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 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. ᮊ 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 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 ᮊ 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 ␣-sequences and Gonshor’s notation. It is this result that allows us to extend the ␣-sequence notation ofwx 10 to all the surreals. But much of the theory of ␣-sequences for the tent map remains to be extended to the set of surreal numbers. It is hoped that the theory of ␣-sequences can be extended to much of the class of surreal numbers. It is for this reason that we present some of the ␣-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 (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. ␣ 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 ( иии ( 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 ␣-sequences 1 and y1 label the lines y s tx1 s ⌽Ž. 11Ž.Ž⌽ . 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 ⌽Ž.0; 1 s 1. 336 BEYER AND LOUCK
Ž. Ž . Ž . Ž . FIG. 2. The inverse graph G2 . The ␣-sequences 2 ) 1, 1 ) y1, y1 ) y2 label the 2 2 lines y s ⌽Ž.2; x s 2 y Ž1r2 .Ž.Žx, y s ⌽ 1, 1; x s 1 q 1r2 .Ž.x, y s ⌽ y1 y1; x s 1 y 2 2 Ž1r2.Ž.Žx, and y s ⌽ y2; x s 1r2 .Žx. The ␣-sequences 1.) Ž0.) Žy1. label the bound- ary abscissae y s ⌽Ž.1; 0 s 3r2, y s ⌽ Ž.0, 1 s 1, and y s ⌽ Žy1; 1 .s 1r2.
An ␣-sequence is a finite sequence of positive integers:
␣ ␣ ␣ ␣ ␣ N s Ž.01, ,..., ki, g q, ig Ž.0,1,2,..., k , or an infinite sequence of positive integers:
␣ ␣ ␣ ␣ N s Ž.01, ,... , ig q, ig Ž.0,1,2,... . Corresponding negative ␣-sequences are defined as the negative of the above:
␣ ␣ ␣ ␣ ␣ N syŽ.Ž.01,y ,...,y ki, g q, ig 0,1,2,..., k , ␣␣ ␣ ␣ N syŽ.01,y ,... , ig q, ig Ž.0,1,2,... . An order relation among the ␣-sequences is given in the third paragraph of the introduction inwx 10 . Here we paraphrase this relation. Consider two ␣-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 ␣-sequence or its negative is defined by:
DŽ.␣ s D Ž.␣ s Ý␣i. i
We also define the sets of sequences:
A ns Ä4␣␣< is positive, DŽ.␣ s n ,
Bn sA01jAjиии j A n.
The inverse graph Gn will now be described. First define the following function for x g wx0, 2 and ␣ g A n:
11␣ 01y1 11␣ y1 ⌽Ž.␣;x'žRy( Ž.Ly(Ry( Ž.Ly(иии
1 1␣ky1 (Ry(Ž.Lxy / Ž.Ž..2.7
We also need the value of the function ⌽Ž.␣; x for a negative ␣ sequence:
␣ 1 0 1 1 ␣ 1y1 1 1 ␣ ky1 ⌽Ž.Ž.␣ ; x ' ž/Ly ( Ry ( Ž.Ly ( иии ( 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 ⌽Ž.␣ ; x , ␣ g A n,2.9Ž.
