1 a ky1 FŽ.Ž.a ; x ' ž/Ly ( Ry ( Ž.Ly ( ??? ( Ry (Ž.Lxy Ž.. Ž.2.8 n The inverse graph Gn then consists of a set of 2 straight line segments, each defined on the intervalwx 0, 2 , and is given by the functions onwx 0, 2 : y s FŽ.a ; x , a g A n,2.9Ž. ysFŽ.a;x,agAn. Ž.2.10 These functions each define separately 2 ny1 lines. We call the combined set a graph. The inverse graph is symmetric about the line segment Ž.0FxF2, y s 1 ; that is, FŽ.a ; x s 2 y F Ž.a ; x , a g A n.
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