2. THE FUNDAMENTAL THEOREM: n\n(n)

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1 OBTAINING DIVIDING FORMULAS n\q(ri) FROM ITERATED MAPS Chyi-Lung Lin Departent of Physics, Soochow University, Taipei, Taiwan, 111, R.O.C. {SubittedMay 1996-Final Revision October 1997) 1. INTRODUCTION In this work we show that we ay use iterated aps to underst generate a dividing forula n\q(n), where n is any positive integer. A well-known exaple of a dividing forula concerning Fibonacci nubers, for instance, is n\(f n+x+ F n _ x -\), (1.1) where n is a prie F n is the /?* Fibonacci nuber. We will show that fro iterated aps we have a systeatic way to construct functions Q(n) such that n\q{ri). In this paper, we show how to derive the above dividing forula fro an iterated ap. We also generalize the result to the case in which /? is any positive integer to the case of Fibonacci nubers of degree. We begin with Theore 2.1 below. Theore 2.1: For an iterated ap, 2. THE FUNDAMENTAL THEOREM: n\n(n) where N(n) is the nuber of period-?? points for the ap. n\n(n), (2.1) Proof: If N(ri) - 0, forula (2.1) is obvious. If N(n) ^ 0, then the orbit of a period-/? point is an /i-cycle containing n distinct period-/? points. Since there are no coon eleents in any two distinct /?-cycles, N{ri) ust be a ultiple of/?, i.e., n\n(ri), N{n)ln is an integer representing the nuber of /?-cycles for the ap. As a consequence of this fundaental theore, each iterated ap, in principle, offers a desired Q(n) function such that n\q(ri), where Q(n)= N{n), the nuber of period-/? points of an iterated ap. Therefore, we have an additional way to underst the dividing forula n\q(n) fro the point of view of iterated aps. 3. THE N(n) OF AN ITERATED MAP For a general discussion, we consider a ap f(x) in soe interval. The fixed points off are deterined fro the forula /(*) = * (3-1) The nuber of fixed points for/can be deterined fro the nuber of intersections of the curve y - f(x) with the diagonal line y = x in the interval. We define fw(x) for the w* iterate of x for /, then f {n \x) = f(f [n ~ l] (x)). We should distinguish a fixed point of f [n \ a period-/? point of / The fixed points of / M are deterined fro the forula fl"\x) = x; however, the period-/? points of/are deterined fro the following two equations: 118 [MAY

2 OBTAINING DIVIDING FORMULAS n \Q{n) FROM ITERATED MAPS f [n] (x) = x, (3.2) fi\x)*x for/ = l,2,...,/?-!. (3.3) We need (3.3) because an x satisfying only (3.2). is not necessarily a period-/? point off since it could be a fixed point of/or, in general, a period-//? point of/, where < n \n. Forulas (3.2) (3.3) together ensure that x is a period-/? point of/ Then N(ri) represents the nuber of points satisfying both (3.2) (3.3). We let N^(n) represent the nuber of points satisfying only (3.2); hence, JV (w) represents the nuber of fixed points for f [n]. Accordingly, we have where the su is over all the divisors of/? (including 1 n). N^(n) is siply deterined fro intersections of the curve y = f [n \x) with the diagonal line. Therefore, what we obtain directly fro an iterated ap is not N(n) but N^(n). We need a reverse forula expressing N(n) in ters of N%(d). This has already been done, because we know the following two forulas fro [2] [7]: tfi(") = 2X<*X (3-4) N{n) = ^M(p'd)N^d) ^juidwxin/dl (3.5) where ju(d) is the Mobius function. N%(ri) is called the Mobius transfor of N(ri), N(n) is the inverse Mobius transfor of N^(n). Hence, after calculating the N^Qi) of an iterated ap, we obtain a dividing forula n\n(n) fro (3.5). There are exaples of iterated aps for which Nx(ri) are calculated (see [1], [5]). We suarize these in the following theore. Theore 3.1: For an iterated ap,, especially, We consider the B{fi\ x) ap defined by n\n(n), with N(n) = X ^ 7 d ) ^ { d ) (3.6) n\{n z {n)-n^{\)) for/? a prie. (3.7) 4. APPLICATIONS [tix for0<x<l/2, B(u;x) = i (4.1) W ; J [//(*-1/2) f o r l / 2 < x < l, \. where ju is the paraeter whose value is restricted to the range 0 < ju < 2 so that an x in the interval [0,1] is apped to the sae interval. We now have the following theore. Theore 4,1: Line segents in B^n\ju; x) are all parallel with slope ju n. Proof: In the beginning, for a given ju, B(ju;x) contains two parallel line segents with slope ju. Fro (4.1), we see that each line segent will ultiply its slope by a factor ju after one iteration. Q.E.D. 1998] 119