ys⌽Ž.␣;x,␣gAn. 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,
⌽Ž.␣ ; x s 2 y ⌽ Ž.␣ ; x , ␣ g A n. Ž.2.11 Ž. Ž . The explicit expression for the values ⌽ ␣; 1 for ␣ s ␣01, ␣ ,...,␣kwas first given, to our knowledge, inwx 11Ž using slightly different notation in Ž.11.11a on page 192 and Ž. 5.5 on page 84 . in the study of properties of the trapezoid map for arbitrary slope parameter . In this work, s 2. The relation between ⌽Ž.␣, x and ⌽ Ž.␣, 1 is given by
Ž.x 1 kq1 y ⌽Ž.␣;xs⌽ Ž.Ž.␣;1 qy1 .Ž. 2.12 2␣01q␣qиии q ␣ k 338 BEYER AND LOUCK
The following are the values of ⌽Ž.␣;1 : 1 ⌽Ž.␣0;1 s2y , ks0,Ž. 2.13 2␣0 21 ⌽Ž.␣01,␣;1 s2yq ,ks1Ž. 2.14 22␣␣001q␣ k 22 Ž.y12 ⌽Ž.␣;1 s2yq␣ ␣␣␣ yиии q ␣иии ␣ 22001q 201qqq ky1
kq1 Ž.y1 q , kG2.Ž. 2.15 2␣01q␣qиии q ␣ k The set of straight line segmentsŽ. 2.9 and Ž 2.10 . constituting the graph
Gn is ordered by the order relation of the ␣-sequences themselvesŽ see wx10.ŽŽ. ; that is, the upper half above the line segment 0 F x F 2, y s 1of the graph consists of the straight line segmentsŽ. 2.9 and satisfy
⌽Ž.␣ ; x ) ⌽ Ž␣ Ј; x .m ␣ ) ␣ Ј᭙xgŽ.0,2 . Ž. 2.16
The line segmentsŽ. 2.10 constituting the lower halfŽŽ below 0 F x F 2, y s 1. of the graph likewise satisfy
⌽Ž.␣ ; x ) ⌽Ž.␣ Ј; x m ␣ - ␣ Ј᭙xgŽ.0,2 . Ž. 2.17 These line segments join pairwise at the two vertical boundaries
B1s Ž.x s 0, 0 F y F 2,
B2sŽ.xs2, 0 F y F 2, between the two straight line segments corresponding to adjacent se- quences in An. The boundary ordinate B1 is labelled by the set of positive sequencesŽŽ including also 0..
B ␣ ␣ ny1s Ä4N 0 F DŽ.F n y 1 ,Ž. 2.18 together with the corresponding set of negative sequences. Explicitly, these boundary ordinate values are
⌽ ␣ ␣ B Ž.;1 , g ny1,Ž. 2.19 ⌽␣ ⌽ ␣ ␣ B Ž.;1 s2y Ž.;1 , g ny1,Ž. 2.20
n which are 2 y 1 in number. These ordinate values occur at the y-level in accordance with the order relation of the ␣-sequence themselves, just as in FUNCTION ITERATION AND SURREAL NUMBERS 339
Ž.2.16 and Ž. 2.17 . It is also convenient to include the horizontal lines y s 0 and y s 2 as boundaries. Between each pair of adjacent boundaries labeled say by ␣ and ␣ Ј with ␣ - ␣ Ј, there lies exactly one of the lines Ž.2.12 , namely, the one labelled by  with ␣ -  - ␣ Ј. The situation is similar for the corresponding boundaries in the lower half of the graph. We call these boundaries conjugate boundaries. The distance between n adjacent boundary lines is 1r2 . Knowledge of the general function ⌽Ž.␣;xfor a positive sequence of arbitrary degree thus yields a complete description of the inverse graph Gn, since the boundaries are obtained from such functions by evaluating them at x s 1.
3. ENUMERATION OF TENT MAP FIXED POINTS AND CYCLE CLASSES
The complete construction of the inverse graph Gn of the nth iterate of the inverse tent map now allows us to give the complete enumeration of the set of fixed points of the iterated tent map function. The set consists of the following 2 n points:
DŽ␣. lŽ.␣ 2 ⌽Ž.Ž.␣;1 qy1 xŽ.␣ ,␣A,3.1Ž. s DŽ␣. lŽ.␣ gn 2 qyŽ.1
xŽ.␣s2yx Ž.␣, ␣gAn,3.2Ž.
Ž. Ž . where l ␣ s k q 1 denotes the number of parts of ␣ s ␣01, ␣ ,...,␣k. The relation between positive dyadic Conway numbers ²:␣ and the functions ⌽Ž␣; 1 .Ž see Eqs. Ž 1.13 .Ž , 1.20 .Ž , 1.21 . inwx 10. is given by
²: ␣0 ␣s␣0q1q2Ž.⌽Ž.␣;1 y2. Ž.3.3
The ⌽Ž.␣; 1 are given in Ž.Ž.Ž. 2.13 , 2.14 , and 2.15 . The Conway numbers Ž:␣ are expressed directly in terms of the fixed pointsŽ.Ž. 3.1 , 3.2 , and the degree D Ž.␣ by:
lŽ.␣ Ž.y1 ²: ␣00␣ ␣s␣0q1y2 q21q Ž.xŽ.␣y1, Ž 3.4 . 2DŽ␣.
