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Transfinite Function Iteration and Surreal Numbers
ADVANCES IN APPLIED MATHEMATICS ARTICLE NO.
18, 333]350 Ž1997.
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 transfinite 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 addition, 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 Conway w3x and Knuth w6x. In this paper we follow the development given in Gonshor’s book w4x which is quite different from that of Conway and Knuth. In w10x, a relation was shown between surreal numbers up to the first transfinite ordinal v and aspects of iterated maps of the interval w0, 2x. In this paper, we specialize the results in w10x to the graph of the nth iterate of the inverse tent map and extend the results of w10x 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 Conway w3x. The Gonshor method was expounded by Kruskal in two popular articles, one with Matthews w7x and a second one with Shulman w8x. In those two articles 333 0196-8858r97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
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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 of w10x 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 Rechard w12x with attribution to Ulam. ŽFor this reason, the tent map could well be called the Ulam map.. We follow the notation of w10x. The direct tent map has the form on w0, 2x: L: x ª 2 x ' t Ž x . ,
0 F x F 1,
Ž 2.1.
R: x ª 2 Ž 2 y x . ' t Ž x . ,
1 F x F 2.
Ž 2.2.
The corresponding two inverse maps on w0, 2x have the form: Ly1 : x ª
x 2
' t1Ž x . ,
Ry1 : x ª 2 y
x 2
' ty1 Ž x . ,
0FxF2
Ž 2.3.
0 F x F 2.
Ž 2.4.
The tent map is the simplest map of the interval w0, 2x realizing certain properties, but not all, of the more general theory given in w10x. 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 in w10x and w11x, 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 topological equivalence between the direct tent map t Ž x . and the quadratic map q Ž x . s 2 x Ž2 y x . on w0, 2x, we also obtain the same detailed description of this latter map. This topological equivalence is expressed by q s hy1 ( t ( h,
Ž 2.5.
FUNCTION ITERATION AND SURREAL NUMBERS
335
where hŽ x . s
4
p
arcsin
(
x 2
,
0 F x F 2.
Ž 2.6.
The homeomorphism hŽ x . on w0, 2x preserves fixed points and order of points on the line. What is missing from these special unimodal maps of the interval w0, 2x in the general theory for the direct map is the cycle containing x s 1, which degenerates to the three points 1, 2, 04 . 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. The sets of a-sequences, denoted by B ny 1 and A n in w10x, 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 by Ž2.1. and Ž2.2.. We denote the inverse graph by Gn , Gn : Ž x, P Ž1. ( P Ž2. ( ??? ( P Ž n. Ž x . . ,
0 F x F 2,
where P Ž i. is either Ly1 or Ry1 . The graphs analogous to those given in Figs. 2, 3 in w10x 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.
FIG. 1. The inverse graph G1 . The a-sequences Ž1. and Žy1. label the lines y s t1Ž x . s F Ž1; x . s 2 y 12 x and y s ty1 Ž x . s F Žy1; x . s 12 x. The sequence Ž0. labels the central boundary abscissa y s F Ž0; 1. s 1.
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FIG. 2. The inverse graph G 2 . The a-sequences Ž2. ) Ž1, 1. ) Žy1, y1. ) Žy2. label the lines y s F Ž2; x . s 2 y Ž1r2 2 . x, y s F Ž1, 1; x . s 1 q Ž1r2 2 . x, y s F Žy1 y1; x . s 1 y Ž1r2 2 . x, and y s F Žy2; x . s Ž1r2 2 . x. The a-sequences Ž1. ) Ž0. ) Žy1. label the boundary abscissae y s F Ž1; 0. s 3r2, y s F Ž0, 1. s 1, and y s F Žy1; 1. s 1r2.
An a-sequence is a finite sequence of positive integers:
a s Ž a0 , a1 , . . . , ak . ,
a i g Nq , i g Ž 0, 1, 2, . . . , k . ,
or an infinite sequence of positive integers:
a s Ž a0 , a1 , . . . . ,
a i g Nq , i g Ž 0, 1, 2, . . . . .
Corresponding negative a-sequences are defined as the negative of the above:
a s Ž ya 0 , ya 1 , . . . , ya k . , a s Ž ya 0 , ya 1 , . . . . ,
a i g Nq , i g Ž 0, 1, 2, . . . , k . , a i g Nq , i g Ž 0, 1, 2, . . . . .
An order relation among the a-sequences is given in the third paragraph of the introduction in w10x. 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.
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FUNCTION ITERATION AND SURREAL NUMBERS
The degree of a finite a-sequence or its negative is defined by: DŽ a . s DŽ a . s
Ý ai . i
We also define the sets of sequences: A n s a < a is positive, D Ž a . s n4 , B n s A 0 j A 1 j ??? j A n . The inverse graph Gn will now be described. First define the following function for x g w0, 2x and a g A n : F Ž a ; x . ' Ry1 ( Ž Ly1 .
ž
a 0 y1
( Ry1 ( Ž Ly1 .
a 1 y1
( ???
( Ry1 ( Ž Ly1 .
a k y1
/ Ž x. .
Ž 2.7.
We also need the value of the function F Ž a ; x . for a negative a sequence: F Ž a ; x . ' Ž Ly1 .
ž
a0
( Ry1 ( Ž Ly1 .
a 1 y1
( ??? ( Ry1 ( Ž Ly1 .
a k y1
/ Ž x. . Ž 2.8.
The inverse graph Gn then consists of a set of 2 n straight line segments, each defined on the interval w0, 2x, and is given by the functions on w0, 2x: y s FŽ a ; x. ,
a g An,
Ž 2.9.
y s FŽ a ; x. ,
a g An.
Ž 2.10.
These functions each define separately 2 ny 1 lines. We call the combined set a graph. The inverse graph is symmetric about the line segment Ž0 F x F 2, y s 1.; that is, FŽ a ; x. s 2 y FŽ a ; x. ,
a g An.
Ž 2.11.
The explicit expression for the values F Ž a ; 1. for a s Ž a 0 , a 1 , . . . , a k . was first given, to our knowledge, in w11x Žusing slightly different notation in Ž11.11a. on page 192 and Ž5.5. on page 84. in the study of properties of the trapezoid map for arbitrary slope parameter z . In this work, z s 2. The relation between F Ž a , x . and F Ž a , 1. is given by F Ž a ; x . s F Ž a ; 1 . q Ž y1 .
Ž x y 1.
kq 1
2
a 0 q a 1 q ??? q a k
.
Ž 2.12.
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BEYER AND LOUCK
The following are the values of F Ž a ; 1.: F Ž a 0 ; 1. s 2 y F Ž a 0 , a 1 ; 1. s 2 y F Ž a ; 1. s 2 y
2 2 a0
q
2 2 a 0q a 1
1
2 2
a0
q
y ??? q q
k s 0,
,
2a0
1 2
,
a 0q a 1
Ž 2.13. ks1
Ž 2.14.
k Ž y1. 2
2 a 0q a 1q ??? q a ky 1
Ž y1.
kq 1
2 a 0q a 1q ??? q a k
,
k G 2. Ž 2.15.
The set of straight line segments Ž2.9. and Ž2.10. constituting the graph Gn is ordered by the order relation of the a-sequences themselves Žsee w10x.; that is, the upper half Žabove the line segment Ž0 F x F 2, y s 1. of the graph consists of the straight line segments Ž2.9. and satisfy F Ž a ; x . ) F Ž a 9; x . m a ) a 9
; x g Ž 0, 2 . .
Ž 2.16.
The line segments Ž2.10. constituting the lower half Žbelow Ž0 F x F 2, y s 1. of the graph likewise satisfy F Ž a ; x . ) F Ž a 9; x . m a - a 9
; x g Ž 0, 2 . .
Ž 2.17.