3 OBTAINING DIVIDING FORMULAS n \Q(ri) FROM ITERATED MAPS We consider the following cases. 4,1 The Case in Which JC = 1/ 2 Is a Period-2 Point If x = 1 / 2 is a period-2 point of the B(ju) ap, it requires that B 2 (ju,l/2) = l/2. This then requires that ju>\ ju 2 -ju-l = 0. Solving this, we have // = (1 + V 5 ) / 2 « , the wellknown golden ean. We denote this ju by I 2, indicating that for this paraeter value x = 1/2 is a period-2 point. To obtain N^(n), we need to count the nuber of line segents in B^n\ju) that intersect the diagonal line. Detailed discussions of this ap can be seen in [3] [4]. Briefly, we see that starting fro x 0 = 1/2, we have a 2-cycle, {x 0, x x }, where x x - ju/2. It follows that there are two types of line segents. We denote by Z the type of line segents connecting points (x a, 0) (x b,ju/ 2) with 0 < x a < x b < 1, denote by S the type of line segents connecting points (x c, 0) (x d,l/ 2) with 0 < x c < x d < 1. Since x 0 -» x x x } -» x 0 under an iteration, it follows that the behavior of line segents of these two types under iteration is L-+L + S S-+L. (4.2) Using the sybols Z S, we see that the graph of B(ju) contains two Z. (4.2) shows how the nuber of line segents increases under the action of iteration. Let L(n) S(n) be the nuber of line segents of type Z S in B^n\ju), respectively. (4.2) shows siply that each Z is fro previous Z S, $ol(n) = L(n -1) + S(n - 1 ), each S is fro previous Z, so S(n) - L(n -1). Fro these, we conclude that L(n) = L(n -1) + L(n - 2) S(n) = is(w -1) + is(w - 2). That is, L(ri) (?2) are both the type of sequences of which each eleent is the su of its previous two eleents. Starting with an Z, according to (4.2), the orbit of which is L -> LS -> 2LS -> 3L2S -> 5Z3S -> 815^ ->, we easily see that L{ri) = F n S(ri) = F n _ l. In conclusion, starting fro an L, under the action of iteration there are F (L-type) S(n) (S-typ6) parallel line segents generated in B^n\ju). We will use this result. N z (n) is then deterined fro the nuber of intersections of these line segents with the diagonal line. Consider first the is-type line segents. We note that all 5-type line segents in B [n \fi) are parallel can be divided into two parts, one part in the range 0 < x < 1 / 2 the other in the range 1 / 2 < x < 1. We easily see that each line segent in the range 0 < x < 1 / 2 intersects the diagonal line once, others in the range 1 / 2 < x < 1 cannot intersect the diagonal line. The original line segent in the range 0< x < 1/2 is an Z,, so the nuber of S-type line segents in BW(ju) in this range is S{n) = F n _ v Consider next the L-type line segent. Siilarly, line segents of this type in B^n\ju) are parallel, each of those that is in the range 0<x< /J/2 intersects the diagonal line once. Others in the range jul2<x<\ cannot intersect the diagonal line. We divide the range 0 < x < ju/2 into 0 < x < 1/2 l/2<x<ju/2. The original line segent in the range 0 < x < 1 / 2 is an Z, so the nuber of Z-type line segents in B^n\ju) in this range is L(n) - F n. Next, the original line segent in the range l / 2 < x < / / / 2 i s a n A S '. After one iteration, S-^ L; the L in the right-h side, after n -1 iterations, generates all the Z-type line segents in B^n\ju) in this range, the nuber of which is therefore L{n -1) = F n _ l. In all, the total nuber of intersections of line segents of types Z S with the diagonal line is thus F n _ l + F n + F n _ x - F n+l + F n _ v We conclude that 120 [MAY