lŽ.␣ Ž.y1 ²: ␣00␣ ␣syŽ.␣0q1 q2 q21q Ž.xŽ.Ž.␣y1, 3.5 2DŽ␣. 340 BEYER AND LOUCK for each ␣ g An. The set of fixed points
Ä4xŽ.␣ , x Ž.␣␣gAn Ž.3.6 decomposes into cyclesŽwx 10 , Eqs.Ž.Ž.. 3.36 ᎐ 3.38 . Correspondingly, the Conway numbers ²:␣ may be decomposed into cycles. The explicit decomposition of the fixed points xŽ.␣ and x Ž.␣ into cycles is obtained by decomposing the set of sequences in A nnj A into cycles with respect to the action of the cyclic group Cn of permuta- tions in this set. This action is defined from the correspondence between ŽŽ... the elements of the set A nnj A and LR-words see Eqs. 1.3 inwx 10 :
␣ 0y1 ␣1y1 ␣ky1 ␣sŽ.␣01,␣,...,␣kªRL RL иии RL ,3.7Ž.
␣01␣y1␣ 2y1 ␣ky1 ␣syŽ.␣01,y␣,...,y␣kªLRL RL иии RL .3.8Ž. The mapsŽ. 3.7 and Ž. 3.8 are maps to LR words. We select any element
g A nnj A and determine the LR-word to which it corresponds. The action of the cyclic group Cn on the ␣-sequence is then defined as the set of ␣-sequences CŽ. obtained by moving letters, one at a time, from one end of the LR-word to the other end, until the original word is reproduced, mapping at each step the word so obtained back into the ␣-sequence notation by use ofŽ. 3.7 and Ž. 3.8 . We use the term cycle class Ž. of for the subset C ; A nnj A . We then select Ј g A nnj A , ЈfCŽ., and repeat the process to construct CŽ. Ј . This procedure is continued until the set A nnj A is decomposed into the union of a finite family of disjoint cycle classes CŽ. , C Ž Ј ., C Ž Љ ., . . . . This procedure is standard in the theory of words, where CŽ. is called the class of sequences conjugate to Žseewx 9, p. 7 and p. 188. . We have used the term ‘‘cycle class of ’’ because the term conjugate has been assigned at least two different meanings in this work.
The detailed decomposition of A nnj A into cycle classes with respect to the cyclic group Cnnmay be given. An ␣-sequence g A j A nis called primiti¨e if it cannot be written as the proper concatenation power of another ␣-sequence.Ž For the definition of concatenation of two se- qy quences, seewx 10 .. We define the set of primitive ␣-sequences Pddand P by
q Pdds Ä4Ž.1,␥␥gL,1, Ž.␥ primitive , Ž.3.9
y PddsÄ4Ž.1,␥␥gL,1, Ž.␥ primitive ,Ž. 3.10 where L d denotes the set of lexical sequences of degree d y 1. Lexical sequences are defined in Louck and Metropoliswx 11, p. 93 . The major result for the labeling of cycle classes is the following theorem whose proof will appear inwx 2 . FUNCTION ITERATION AND SURREAL NUMBERS 341
THEOREM 1. The cycle classes of A nnj A with respect to the cyclic group q y Cnn may be labelled by the sequences in the set R s Rnj Rn, where the definition of the two sets in this union is gi¨en, respecti¨ely, by
qqnrd Rnds½5Ž.1,␥ dn<;1, Ž.␥ gP , Ž.3.11
yynrd RndsÄ4Ž.1,␥ dn<;1, Ž.␥gP .Ž. 3.12
Thus, we ha¨e
A A CŽ. . Ž.3.13 nnj s D gRn
The significance of Theorem 1 for the decomposition of the set of
␣-sequences in A nnj A into cycle classes is its one-to-one correspondence with the decomposition of the set of fixed points of the nth iterate of the inverse tent function, i.e., the intersection of the graph Gn with the line y s x, into cycles of the nth iterate of the inverse tent map, as described in the second major theorem:
THEOREM 2. The set of cycle classes of A nnj A with respect to the cyclic group Cn is in one-to-one correspondence with the set of cycles of the nth iterate of the in¨erse tent map. The points in a gi¨en cycle are gi¨en by the set
XŽ. s Ä4x Ž Ј . Ј g C Ž. , each g RnŽ.3.14
Ž.recall that Ј is an ␣-sequence and the decomposition of the set of fixed points I nnof G into these cycles is gi¨en by
I XŽ.. Ž.3.15 nsD gRn
Remark. The complete proof of this result is given inwx 10 , as a special case of a more general theorem. The application of Theorem 1 to the set
A 33j A s ÄŽ.Ž3 , 2,1 .Ž , 1,1,1 .Ž , 1,2 .Ž , y1,y2, . Ž.Ž.Ž.Ž.y1,y1,y1,y2,y1,y34 3.16 gives
A 33j A s CŽ.Ž.Ž1,1,1 j C 1,2 j C y1,y2 .Ž.Ž.j C y3 , 3.17 342 BEYER AND LOUCK where
CŽ1,1,1 .s Ä4 Ž1,1,1 . , C Ž.1,2 s Ä Ž.Ž2,1 , y1,y1,y1 .Ž4 3.18 . CŽ.Ž.Ž.Ž.Žy1,y2 syÄ41,y2,y2,y1,3 , Cy1,y1,y1 .syÄ43. Ž.3.19
The corresponding result, Theorem 2, giving the decomposition of the set
I 3 of fixed points of the third iterate of the tent map into cycles, is
I 3s XŽ.Ž.Ž1,1,1 j X 1,2 j X y1,y2 .Ž.Ž.j X y3 , 3.20 where
XŽ.Ž.1,1,1 s Ä4x 1,1,1 s 11r8 , Ž.3.21 XŽ.1,2 sÄ4x Ž.1,2 s9r8, x Ž.2,1 xs13r8, x Žy1,y1,y1 .s5r8, Ž.3.22
XŽ.Ž.Ž.y1,y2 sÄ4x y1,y2 s7r8, x y2,y1 s3r8, x Ž.3 s15r8, Ž.3.23
XŽ.y3sÄ4x Ž.y3s1r8 .Ž. 3.24
Thus, the set of fixed points of G3 as shown in Fig. 3 decomposes into two 1-cycles and two 3-cycles. Let us note also the following cardinality relations applying to the decompositionsŽ. 3.13 and Ž. 3.15 :
n qy The Mobius¨ inversion formula applies to this relation and yields the following expression for the number of cycles of degree d, which is also n equal to the number Nd of d-cycles contained in the set of fixed points I n, 1 n qy drr Ndds<<< FIG. 3. The inverse graph G3. The ␣-sequences Ž.Ž.Ž3 ) 2,1 ) 1,1,1 .Ž.Ž) 1,2 ) y1,y2 .Ž) y1,y1,y1 .Ž) y2,y1 .Ž.) y3 label the graphs of the lines: 333 ys⌽Ž.3; x s 2 y Ž.1r2 x, y s ⌽Ž.2,1; x sq2 Ž.1r2 x 3 33 ys⌽Ž.1,1,1, x sy2 Ž.1r2 x, ys⌽Ž.1,2; x s 1 q Ž.1r2 x 331 ys⌽Ž.y1,y2; x s 1 y Ž.1r2 x, y s ⌽Ž.y1,y1,y1; x sq2 Ž.1r2 x 1 33 ys⌽Ž.y2,y1; x sy2 Ž.1r2 x, ys⌽Ž.y3; x s Ž.1r2 x. The ␣-sequencesŽ. 2 ) Ž.1 ) Ž1, 1 .) Ž.0 ) Žy1, y1 .) Žy2 . label the boundary abscissae Ž.73 Ž. Ž . 5 Ž . 3 as: y s ⌽ 2; 1 s 42, y s ⌽ 1; 1 s , y s ⌽ 1, 1; 1 s 4, y s ⌽ y1, y1; 1 s 4, y s Ž.1 ⌽y2; 1 s 4 . 4. THE TOPOLOGICALLY EQUIVALENT QUADRATIC MAP It is quite remarkable that because the tent map is topological equiva- lent to the parabolic map 2 xŽ.1 y x , one also obtains from the above results the complete construction of the inverse map of the nth iterate of the inverse quadratic map, giving all of its fixed points and the decomposi- tion of this set of fixed points into cycles. This equivalence is effected by the arcsin function given inŽ. 2.6 above. We also need its inverse: 12 hxy Ž.s2 sin x ,0FxF2.Ž. 4.1 ž/4 344 BEYER AND LOUCK For a succinct description of this topological equivalence, it is conve- nient to introduce the following table of notations, where the range of x is wx0, 2 . The function ⌿ is defined analogously to ⌽ inŽ. 2.7 and Ž. 2.8 . For the quadratic map, q1 and qy11replaces t and ty1of the tent map. Obviously, q stands for quadratic and t stands for tent. Quadratic Map Tent Map Type qxŽ.s2x Ž1yxtx . Ž.s2x,0FxF1 Direct Map txŽ.s21 Žyx .,1FxF2 xx Ž. Ž. qx11s1q 1y txs2y Inverse Map '22 xx Ž. Ž. qx1 s1y 1y tx1 s Inverse Map y ' 22y q Ž. Ä Ž. Ž.4t Ž. ÄŽ. Ž.4 y11xsqx,qxy1 y11xstx,txy1 Paired Maps ⌿Ž.