These line segments join pairwise at the two vertical boundaries B1 s Ž x s 0, 0 F y F 2 . , B2 s Ž x s 2, 0 F y F 2 . , between the two straight line segments corresponding to adjacent sequences in A n . The boundary ordinate B1 is labelled by the set of positive sequences Žincluding also Ž0.. B ny 1 s a N 0 F D Ž a . F n y 1 4 ,
Ž 2.18.
together with the corresponding set of negative sequences. Explicitly, these boundary ordinate values are F Ž a ; 1. ,
a g B ny 1 ,
F Ž a ; 1. s 2 y F Ž a ; 1. ,
a g B ny 1 ,
Ž 2.19. Ž 2.20.
which are 2 n y 1 in number. These ordinate values occur at the y-level in accordance with the order relation of the a-sequence themselves, just as in
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FUNCTION ITERATION AND SURREAL NUMBERS
Ž2.16. and Ž2.17.. 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 a and a 9 with a - a 9, there lies exactly one of the lines Ž2.12., namely, the one labelled by b with a - b - a 9. The situation is similar for the corresponding boundaries in the lower half of the graph. We call these boundaries conjugate boundaries. The distance between adjacent boundary lines is 1r2 n. Knowledge of the general function F Ž a ; x . for 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:
xŽ a . s
2 DŽ a . F Ž a ; 1 . q Ž y1 .
l Ža.
xŽ a . s 2 y xŽ a . ,
a g An,
,
l Ža.
2 DŽ a . q Ž y1 .
a g An,
Ž 3.1. Ž 3.2.
where l Ž a . s k q 1 denotes the number of parts of a s Ž a 0 , a 1 , . . . , a k .. The relation between positive dyadic Conway numbers ² a : and the functions F Ž a ; 1. Žsee Eqs. Ž1.13., Ž1.20., Ž1.21. in w10x. is given by ² a : s a 0 q 1 q 2 a 0 Ž F Ž a ; 1. y 2. .
Ž 3.3.
The F Ž a ; 1. are given in Ž2.13., Ž2.14., and Ž2.15.. The Conway numbers Ž a : are expressed directly in terms of the fixed points Ž3.1., Ž3.2., and the degree DŽ a . by: ² a : s a0 q 1 y 2
a0
² a : s y Ž a 0 q 1. q 2
q2
a0
a0
q2
1q
a0
Ž y1. l
1q
Ža.
Ž x Ž a . y 1. ,
2 DŽ a .
Ž y1. l
Ž 3.4.
Ža.
2 DŽ a .
Ž x Ž a . y 1 . , Ž 3.5.
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BEYER AND LOUCK
for each a g A n . The set of fixed points
xŽ a . , xŽ a .
a g A n4
Ž 3.6.
decomposes into cycles Žw10x, Eqs. Ž3.36. ] Ž3.38... Correspondingly, the Conway numbers ² a : may be decomposed into cycles. The explicit decomposition of the fixed points x Ž a . and x Ž a . into cycles is obtained by decomposing the set of sequences in A n j A n into cycles with respect to the action of the cyclic group Cn of permutations in this set. This action is defined from the correspondence between the elements of the set A n j A n and LR-words Žsee Eqs. Ž1.3.. in w10x.:
a s Ž a 0 , a 1 , . . . , a k . ª RLa 0y1 RLa 1y1 ??? RLa ky1 , a0
a 1 y1
a s Ž ya 0 , ya 1 , . . . , ya k . ª L RL
a 2 y1
RL
a k y1
??? RL
Ž 3.7. . Ž 3.8.
The maps Ž3.7. and Ž3.8. are maps to LR words. We select any element t g A n j A n and determine the LR-word to which it corresponds. The action of the cyclic group Cn on the a-sequence t is then defined as the set of a-sequences CŽt . 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 a-sequence notation by use of Ž3.7. and Ž3.8.. We use the term cycle class of t for the subset CŽt . ; A n j A n . We then select t 9 g A n j A n , t 9 f CŽt ., and repeat the process to construct CŽt 9.. This procedure is continued until the set A n j A n is decomposed into the union of a finite family of disjoint cycle classes CŽt ., CŽt 9., CŽt 0 ., . . . . This procedure is standard in the theory of words, where CŽt . is called the class of sequences conjugate to t Žsee w9, p. 7 and p. 188x.. We have used the term ‘‘cycle class of t ’’ because the term conjugate has been assigned at least two different meanings in this work. The detailed decomposition of A n j A n into cycle classes with respect to the cyclic group Cn may be given. An a-sequence t g A n j A n is called primiti¨ e if it cannot be written as the proper concatenation power of another a-sequence. ŽFor the definition of concatenation of two sequences, see w10x.. We define the set of primitive a-sequences Pdq and Pdy by Pdq s Ž 1, g . g g L d , Ž 1, g . primitive 4 ,
Ž 3.9.
Pdys Ž 1, g . g g L d , Ž 1, g . primitive 4 ,
Ž 3.10.
where L d denotes the set of lexical sequences of degree d y 1. Lexical sequences are defined in Louck and Metropolis w11, p. 93x. The major result for the labeling of cycle classes is the following theorem whose proof will appear in w2x.
FUNCTION ITERATION AND SURREAL NUMBERS
341
THEOREM 1. The cycle classes of A n j A n with respect to the cyclic group Cn may be labelled by the sequences in the set Rn s Rnq j Rny , where the definition of the two sets in this union is gi¨ en, respecti¨ ely, by Rnq s Ž 1, g .
nrd
d < n; Ž 1, g . g Pdq ,
Ž 3.11.
Rny s Ž 1, g .
nrd
d < n; Ž 1, g . g Pdy .
Ž 3.12.
½
5
4
Thus, we ha¨ e An j An s
D CŽ t . .
Ž 3.13.
tg Rn
The significance of Theorem 1 for the decomposition of the set of a-sequences in A n j A n 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 n j A n 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 X Ž t . s x Ž t 9 . t 9 g C Ž t . , each t g Rn 4
Ž 3.14.
Ž recall that t 9 is an a-sequence. and the decomposition of the set of fixed points I n of Gn into these cycles is gi¨ en by In s
D XŽ t . .
Ž 3.15.
tg Rn
Remark. The complete proof of this result is given in w10x, as a special case of a more general theorem. The application of Theorem 1 to the set A 3 j A 3 s Ž 3 . , Ž 2, 1 . , Ž 1, 1, 1 . , Ž 1, 2 . , Ž y1, y2 . ,
Ž y1, y1, y1. , Ž y2, y1. , Ž y3. 4
Ž 3.16.
gives A 3 j A 3 s C Ž 1, 1, 1 . j C Ž 1, 2 . j C Ž y1, y2 . j C Ž y3 . , Ž 3.17.
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BEYER AND LOUCK
where C Ž 1, 1, 1 . s Ž 1, 1, 1 . 4 , C Ž 1, 2 . s Ž 2, 1 . , Ž y1, y1, y1 . 4 Ž 3.18. C Ž y1, y2 . s Ž y1, y2 . , Ž y2, y1 . , Ž 3 . 4 , C Ž y1, y1, y1 . s y3 4 . 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 3 s X Ž 1, 1, 1 . j X Ž 1, 2 . j X Ž y1, y2 . j X Ž y3 . ,
Ž 3.20.
where X Ž 1, 1, 1 . s x Ž 1, 1, 1 . s 11r8 4 ,
Ž 3.21.
X Ž 1, 2 . s x Ž 1, 2 . s9r8, x Ž 2, 1 . xs13r8, x Ž y1, y1, y1 . s5r8 4 , Ž 3.22. X Ž y1, y2 . s x Ž y1, y2 . s7r8, x Ž y2, y1 . s3r8, x Ž 3. s15r8 4 , Ž 3.23. X Ž y3 . s x Ž y3 . s 1r8 4 .