␣;x ⌽Ž.␣;x Iteration function ⌿Ž.␣;1 ⌽ Ž.␣; 1 Positive Conway numbers ⌿Ž.␣;1 ⌽Ž.␣; 1 Negative Conway numbers Each quantity in the left-hand column, denoted generically by zxqŽ.,is related to the corresponding quantity zxtŽ.in the second column by y1 zxqtŽ.sŽ.h(z(hxŽ..4.2Ž. In particular, this relation is valid at x s 1 and gives 2 ⌿Ž. ;1 s2 sin ⌽ Ž. ; 1 , each g A nnj A ,4.3Ž. ž/4 which gives the relation between the Conway numbers ⌽Ž. ; 1 defined by the inverse tent map and the corresponding numbers ⌿Ž. ; 1 defined in terms of the inverse quadratic map. Finally, the fixed points x qŽ. of the nth iterate of qx Ž.are related to the fixed points xtŽ. of nth iterate of txŽ.by 2 x qtnŽ. s 2 sin x Ž. , g A j A n.4.4Ž. ž/4 Ž. There is, of course, a decomposition of x qnn , g A j A , into cycles, exactly as inŽ. 3.14 . 5. SURREAL DYADIC NUMBERS Gonshor, in his bookwx 4 , defines a surreal number as follows: ‘‘A surreal number is a function from an initial segment of the ordinals into the set Ä4q,y, i.e., informally, an ordinal sequence consisting of pluses and FUNCTION ITERATION AND SURREAL NUMBERS 345 minuses which terminate. The empty sequence is included as a possibility.’’ So we begin with Ä< 4 which is defined as 0 in the ␣-sequence notation. To connect Gonshor’s representation of surreal numbers with the pre- sent work, we need the function ²:␣ of ␣ introduced in Louckwx 10 . This function is derived form ⌽Ž.␣; 1 , given above, and is defined by ²: ␣ 0 ␣s␣0q1q2 ⌽Ž.␣;1 y2, Ž.5.1 so that the values of ²:␣ are: ²: ␣00s␣, ks0,Ž. 5.2 1 ²: ␣01,␣s␣ 0y1q,ks1,Ž. 5.3 2␣1 k 22 Ž.y12 ²: ␣s␣0 y1qy␣␣␣␣ qиии q ␣иии ␣ 22112q 201qqq ky1 kq1 Ž.y1 q , kG2.Ž. 5.4 2␣01q␣qиии q ␣ k The transformationŽ. 5.1 was found by Louckwx 10 as a way of transforming a function giving surreal numbers onŽ. 0, 2 to a function that gives surreal numbers onŽ. 0, ϱ . For ordinals less than , we give the relation between ␣-sequences of length - and surreal numbers defined by a sequence of pluses and minuses of length - . Let  be a positive number of finite length in Gonshor’s notation. Then  represents a positive dyadic number and the initial sign of  is plus. We can write for k G 0, k  sqŽ.Ž.Ž.␣012 y␣␣ q ... Žy . ␣k,5Ž..5 k where y denotes q if k is even and y otherwise and where the ␣i are positive integers that count the number of signs in the ith maximal sequence of like signs. This is often called a finite signed binary expansion. The dyadic number ²: corresponding to the Gonshor sequence  defined byŽ. 5.5 is given by: ²: s␣0, ks05.6Ž. ²: ␣ S, k 15.7Ž. s0y␣1 s 11 ²: s␣0 yS␣␣q Sy S ␣qиии 1222␣␣112q␣ 3 k Ž.y1 q␣ иии ␣ S␣,kG2,Ž. 5.8 2 1 qq k y 1 k 346 BEYER AND LOUCK where 11 1 S␣sq qиии q .5.9Ž. i 2222 ␣i Here we have used Gonshor’s informal prescription on page 29 ofwx 4 that plus is counted as a 1 until a sign change occurs, at which point the sequence of pluses and minuses is treated as a binary decimal except that q is treated as 1 and y is treated as y1. If  is negative, we put  syЈ and expand ²:Ј into a finite sign expansion in powers of 1r2 and prefix it with a negative sign. This has the effect of changing all signs. We use the identity 1 S␣s 1 y Ž.5.10 i 2␣i to obtain fromŽ. 