Ž 3.24.
Thus, the set of fixed points of G 3 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 decompositions Ž3.13. and Ž3.15.: < A3 j A3< s 2n s
Ý
CŽ t . s
Ý d Ž
Ž 3.25.
d< n
tg Rn
The Mobius inversion formula applies to this relation and yields the ¨ following expression for the number of cycles of degree d, which is also equal to the number Ndn of d-cycles contained in the set of fixed points I n , Ndn s
1 d
Ý mŽ r .2 dr r,
Ž 3.26.
r
where m Ž r . denotes the well-known Mobius function. ¨ The results given in this section solve the problem of constructing the inverse graph of the nth iterate of the tent map, of giving its fixed points, and of decomposing this set of fixed points into cycles.
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FUNCTION ITERATION AND SURREAL NUMBERS
FIG. 3. The inverse graph G 3 . The a-sequences
Ž 3 . ) Ž 2, 1 . ) Ž 1, 1, 1 . ) Ž1, 2 . ) Žy1, y2 . ) Žy1, y1, y1 . ) Žy2, y1 . ) Žy3 . label the graphs of the lines: y s F Ž 3; x . s 2 y Ž 1r2 3 . x, y s F Ž 1, 1, 1, x . s
3 2
y s F Ž 2, 1; x . s
y Ž 1r2 3 . x,
y s F Ž y1, y2; x . s 1 y Ž 1r2 3 . x, y s F Ž y2, y1; x . s
1 2
3 2
q Ž 1r2 3 . x
y s F Ž 1, 2; x . s 1 q Ž 1r2 3 . x y s F Ž y1, y1, y1; x . s
y Ž 1r2 3 . x,
1 2
q Ž 1r2 3 . x
y s F Ž y3; x . s Ž 1r2 3 . x.
The a-sequences Ž2. ) Ž1. ) Ž1, 1. ) Ž0. ) Žy1, y1. ) Žy2. label the boundary abscissae as: y s F Ž2; 1. s 74 , y s F Ž1; 1. s 32 , y s F Ž1, 1; 1. s 54 , y s F Žy1, y1; 1. s 34 , y s F Žy2; 1. s 41 .
4. THE TOPOLOGICALLY EQUIVALENT QUADRATIC MAP It is quite remarkable that because the tent map is topological equivalent to the parabolic map 2 x Ž1 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 decomposition of this set of fixed points into cycles. This equivalence is effected by the arcsin function given in Ž2.6. above. We also need its inverse: hy1 Ž x . s 2 sin 2
p
ž / 4
x ,
0 F x F 2.
Ž 4.1.
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BEYER AND LOUCK
For a succinct description of this topological equivalence, it is convenient to introduce the following table of notations, where the range of x is w0, 2x. The function C is defined analogously to F in Ž2.7. and Ž2.8.. For the quadratic map, q1 and qy1 replaces t 1 and ty1 of the tent map. Obviously, q stands for quadratic and t stands for tent. Quadratic Map
Tent Map
q Ž x . s 2 x Ž1 y x . q1Ž x . s 1 q
' '
qy1 Ž x . s 1 y
1y
x 2
1y
x
2 qy1 Ž x . s q1Ž x ., qy1Ž x .4 CŽ a ; x . C Ž a ; 1. C Ž a ; 1.
Type
t Ž x . s 2 x, 0 F x F 1 t Ž x . s 2Ž1 y x ., 1 F x F 2 x t1Ž x . s 2 y 2 x ty1Ž x . s 2 ty1Ž x . s t1Ž x ., ty1Ž x .4 FŽ a ; x . F Ž a ; 1. F Ž a ; 1.
Direct Map Inverse Map Inverse Map Paired Maps Iteration function Positive Conway numbers Negative Conway numbers
Each quantity in the left-hand column, denoted generically by z q Ž x ., is related to the corresponding quantity z t Ž x . in the second column by z q Ž x . s Ž hy1 ( z t ( h . Ž x . .
Ž 4.2.
In particular, this relation is valid at x s 1 and gives C Ž t ; 1 . s 2 sin 2
ž
p 4
F Ž t ; 1. ,
/
each t g A n j A n ,
Ž 4.3.
which gives the relation between the Conway numbers F Žt ; 1. defined by the inverse tent map and the corresponding numbers C Žt ; 1. defined in terms of the inverse quadratic map. Finally, the fixed points x q Žt . of the nth iterate of q Ž x . are related to the fixed points x t Žt . of nth iterate of t Ž x . by x q Ž t . s 2 sin 2
ž
p 4
xt Žt . ,
/
t g An j An.
Ž 4.4.
There is, of course, a decomposition of x q Žt ., t g A n j A n , into cycles, exactly as in Ž3.14..
5. SURREAL DYADIC NUMBERS Gonshor, in his book w4x, defines a surreal number as follows: ‘‘A surreal number is a function from an initial segment of the ordinals into the set q, y4 , i.e., informally, an ordinal sequence consisting of pluses and
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FUNCTION ITERATION AND SURREAL NUMBERS
minuses which terminate. The empty sequence is included as a possibility.’’ So we begin with f < f 4 which is defined as 0 in the a-sequence notation. To connect Gonshor’s representation of surreal numbers with the present work, we need the function ² a : of a introduced in Louck w10x. This function is derived form F Ž a ; 1., given above, and is defined by ² a : s a 0 q 1 q 2 a 0 F Ž a ; 1. y 2 ,
Ž 5.1.
so that the values of ² a : are: ² a0 : s a0 ,
k s 0, 1
² a0 , a1: s a0 y 1 q ² a : s a0 y 1 q
2 2 a1
y
2 2 a 1q a 2
2 a1
q ??? q q
Ž 5.2. k s 1,
,
Ž 5.3.
k Ž y1. 2
2 a 0q a 1q ??? q a ky 1
Ž y1.
kq 1
2 a 0q a 1q ??? q a k
,
k G 2. Ž 5.4.
The transformation Ž5.1. was found by Louck w10x as a way of transforming a function giving surreal numbers on Ž0, 2. to a function that gives surreal numbers on Ž0, `.. For ordinals less than v , we give the relation between a-sequences of length - v and surreal numbers defined by a sequence of pluses and minuses of length - v . Let b be a positive number of finite length in Gonshor’s notation. Then b represents a positive dyadic number and the initial sign of b is plus. We can write for k G 0,
b s Ž q . a 0 Ž y . a 1 Ž q . a 2 . . . Ž yk . a k ,
Ž 5.5.
where yk denotes q if k is even and y otherwise and where the a 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 b : corresponding to the Gonshor sequence b defined by Ž5.5. is given by: ² b : s a0 ,
ks0
² b : s a 0 y Sa , 1 ² b : s a 0 y Sa q 1
1 2
a1
Sa 2 y q
Ž 5.6.
ks1 1 2
a 1q a 2
Ž 5.7.
Sa 3 q ???
Ž y1.
k
2 a 1q ??? q a ky 1
Sa k , k G 2,
Ž 5.8.
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where Sa i s
1 2
q
1 2
2
q ??? q
1 2ai
.
Ž 5.9.
Here we have used Gonshor’s informal prescription on page 29 of w4x 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 b is negative, we put b s yb 9 and expand ² b 9: 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 Sa i s 1 y
1
Ž 5.10.
2ai
to obtain from Ž5.8. the following forms for the dyadic number ² b : corresponding to the Gonshor sequence b . ² b : s a0 , ² b : s a0 y 1 q
k s 0, 1 2 a1
,
k s 1,
² b : s a0 y 1 q q ??? q
Ž y1.
Ž 5.11.
2 2
a1
y
Ž 5.12. 2 2
a 1q a 2
kq 1
2 a 1q? ? ?q a k
,
k G 2.
Ž 5.13.
Thus, we have the identity ² b : s ² a :.
Ž 5.14.
Further, every positive dyadic number has the form Ž5.11., Ž5.12., or Ž5.13. depending on k. A zero or negative value for a dyadic number can be treated similarly to give y² b : s y² a : ,
Ž 0. F b .
Ž 5.15.
The one-to-one correspondence ² b : s ² a : Ž b positive. and y² b : s y² a : Ž b positive. established above is between real dyadic numbers. In Gonshor’s notation, it is the sequence b itself that is called a surreal number. Correspondingly, we now take the sequence a itself to denote a surreal number. The relation
a ) a 9 and a - a 9 m ² a : ) ² a 9:
Ž 5.16.
FUNCTION ITERATION AND SURREAL NUMBERS
347
assures the consistency of using the sequence themselves. We have established then the one-to-one relation between finite a-sequences and finite Gonshor sequences given by
a s Ž a 0 , a 1 , . . . , a k . m Ž q . a 0 , Ž y . a 1 , . . . , Ž yk . a k ,
Ž 5.17.
and, similarly,
a l yb .
Ž 5.18.
6. NONDYADIC REAL NUMBERS Let us first consider positive nondyadic real numbers. Gonshor Žp. 33 of w4x. defines a real number as a surreal number which is either of finite length Ža dyadic number. or of length v that does not eventually have constant signs. He then proves in Theorem 4.3, p. 34 of w4x 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 v and positive real numbers expressed as an a-sequence arising from an v-length iteration of the inverse tent function. Suppose b is a non-dyadic positive real number. Then the signed expansion of b is non-terminating and has no tail of constant sign for otherwise b would be dyadic. In this case, Ž5.5., Ž5.8., and Ž5.13. can be extended to an infinite series in a i and thus we obtain a corresponding a-expansion. Again, if b is 0 or negative, corresponding remarks as in §5 can be made. 7. SURREAL NUMBERS OF LENGTH v THAT ARE NOT REAL We now consider the case of surreal numbers whose length in the Gonshor notation is v that are not real numbers. The correspondence between a-sequences and Gonshor sequences is given by
Ž a 0 , a 1 , . . . , a ky1 , v . l Ž q. a 0 Ž y . a 1 . . . Ž yky1 . a ky 1 Ž yk . Ž yk . . . . , Ž 7.1. where a i , 0 F i F k y 1, are positive integers and yky1 denotes q if k is odd and y if k is even. The function F Ž a ; 1. associated with the a-sequence in Ž7.1. is defined formally as the following infinite sequence of
348
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iterations of the inverse tent map evaluated at x s 1 for k G 1: ² a 0 , a 1 , . . . , a ky1 , v : s F Ž a 0 , a 1 , . . . , a ky1 , v ; 1 . s Ry1 ( Ž Ly1 .
a 0 y1
( Ry1 ( Ž Ly1 .
( ??? ( Ry1 ( Ž Ly1 .
a ky 1 y1
a 1 y1
( Ry1 ( Ly1 ( Ly1 . . . Ž 1 .
s lim F Ž a 0 , a 1 , . . . , a k ; 1 . a kª`
s F Ž a 0 , a 1 , . . . , a ky1 ; x .
xs2 .
Ž 7.2.
We remark again that it is the sequences Ž7.1. themselves that are the surreal numbers and not the values given by the real numbers Ž7.2..
8. THE REMAINING SURREAL NUMBERS The invertible mapping from surreal numbers a of length F v to members of the a-sequences was defined in Sections 5]7. This function is now generalized to transfinite a-sequences associated with transfinite surreals. Let a be a surreal number in the sense of Conway w3x and r g Ž a. its Gonshor representation as given in the first paragraph of Section 5. Define ra Ž a. as the a-sequence representation of a. Let w Ž r g Ž a.. be the map from the Gonshor representation of a to the a-sequence representation of a, that is, w Ž r g Ž a . . s ra Ž a . .
Ž 8.1.
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 v . 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 example w5x. But we prefer to state it explicitly as a lemma. LEMMA 1.
Let On be the class of all ordinals. Then On s
D Ee
Ž 8.2.
e
where the union is o¨ er all limit ordinal numbers e and Ee s
`
D Ž e q n. .
ns0
Ž 8.3.
FUNCTION ITERATION AND SURREAL NUMBERS
349
Proof. It is clear that
D Ee ; On.
Ž 8.4.
e
Let a be a nonlimit ordinal number. Then there must be a maximal limit ordinal e - a so that a y e is finite. This is so because if there is not such an e , there would be an infinite sequence of direct predecessors to a. But then the ordinal a would have a subset of type v * and thus a cannot be well-ordered. See Theorem 1 on page 262 of Sierpinski w13x. So for some e , a g Ee . Hence On ;
D Ee .
Ž 8.5.
D Ee .
Ž 8.6.
e
Hence, On s
e
This completes the proof. We now extend the 1-to-1 mapping between surreal numbers of length F v and a-sequences of length F v to a 1-to-1 mapping between surreal numbers of length ) v and a-sequences of length ) v . Every surreal number a is a map from an initial segment of the ordinals onto the set q, y4 . 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 Ee of ordinals and Ee is either of finite length or length v . Conversely, every nonlimit ordinal is a member of such an Ee . For a sequence Ee of length F v , we construct as was done in Sections 5]7 an a-sequence corresponding to the Gonshor values of the surreal number on Ee . We denote this a-sequence by
a 0Ee , a 1Ee , a 2Ee , . . . .
Ž 8.7.
This sequence may be of finite length or of length v . If it is of finite length, we append v blanks. This construction is carried out for every limit ordinal appearing in a. We obtain from this construction an a-sequence of possibly transfinite length:
a 0 , a 1 , a 2 , . . . av , av q1 , av q2 , . . . av 2 , av 2q1 , av 2q2 , . . . av 3 , . . . . Ž 8.8. The function w is invertible. So the mapping w between surreals and a-sequences is 1]1. This completes the transfinite induction definition and completes the statement of correspondence between surreals and the now generalized a-sequences. These generalized a-sequences can be thought of as a-sequences of length v concatenated an ordinal number of times and possibly terminating with a finite a-sequence.
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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 Ž1991., 3]27. 2. R. L. Bivins, J. D. Louck, N. Metropolis, and M. L. Stein, Classification of all cycles of the parabolic map: The complete solution, to appear. 3. J. H. Conway, ‘‘On Numbers and Games,’’ London Math. Society Monographs, Academic Press, London, 1976. 4. H. Gonshor, ‘‘An Introduction to the Theory of Surreal Numbers,’’ London Mathematical Society Lecture Notes Series, Vol. 110, Cambridge Univ. Press, Cambridge, 1986. 5. F. Hausdorff, ‘‘Set Theory,’’ p. 83, lines 11, 12, Chelsea, New York, 1957. 6. D. E. Knuth, Surreal Numbers, Addison]Wesley, Reading, MA, 1974. 7. M. Kruskal and R. Matthews, Surreal numbers, New Scientist 2 ŽSept. 1995., 37]40. 8. M. Kruskal and P. Shulman, Infinity plus one, and other surreal numbers, Disco¨ er ŽDec. 1995., 96]105. 9. M. Lothaire, Combinatorics on words, in ‘‘Encyclopedia of Mathematics and its Applications,’’ vol. 17, Addison]Wesley, Reading, MA, 1983. 10. J. D. Louck, Conway numbers and iteration theory, Ad¨ . Appl. Math., to appear. 11. J. D. Louck and N. Metropolis, ‘‘Symbolic Dynamics of Trapezoidal Maps,’’ Reidel, Dordrecht, 1986. 12. O. W. Rechard, Invariant measures for many]one transformations, Duke Math. J. 23 Ž1956., 477]488. 13. W. Sierpinski, ‘‘Cardinal and Ordinal Numbers,’’ PWN]Polish Scientific, Warsaw, 1965.