5.8 the following forms for the dyadic number ²: corresponding to the Gonshor sequence . ²: s␣0, ks0,Ž. 5.11 1 ²: s␣0y1q , ks1,Ž. 5.12 2␣1 22 ²: s␣0y1qy 22␣␣112q␣ kq1 Ž.y1 qиии q , k G 2.Ž. 5.13 2␣1qиииq␣k Thus, we have the identity ²:s ²:␣. Ž.5.14 Further, every positive dyadic number has the formŽ.Ž.Ž. 5.11 , 5.12 , or 5.13 depending on k. A zero or negative value for a dyadic number can be treated similarly to give y²:sy ²:␣ ,0Ž.F. Ž.5.15 The one-to-one correspondence ²: s ²:Ž␣positive . and y ²: s y²:Ž␣positive . established above is between real dyadic numbers. In Gonshor’s notation, it is the sequence  itself that is called a surreal number. Correspondingly, we now take the sequence ␣ itself to denote a surreal number. The relation ␣)␣Јand ␣ - ␣ Ј m ²:␣ ) ²␣ Ј : Ž.5.16 FUNCTION ITERATION AND SURREAL NUMBERS 347 assures the consistency of using the sequence themselves. We have estab- lished then the one-to-one relation between finite ␣-sequences and finite Gonshor sequences given by ␣ ␣ ␣ ␣ k s Ž01, ,..., k.m Ž.q ␣01, Ž.y ␣␣,..., Žy .k, Ž 5.17 . and, similarly, ␣ l y. Ž.5.18 6. NONDYADIC REAL NUMBERS Let us first consider positive nondyadic real numbers. GonshorŽ p. 33 of wx4. defines a real number as a surreal number which is either of finite lengthŽ. a dyadic number or of length that does not eventually have constant signs. He then proves in Theorem 4.3, p. 34 ofwx 4 that these real numbers are essentially the same as those defined in more traditional ways. From this we can make a correspondence between positive real numbers expressed in the Gonshor form for surreals of length and positive real numbers expressed as an ␣-sequence arising from an -length iteration of the inverse tent function. Suppose  is a non-dyadic positive real number. Then the signed expansion of  is non-terminating and has no tail of constant sign for otherwise  would be dyadic. In this case,Ž.Ž. 5.5 , 5.8 , and Ž.5.13 can be extended to an infinite series in ␣i and thus we obtain a corresponding ␣-expansion. Again, if  is 0 or negative, corresponding remarks as in §5 can be made. 7. SURREAL NUMBERS OF LENGTH THAT ARE NOT REAL We now consider the case of surreal numbers whose length in the Gonshor notation is that are not real numbers. The correspondence between ␣-sequences and Gonshor sequences is given by ky1 k k Ž.␣ , ␣ ,...,␣ , Ž.␣Ž.␣␣...Ž.Ž.Ž...., 01 ky1 l qy01y ky1y y Ž.7.1 ky1 where ␣i,0FiFky1, are positive integers and y denotes q if k is odd and y if k is even. The function ⌽Ž.␣; 1 associated with the ␣-sequence inŽ. 7.1 is defined formally as the following infinite sequence of 348 BEYER AND LOUCK iterations of the inverse tent map evaluated at x s 1 for k G 1: ²:␣␣␣ ⌽ ␣ ␣ ␣ 01,,..., ky101, s Ž., ,..., ky1, ;1 11␣01y111␣y1 sRy(Ž.Ly(Ry( Ž.Ly ␣ 1 иии y1 ky1y y1 y1 y1 (( R (Ž.Ly1 ( R ( L ( L ...Ž. 1 slim ⌽Ž.␣01, ␣ ,...,␣k;1 ␣kªϱ ⌽␣,␣,...,␣ ; x .7.2 sŽ.01 ky1 xs2 Ž. We remark again that it is the sequencesŽ. 7.1 themselves that are the surreal numbers and not the values given by the real numbers Ž.7.2 . 