18, 333]350 Ž1997.
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 transfinite 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 addition, 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 Conway w3x and Knuth w6x. In this paper we follow the development given in Gonshor’s book w4x which is quite different from that of Conway and Knuth. In w10x, a relation was shown between surreal numbers up to the first transfinite ordinal v and aspects of iterated maps of the interval w0, 2x. In this paper, we specialize the results in w10x to the graph of the nth iterate of the inverse tent map and extend the results of w10x 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 Conway w3x. The Gonshor method was expounded by Kruskal in two popular articles, one with Matthews w7x and a second one with Shulman w8x. In those two articles 333 0196-8858r97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
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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 of w10x 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 Rechard w12x with attribution to Ulam. ŽFor this reason, the tent map could well be called the Ulam map.. We follow the notation of w10x. The direct tent map has the form on w0, 2x: L: x ª 2 x ' t Ž x . ,
0 F x F 1,
Ž 2.1.
R: x ª 2 Ž 2 y x . ' t Ž x . ,
1 F x F 2.
Ž 2.2.
The corresponding two inverse maps on w0, 2x have the form: Ly1 : x ª
x 2
' t1Ž x . ,
Ry1 : x ª 2 y
x 2
' ty1 Ž x . ,
0FxF2
Ž 2.3.
0 F x F 2.
Ž 2.4.
The tent map is the simplest map of the interval w0, 2x realizing certain properties, but not all, of the more general theory given in w10x. 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 in w10x and w11x, 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 topological equivalence between the direct tent map t Ž x . and the quadratic map q Ž x . s 2 x Ž2 y x . on w0, 2x, we also obtain the same detailed description of this latter map. This topological equivalence is expressed by q s hy1 ( t ( h,
Ž 2.5.
FUNCTION ITERATION AND SURREAL NUMBERS
335
where hŽ x . s
4
p
arcsin
(
x 2
,
0 F x F 2.
Ž 2.6.
The homeomorphism hŽ x . on w0, 2x preserves fixed points and order of points on the line. What is missing from these special unimodal maps of the interval w0, 2x in the general theory for the direct map is the cycle containing x s 1, which degenerates to the three points 1, 2, 04 . 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. The sets of a-sequences, denoted by B ny 1 and A n in w10x, 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 by Ž2.1. and Ž2.2.. We denote the inverse graph by Gn , Gn : Ž x, P Ž1. ( P Ž2. ( ??? ( P Ž n. Ž x . . ,
0 F x F 2,
where P Ž i. is either Ly1 or Ry1 . The graphs analogous to those given in Figs. 2, 3 in w10x 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.
FIG. 1. The inverse graph G1 . The a-sequences Ž1. and Žy1. label the lines y s t1Ž x . s F Ž1; x . s 2 y 12 x and y s ty1 Ž x . s F Žy1; x . s 12 x. The sequence Ž0. labels the central boundary abscissa y s F Ž0; 1. s 1.
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FIG. 2. The inverse graph G 2 . The a-sequences Ž2. ) Ž1, 1. ) Žy1, y1. ) Žy2. label the lines y s F Ž2; x . s 2 y Ž1r2 2 . x, y s F Ž1, 1; x . s 1 q Ž1r2 2 . x, y s F Žy1 y1; x . s 1 y Ž1r2 2 . x, and y s F Žy2; x . s Ž1r2 2 . x. The a-sequences Ž1. ) Ž0. ) Žy1. label the boundary abscissae y s F Ž1; 0. s 3r2, y s F Ž0, 1. s 1, and y s F Žy1; 1. s 1r2.
An a-sequence is a finite sequence of positive integers:
a s Ž a0 , a1 , . . . , ak . ,
a i g Nq , i g Ž 0, 1, 2, . . . , k . ,
or an infinite sequence of positive integers:
a s Ž a0 , a1 , . . . . ,
a i g Nq , i g Ž 0, 1, 2, . . . . .
Corresponding negative a-sequences are defined as the negative of the above:
a s Ž ya 0 , ya 1 , . . . , ya k . , a s Ž ya 0 , ya 1 , . . . . ,
a i g Nq , i g Ž 0, 1, 2, . . . , k . , a i g Nq , i g Ž 0, 1, 2, . . . . .
An order relation among the a-sequences is given in the third paragraph of the introduction in w10x. 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.
337
FUNCTION ITERATION AND SURREAL NUMBERS
The degree of a finite a-sequence or its negative is defined by: DŽ a . s DŽ a . s
Ý ai . i
We also define the sets of sequences: A n s a < a is positive, D Ž a . s n4 , B n s A 0 j A 1 j ??? j A n . The inverse graph Gn will now be described. First define the following function for x g w0, 2x and a g A n : F Ž a ; x . ' Ry1 ( Ž Ly1 .
ž
a 0 y1
( Ry1 ( Ž Ly1 .
a 1 y1
( ???
( Ry1 ( Ž Ly1 .
a k y1
/ Ž x. .
Ž 2.7.
We also need the value of the function F Ž a ; x . for a negative a sequence: F Ž a ; x . ' Ž Ly1 .
ž
a0
( Ry1 ( Ž Ly1 .
a 1 y1
( ??? ( Ry1 ( Ž Ly1 .
a k y1
/ Ž x. . Ž 2.8.
The inverse graph Gn then consists of a set of 2 n straight line segments, each defined on the interval w0, 2x, and is given by the functions on w0, 2x: y s FŽ a ; x. ,
a g An,
Ž 2.9.
y s FŽ a ; x. ,
a g An.
Ž 2.10.
These functions each define separately 2 ny 1 lines. We call the combined set a graph. The inverse graph is symmetric about the line segment Ž0 F x F 2, y s 1.; that is, FŽ a ; x. s 2 y FŽ a ; x. ,
a g An.
Ž 2.11.
The explicit expression for the values F Ž a ; 1. for a s Ž a 0 , a 1 , . . . , a k . was first given, to our knowledge, in w11x Žusing slightly different notation in Ž11.11a. on page 192 and Ž5.5. on page 84. in the study of properties of the trapezoid map for arbitrary slope parameter z . In this work, z s 2. The relation between F Ž a , x . and F Ž a , 1. is given by F Ž a ; x . s F Ž a ; 1 . q Ž y1 .
Ž x y 1.
kq 1
2
a 0 q a 1 q ??? q a k
.
Ž 2.12.
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BEYER AND LOUCK
The following are the values of F Ž a ; 1.: F Ž a 0 ; 1. s 2 y F Ž a 0 , a 1 ; 1. s 2 y F Ž a ; 1. s 2 y
2 2 a0
q
2 2 a 0q a 1
1
2 2
a0
q
y ??? q q
k s 0,
,
2a0
1 2
,
a 0q a 1
Ž 2.13. ks1
Ž 2.14.
k Ž y1. 2
2 a 0q a 1q ??? q a ky 1
Ž y1.
kq 1
2 a 0q a 1q ??? q a k
,
k G 2. Ž 2.15.
The set of straight line segments Ž2.9. and Ž2.10. constituting the graph Gn is ordered by the order relation of the a-sequences themselves Žsee w10x.; that is, the upper half Žabove the line segment Ž0 F x F 2, y s 1. of the graph consists of the straight line segments Ž2.9. and satisfy F Ž a ; x . ) F Ž a 9; x . m a ) a 9
; x g Ž 0, 2 . .
Ž 2.16.
The line segments Ž2.10. constituting the lower half Žbelow Ž0 F x F 2, y s 1. of the graph likewise satisfy F Ž a ; x . ) F Ž a 9; x . m a - a 9
; x g Ž 0, 2 . .
Ž 2.17.
These line segments join pairwise at the two vertical boundaries B1 s Ž x s 0, 0 F y F 2 . , B2 s Ž x s 2, 0 F y F 2 . , between the two straight line segments corresponding to adjacent sequences in A n . The boundary ordinate B1 is labelled by the set of positive sequences Žincluding also Ž0.. B ny 1 s a N 0 F D Ž a . F n y 1 4 ,
Ž 2.18.
together with the corresponding set of negative sequences. Explicitly, these boundary ordinate values are F Ž a ; 1. ,
a g B ny 1 ,
F Ž a ; 1. s 2 y F Ž a ; 1. ,
a g B ny 1 ,
Ž 2.19. Ž 2.20.
which are 2 n y 1 in number. These ordinate values occur at the y-level in accordance with the order relation of the a-sequence themselves, just as in
339
FUNCTION ITERATION AND SURREAL NUMBERS
Ž2.16. and Ž2.17.. 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 a and a 9 with a - a 9, there lies exactly one of the lines Ž2.12., namely, the one labelled by b with a - b - a 9. The situation is similar for the corresponding boundaries in the lower half of the graph. We call these boundaries conjugate boundaries. The distance between adjacent boundary lines is 1r2 n. Knowledge of the general function F Ž a ; x . for 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:
xŽ a . s
2 DŽ a . F Ž a ; 1 . q Ž y1 .
l Ža.
xŽ a . s 2 y xŽ a . ,
a g An,
,
l Ža.
2 DŽ a . q Ž y1 .
a g An,
Ž 3.1. Ž 3.2.
where l Ž a . s k q 1 denotes the number of parts of a s Ž a 0 , a 1 , . . . , a k .. The relation between positive dyadic Conway numbers ² a : and the functions F Ž a ; 1. Žsee Eqs. Ž1.13., Ž1.20., Ž1.21. in w10x. is given by ² a : s a 0 q 1 q 2 a 0 Ž F Ž a ; 1. y 2. .
Ž 3.3.
The F Ž a ; 1. are given in Ž2.13., Ž2.14., and Ž2.15.. The Conway numbers Ž a : are expressed directly in terms of the fixed points Ž3.1., Ž3.2., and the degree DŽ a . by: ² a : s a0 q 1 y 2
a0
² a : s y Ž a 0 q 1. q 2
q2
a0
a0
q2
1q
a0
Ž y1. l
1q
Ža.
Ž x Ž a . y 1. ,
2 DŽ a .
Ž y1. l
Ž 3.4.
Ža.
2 DŽ a .
Ž x Ž a . y 1 . , Ž 3.5.
340
BEYER AND LOUCK
for each a g A n . The set of fixed points
xŽ a . , xŽ a .
a g A n4
Ž 3.6.
decomposes into cycles Žw10x, Eqs. Ž3.36. ] Ž3.38... Correspondingly, the Conway numbers ² a : may be decomposed into cycles. The explicit decomposition of the fixed points x Ž a . and x Ž a . into cycles is obtained by decomposing the set of sequences in A n j A n into cycles with respect to the action of the cyclic group Cn of permutations in this set. This action is defined from the correspondence between the elements of the set A n j A n and LR-words Žsee Eqs. Ž1.3.. in w10x.:
a s Ž a 0 , a 1 , . . . , a k . ª RLa 0y1 RLa 1y1 ??? RLa ky1 , a0
a 1 y1
a s Ž ya 0 , ya 1 , . . . , ya k . ª L RL
a 2 y1
RL
a k y1
??? RL
Ž 3.7. . Ž 3.8.
The maps Ž3.7. and Ž3.8. are maps to LR words. We select any element t g A n j A n and determine the LR-word to which it corresponds. The action of the cyclic group Cn on the a-sequence t is then defined as the set of a-sequences CŽt . 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 a-sequence notation by use of Ž3.7. and Ž3.8.. We use the term cycle class of t for the subset CŽt . ; A n j A n . We then select t 9 g A n j A n , t 9 f CŽt ., and repeat the process to construct CŽt 9.. This procedure is continued until the set A n j A n is decomposed into the union of a finite family of disjoint cycle classes CŽt ., CŽt 9., CŽt 0 ., . . . . This procedure is standard in the theory of words, where CŽt . is called the class of sequences conjugate to t Žsee w9, p. 7 and p. 188x.. We have used the term ‘‘cycle class of t ’’ because the term conjugate has been assigned at least two different meanings in this work. The detailed decomposition of A n j A n into cycle classes with respect to the cyclic group Cn may be given. An a-sequence t g A n j A n is called primiti¨ e if it cannot be written as the proper concatenation power of another a-sequence. ŽFor the definition of concatenation of two sequences, see w10x.. We define the set of primitive a-sequences Pdq and Pdy by Pdq s Ž 1, g . g g L d , Ž 1, g . primitive 4 ,
Ž 3.9.
Pdys Ž 1, g . g g L d , Ž 1, g . primitive 4 ,
Ž 3.10.
where L d denotes the set of lexical sequences of degree d y 1. Lexical sequences are defined in Louck and Metropolis w11, p. 93x. The major result for the labeling of cycle classes is the following theorem whose proof will appear in w2x.
FUNCTION ITERATION AND SURREAL NUMBERS
341
THEOREM 1. The cycle classes of A n j A n with respect to the cyclic group Cn may be labelled by the sequences in the set Rn s Rnq j Rny , where the definition of the two sets in this union is gi¨ en, respecti¨ ely, by Rnq s Ž 1, g .
nrd
d < n; Ž 1, g . g Pdq ,
Ž 3.11.
Rny s Ž 1, g .
nrd
d < n; Ž 1, g . g Pdy .
Ž 3.12.
½
5
4
Thus, we ha¨ e An j An s
D CŽ t . .
Ž 3.13.
tg Rn
The significance of Theorem 1 for the decomposition of the set of a-sequences in A n j A n 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 n j A n 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 X Ž t . s x Ž t 9 . t 9 g C Ž t . , each t g Rn 4
Ž 3.14.
Ž recall that t 9 is an a-sequence. and the decomposition of the set of fixed points I n of Gn into these cycles is gi¨ en by In s
D XŽ t . .
Ž 3.15.
tg Rn
Remark. The complete proof of this result is given in w10x, as a special case of a more general theorem. The application of Theorem 1 to the set A 3 j A 3 s Ž 3 . , Ž 2, 1 . , Ž 1, 1, 1 . , Ž 1, 2 . , Ž y1, y2 . ,
Ž y1, y1, y1. , Ž y2, y1. , Ž y3. 4
Ž 3.16.
gives A 3 j A 3 s C Ž 1, 1, 1 . j C Ž 1, 2 . j C Ž y1, y2 . j C Ž y3 . , Ž 3.17.
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where C Ž 1, 1, 1 . s Ž 1, 1, 1 . 4 , C Ž 1, 2 . s Ž 2, 1 . , Ž y1, y1, y1 . 4 Ž 3.18. C Ž y1, y2 . s Ž y1, y2 . , Ž y2, y1 . , Ž 3 . 4 , C Ž y1, y1, y1 . s y3 4 . 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 3 s X Ž 1, 1, 1 . j X Ž 1, 2 . j X Ž y1, y2 . j X Ž y3 . ,
Ž 3.20.
where X Ž 1, 1, 1 . s x Ž 1, 1, 1 . s 11r8 4 ,
Ž 3.21.
X Ž 1, 2 . s x Ž 1, 2 . s9r8, x Ž 2, 1 . xs13r8, x Ž y1, y1, y1 . s5r8 4 , Ž 3.22. X Ž y1, y2 . s x Ž y1, y2 . s7r8, x Ž y2, y1 . s3r8, x Ž 3. s15r8 4 , Ž 3.23. X Ž y3 . s x Ž y3 . s 1r8 4 .