8. THE REMAINING SURREAL NUMBERS The invertible mapping from surreal numbers a of length F to members of the ␣-sequences was defined in Sections 5᎐7. This function is now generalized to transfinite ␣-sequences associated with transfinite surreals. Let a be a surreal number in the sense of Conwaywx 3 and ragŽ. its Gonshor representation as given in the first paragraph of Section 5. Define ra␣Ž.as the ␣-sequence representation of a. Let wr Žg Ž.. a be the map from the Gonshor representation of a to the ␣-sequence representa- tion of a, that is, wrŽ.gŽ. a sra␣ Ž..8.1Ž. We first need a lemma about the class of all ordinals. This lemma states that the class of all ordinal numbers can be decomposed into the union of countably infinite sets that are each of order type . To put it in other terms, every ordinal is of the form of a limit ordinal plus an integer. This fact is well known. See for examplewx 5 . But we prefer to state it explicitly as a lemma. LEMMA 1. Let On be the class of all ordinals. Then On s E⑀ Ž.8.2 D⑀ where the union is o¨er all limit ordinal numbers ⑀ and ϱ E Ž.⑀ n .8.3Ž. ⑀s Dq ns0 FUNCTION ITERATION AND SURREAL NUMBERS 349 Proof. It is clear that E⑀; On.Ž. 8.4 D⑀ Let a be a nonlimit ordinal number. Then there must be a maximal limit ordinal ⑀ - a so that a y ⑀ is finite. This is so because if there is not such an ⑀, there would be an infinite sequence of direct predecessors to a. But then the ordinal a would have a subset of type * and thus a cannot be well-ordered. See Theorem 1 on page 262 of Sierpinskiwx 13 . So for some ⑀, a g E⑀ . Hence On ; E⑀.8.5Ž. D⑀ Hence, On s E⑀.8Ž..6 D⑀ This completes the proof. We now extend the 1-to-1 mapping between surreal numbers of length F and ␣-sequences of length F to a 1-to-1 mapping between surreal numbers of length ) and ␣-sequences of length ) . Every surreal number a is a map from an initial segment of the ordinals onto the set Ä4q,y. Every position in this initial segment is either a limit ordinal or is a non-limit ordinal. Every limit ordinal is the initial position of a sequence E⑀ of ordinals and E⑀ is either of finite length or length . Conversely, every nonlimit ordinal is a member of such an E⑀ . For a sequence E⑀ of length F , we construct as was done in Sections 5᎐7an ␣-sequence corresponding to the Gonshor values of the surreal number on E⑀.We denote this ␣-sequence by E⑀ E⑀ E⑀ ␣012, ␣ , ␣ ,... . Ž.8.7 This sequence may be of finite length or of length . If it is of finite length, we append blanks. This construction is carried out for every limit ordinal appearing in a. We obtain from this construction an ␣-se- quence of possibly transfinite length: ␣ ␣ ␣ ␣ ␣ ␣ ␣ ␣ ␣ ␣ 012, , ,... , q1, q2,... 2, 2q1, 2q2,... 3,... .Ž. 8.8 The function w is invertible. So the mapping w between surreals and ␣-sequences is 1᎐1. This completes the transfinite induction definition and completes the statement of correspondence between surreals and the now generalized ␣-sequences. These generalized ␣-sequences can be thought of as ␣-sequences of length concatenated an ordinal number of times and possibly terminating with a finite ␣-sequence. 350 BEYER AND LOUCK REFERENCES 1. R. L. Bivins, J. D. Louck, N. Metropolis, and M. L. Stein, Classification of all cycles of the parabolic map, Physica D 51 Ž.1991 , 3᎐27. 2. R. L. Bivins, J. D. Louck, N. Metropolis, and M. L. 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