Ž 3.24.
Thus, the set of fixed points of G 3 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 decompositions Ž3.13. and Ž3.15.: < A3 j A3< s 2n s
Ý
CŽ t . s
Ý d Ž
Ž 3.25.
d< n
tg Rn
The Mobius inversion formula applies to this relation and yields the ¨ following expression for the number of cycles of degree d, which is also equal to the number Ndn of d-cycles contained in the set of fixed points I n , Ndn s
1 d
Ý mŽ r .2 dr r,
Ž 3.26.
r
where m Ž r . denotes the well-known Mobius function. ¨ The results given in this section solve the problem of constructing the inverse graph of the nth iterate of the tent map, of giving its fixed points, and of decomposing this set of fixed points into cycles.
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FIG. 3. The inverse graph G 3 . The a-sequences
Ž 3 . ) Ž 2, 1 . ) Ž 1, 1, 1 . ) Ž1, 2 . ) Žy1, y2 . ) Žy1, y1, y1 . ) Žy2, y1 . ) Žy3 . label the graphs of the lines: y s F Ž 3; x . s 2 y Ž 1r2 3 . x, y s F Ž 1, 1, 1, x . s
3 2
y s F Ž 2, 1; x . s
y Ž 1r2 3 . x,
y s F Ž y1, y2; x . s 1 y Ž 1r2 3 . x, y s F Ž y2, y1; x . s
1 2
3 2
q Ž 1r2 3 . x
y s F Ž 1, 2; x . s 1 q Ž 1r2 3 . x y s F Ž y1, y1, y1; x . s
y Ž 1r2 3 . x,
1 2
q Ž 1r2 3 . x
y s F Ž y3; x . s Ž 1r2 3 . x.
The a-sequences Ž2. ) Ž1. ) Ž1, 1. ) Ž0. ) Žy1, y1. ) Žy2. label the boundary abscissae as: y s F Ž2; 1. s 74 , y s F Ž1; 1. s 32 , y s F Ž1, 1; 1. s 54 , y s F Žy1, y1; 1. s 34 , y s F Žy2; 1. s 41 .
4. THE TOPOLOGICALLY EQUIVALENT QUADRATIC MAP It is quite remarkable that because the tent map is topological equivalent to the parabolic map 2 x Ž1 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 decomposition of this set of fixed points into cycles. This equivalence is effected by the arcsin function given in Ž2.6. above. We also need its inverse: hy1 Ž x . s 2 sin 2
p
ž / 4
x ,
0 F x F 2.
Ž 4.1.
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For a succinct description of this topological equivalence, it is convenient to introduce the following table of notations, where the range of x is w0, 2x. The function C is defined analogously to F in Ž2.7. and Ž2.8.. For the quadratic map, q1 and qy1 replaces t 1 and ty1 of the tent map. Obviously, q stands for quadratic and t stands for tent. Quadratic Map
Tent Map
q Ž x . s 2 x Ž1 y x . q1Ž x . s 1 q
' '
qy1 Ž x . s 1 y
1y
x 2
1y
x
2 qy1 Ž x . s q1Ž x ., qy1Ž x .4 CŽ a ; x . C Ž a ; 1. C Ž a ; 1.
Type
t Ž x . s 2 x, 0 F x F 1 t Ž x . s 2Ž1 y x ., 1 F x F 2 x t1Ž x . s 2 y 2 x ty1Ž x . s 2 ty1Ž x . s t1Ž x ., ty1Ž x .4 FŽ a ; x . F Ž a ; 1. F Ž a ; 1.
Direct Map Inverse Map Inverse Map Paired Maps Iteration function Positive Conway numbers Negative Conway numbers
Each quantity in the left-hand column, denoted generically by z q Ž x ., is related to the corresponding quantity z t Ž x . in the second column by z q Ž x . s Ž hy1 ( z t ( h . Ž x . .
Ž 4.2.
In particular, this relation is valid at x s 1 and gives C Ž t ; 1 . s 2 sin 2
ž
p 4
F Ž t ; 1. ,
/
each t g A n j A n ,
Ž 4.3.
which gives the relation between the Conway numbers F Žt ; 1. defined by the inverse tent map and the corresponding numbers C Žt ; 1. defined in terms of the inverse quadratic map. Finally, the fixed points x q Žt . of the nth iterate of q Ž x . are related to the fixed points x t Žt . of nth iterate of t Ž x . by x q Ž t . s 2 sin 2
ž
p 4
xt Žt . ,
/
t g An j An.
Ž 4.4.
There is, of course, a decomposition of x q Žt ., t g A n j A n , into cycles, exactly as in Ž3.14..
5. SURREAL DYADIC NUMBERS Gonshor, in his book w4x, defines a surreal number as follows: ‘‘A surreal number is a function from an initial segment of the ordinals into the set q, y4 , i.e., informally, an ordinal sequence consisting of pluses and
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minuses which terminate. The empty sequence is included as a possibility.’’ So we begin with f < f 4 which is defined as 0 in the a-sequence notation. To connect Gonshor’s representation of surreal numbers with the present work, we need the function ² a : of a introduced in Louck w10x. This function is derived form F Ž a ; 1., given above, and is defined by ² a : s a 0 q 1 q 2 a 0 F Ž a ; 1. y 2 ,
Ž 5.1.
so that the values of ² a : are: ² a0 : s a0 ,
k s 0, 1
² a0 , a1: s a0 y 1 q ² a : s a0 y 1 q
2 2 a1
y
2 2 a 1q a 2
2 a1
q ??? q q
Ž 5.2. k s 1,
,
Ž 5.3.
k Ž y1. 2
2 a 0q a 1q ??? q a ky 1
Ž y1.
kq 1
2 a 0q a 1q ??? q a k
,
k G 2. Ž 5.4.
The transformation Ž5.1. was found by Louck w10x as a way of transforming a function giving surreal numbers on Ž0, 2. to a function that gives surreal numbers on Ž0, `.. For ordinals less than v , we give the relation between a-sequences of length - v and surreal numbers defined by a sequence of pluses and minuses of length - v . Let b be a positive number of finite length in Gonshor’s notation. Then b represents a positive dyadic number and the initial sign of b is plus. We can write for k G 0,
b s Ž q . a 0 Ž y . a 1 Ž q . a 2 . . . Ž yk . a k ,
Ž 5.5.
where yk denotes q if k is even and y otherwise and where the a 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 b : corresponding to the Gonshor sequence b defined by Ž5.5. is given by: ² b : s a0 ,
ks0
² b : s a 0 y Sa , 1 ² b : s a 0 y Sa q 1
1 2
a1
Sa 2 y q
Ž 5.6.
ks1 1 2
a 1q a 2
Ž 5.7.
Sa 3 q ???
Ž y1.
k
2 a 1q ??? q a ky 1
Sa k , k G 2,
Ž 5.8.
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where Sa i s
1 2
q
1 2
2
q ??? q
1 2ai
.
Ž 5.9.
Here we have used Gonshor’s informal prescription on page 29 of w4x 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 b is negative, we put b s yb 9 and expand ² b 9: 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 Sa i s 1 y
1
Ž 5.10.
2ai
to obtain from Ž5.8. the following forms for the dyadic number ² b : corresponding to the Gonshor sequence b . ² b : s a0 , ² b : s a0 y 1 q
k s 0, 1 2 a1
,
k s 1,
² b : s a0 y 1 q q ??? q
Ž y1.
Ž 5.11.
2 2
a1
y
Ž 5.12. 2 2
a 1q a 2
kq 1
2 a 1q? ? ?q a k
,
k G 2.
Ž 5.13.
Thus, we have the identity ² b : s ² a :.
Ž 5.14.
Further, every positive dyadic number has the form Ž5.11., Ž5.12., or Ž5.13. depending on k. A zero or negative value for a dyadic number can be treated similarly to give y² b : s y² a : ,
Ž 0. F b .
Ž 5.15.
The one-to-one correspondence ² b : s ² a : Ž b positive. and y² b : s y² a : Ž b positive. established above is between real dyadic numbers. In Gonshor’s notation, it is the sequence b itself that is called a surreal number. Correspondingly, we now take the sequence a itself to denote a surreal number. The relation
a ) a 9 and a - a 9 m ² a : ) ² a 9:
Ž 5.16.
FUNCTION ITERATION AND SURREAL NUMBERS
347
assures the consistency of using the sequence themselves. We have established then the one-to-one relation between finite a-sequences and finite Gonshor sequences given by
a s Ž a 0 , a 1 , . . . , a k . m Ž q . a 0 , Ž y . a 1 , . . . , Ž yk . a k ,
Ž 5.17.
and, similarly,
a l yb .
Ž 5.18.
6. NONDYADIC REAL NUMBERS Let us first consider positive nondyadic real numbers. Gonshor Žp. 33 of w4x. defines a real number as a surreal number which is either of finite length Ža dyadic number. or of length v that does not eventually have constant signs. He then proves in Theorem 4.3, p. 34 of w4x 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 v and positive real numbers expressed as an a-sequence arising from an v-length iteration of the inverse tent function. Suppose b is a non-dyadic positive real number. Then the signed expansion of b is non-terminating and has no tail of constant sign for otherwise b would be dyadic. In this case, Ž5.5., Ž5.8., and Ž5.13. can be extended to an infinite series in a i and thus we obtain a corresponding a-expansion. Again, if b is 0 or negative, corresponding remarks as in §5 can be made. 7. SURREAL NUMBERS OF LENGTH v THAT ARE NOT REAL We now consider the case of surreal numbers whose length in the Gonshor notation is v that are not real numbers. The correspondence between a-sequences and Gonshor sequences is given by
Ž a 0 , a 1 , . . . , a ky1 , v . l Ž q. a 0 Ž y . a 1 . . . Ž yky1 . a ky 1 Ž yk . Ž yk . . . . , Ž 7.1. where a i , 0 F i F k y 1, are positive integers and yky1 denotes q if k is odd and y if k is even. The function F Ž a ; 1. associated with the a-sequence in Ž7.1. is defined formally as the following infinite sequence of
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iterations of the inverse tent map evaluated at x s 1 for k G 1: ² a 0 , a 1 , . . . , a ky1 , v : s F Ž a 0 , a 1 , . . . , a ky1 , v ; 1 . s Ry1 ( Ž Ly1 .
a 0 y1
( Ry1 ( Ž Ly1 .
( ??? ( Ry1 ( Ž Ly1 .
a ky 1 y1
a 1 y1
( Ry1 ( Ly1 ( Ly1 . . . Ž 1 .
s lim F Ž a 0 , a 1 , . . . , a k ; 1 . a kª`
s F Ž a 0 , a 1 , . . . , a ky1 ; x .
xs2 .
Ž 7.2.
We remark again that it is the sequences Ž7.1. themselves that are the surreal numbers and not the values given by the real numbers Ž7.2..
8. THE REMAINING SURREAL NUMBERS The invertible mapping from surreal numbers a of length F v to members of the a-sequences was defined in Sections 5]7. This function is now generalized to transfinite a-sequences associated with transfinite surreals. Let a be a surreal number in the sense of Conway w3x and r g Ž a. its Gonshor representation as given in the first paragraph of Section 5. Define ra Ž a. as the a-sequence representation of a. Let w Ž r g Ž a.. be the map from the Gonshor representation of a to the a-sequence representation of a, that is, w Ž r g Ž a . . s ra Ž a . .
Ž 8.1.
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 v . 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 example w5x. But we prefer to state it explicitly as a lemma. LEMMA 1.
Let On be the class of all ordinals. Then On s
D Ee
Ž 8.2.
e
where the union is o¨ er all limit ordinal numbers e and Ee s
`
D Ž e q n. .
ns0
Ž 8.3.
FUNCTION ITERATION AND SURREAL NUMBERS
349
Proof. It is clear that
D Ee ; On.
Ž 8.4.
e
Let a be a nonlimit ordinal number. Then there must be a maximal limit ordinal e - a so that a y e is finite. This is so because if there is not such an e , there would be an infinite sequence of direct predecessors to a. But then the ordinal a would have a subset of type v * and thus a cannot be well-ordered. See Theorem 1 on page 262 of Sierpinski w13x. So for some e , a g Ee . Hence On ;
D Ee .
Ž 8.5.
D Ee .
Ž 8.6.
e
Hence, On s
e
This completes the proof. We now extend the 1-to-1 mapping between surreal numbers of length F v and a-sequences of length F v to a 1-to-1 mapping between surreal numbers of length ) v and a-sequences of length ) v . Every surreal number a is a map from an initial segment of the ordinals onto the set q, y4 . 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 Ee of ordinals and Ee is either of finite length or length v . Conversely, every nonlimit ordinal is a member of such an Ee . For a sequence Ee of length F v , we construct as was done in Sections 5]7 an a-sequence corresponding to the Gonshor values of the surreal number on Ee . We denote this a-sequence by
a 0Ee , a 1Ee , a 2Ee , . . . .
Ž 8.7.
This sequence may be of finite length or of length v . If it is of finite length, we append v blanks. This construction is carried out for every limit ordinal appearing in a. We obtain from this construction an a-sequence of possibly transfinite length:
a 0 , a 1 , a 2 , . . . av , av q1 , av q2 , . . . av 2 , av 2q1 , av 2q2 , . . . av 3 , . . . . Ž 8.8. The function w is invertible. So the mapping w between surreals and a-sequences is 1]1. This completes the transfinite induction definition and completes the statement of correspondence between surreals and the now generalized a-sequences. These generalized a-sequences can be thought of as a-sequences of length v concatenated an ordinal number of times and possibly terminating with a finite a-sequence.
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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 Ž1991., 3]27. 2. R. L. Bivins, J. D. Louck, N. Metropolis, and M. L. Stein, Classification of all cycles of the parabolic map: The complete solution, to appear. 3. J. H. Conway, ‘‘On Numbers and Games,’’ London Math. Society Monographs, Academic Press, London, 1976. 4. H. Gonshor, ‘‘An Introduction to the Theory of Surreal Numbers,’’ London Mathematical Society Lecture Notes Series, Vol. 110, Cambridge Univ. Press, Cambridge, 1986. 5. F. Hausdorff, ‘‘Set Theory,’’ p. 83, lines 11, 12, Chelsea, New York, 1957. 6. D. E. Knuth, Surreal Numbers, Addison]Wesley, Reading, MA, 1974. 7. M. Kruskal and R. Matthews, Surreal numbers, New Scientist 2 ŽSept. 1995., 37]40. 8. M. Kruskal and P. Shulman, Infinity plus one, and other surreal numbers, Disco¨ er ŽDec. 1995., 96]105. 9. M. Lothaire, Combinatorics on words, in ‘‘Encyclopedia of Mathematics and its Applications,’’ vol. 17, Addison]Wesley, Reading, MA, 1983. 10. J. D. Louck, Conway numbers and iteration theory, Ad¨ . Appl. Math., to appear. 11. J. D. Louck and N. Metropolis, ‘‘Symbolic Dynamics of Trapezoidal Maps,’’ Reidel, Dordrecht, 1986. 12. O. W. Rechard, Invariant measures for many]one transformations, Duke Math. J. 23 Ž1956., 477]488. 13. W. Sierpinski, ‘‘Cardinal and Ordinal Numbers,’’ PWN]Polish Scientific, Warsaw, 1965.