4 OBTAINING DIVIDING FORMULAS n\q(ri) FROM ITERATED MAPS N z (n) = F n+l + F _, = F + 2F n _ v (4.3) N z (ri) is, in fact, the Lucas nuber L n. Fro (3.6) (3.7), we have n\n{n), with#(») = X / < " / ^ + i + ivi) (4.4) n\{f n + x + F n _ x -X), for n a prie. (4.5) Forula (4.5) Is also a known result [6]; however, we see that it is traceable fro the point of view of iterated aps. For n a coposite nuber, (4.4) offers additional relations for Fibonacci nubers or Lucas nubers. This result sees to be a new one. Consider, as a siple exaple, taking n = 12; since #(12) = 300, we can easily check that The Case In Which JC = 1/ 2 Is a Perlod-#i Point We now consider the general case in which x = 1 / 2 is a period-/w point. In general, there are any values of ju in the range 0 < ju < 2 such that x = 1 / 2 is a period- point. If there are k such < paraeter values, we denote these by fi l < i 2 M3<"' <Mk =^- We will choose the largest one as the paraeter value, i.e., ju = Z. It follows that E w satisfies the equation: -\ / / " - I / / = ( ). (4.6) ;=0 For instance, we have E 3 = 1.839, E 4 = We refer the reader to [4] for details. Briefly, we see that, starting fro x 0 = 1/2, we have an w-cycle, {x 0, x 1?..., x _ x }, where x t = f^(x 0 ) is the I th iterate of x Q. It is convenient also to define x = x 0. We have, for instance, x x = (1/2)//, fro (4.6) we have x l = (1 /2)(1 +1 / /i +1 / /i 2 + +!/ ff 1 " 1 ). We list all the values of x t in this way: x 1 -(l/2)(l + l//i + l/ / i l//i w - 1 ), x 2 = (l/2)(l + l/ / i + l/ / / l/// w - 2 ), C 2 = (l/2)(l + l/// + l//i 2 ), x _ x = (l/2)(l-fl ///), x = x 0 = l/z We see that x x > x 2 > x 3 > > x ^ > x. It follows that there are types of line segents. We denote by L t the type of line segents connecting points (x a,0) (x^x,), where 0<x a < x b <l 1 < / <. The behavior of line segents of these types after one iteration is Z^-^JLJ + 1,2, ( 4 '? ) ^2~^A + ^3> M g ) L _ l > L l + L y 4, - > p (4-9) 1998] 121

5 OBTAINING DIVIDING FORMULAS n \Q{n) FROM ITERATED MAPS We note that L (or L 0 ) is the only type of line segent that does not break into two line segents after one iteration. The original graph of B(ju) contains two L x. Equations (4.8) (4.9) show how the nuber of line segents increases under the action of iteration. Let L t {n) be the nuber of line segents of type L t in B [n] (ju), then (4.8) (4.9) show that L 2 (n) = L l (n-l), L 3 (n) = L 2 (n-l) = L l (n-2x L 4 (n) = L 3 (n-l) = L l (n-3x or especially, A» = L -M ~ 1) = L i( n - + l), Z /.(/i) = i /._ 1 (w-l) = Z 1 (w-7 + l), for2<j<, (4.10) It follows that all these L t {n) are sequences of the following type: AW = SA("-.A l^iz, (4.12) i.e., each eleent of which is the su of its previous eleents. Starting with an Zj, according to (4.8) (4.9), the orbit of L x is L x -» L X L 2 -> 2L X L 2 L 3 ->. We see that L x {n) = F^\ the Fibonacci nubers of degree, whose definition is with the first eleents defined by Conventionally, we define Fj ) = 0 for j < 0. pirn) _ V^ p{) F x () = l F^ = 2 l ~ 2 for 2 <i<. (4.13) We conclude that, starting fro an Z 1? we have L x (n) - F^), in general, L t {ri) = F $ +l, where \<i<. In order to discuss the nuber of intersections of these line segents with the diagonal line, we need to know, starting fro an Z a -type line segent, how any Z^-type line segents there are in B^n\ju). We let L b (n, L a ) represent the nuber of Z^-type line segents generated after n iterations of a starting line segent L a. Using this notation, we have we have the following theore. -a Theore 4.2: 4(w, L a ) = Y* F n-b-s> w h e r e L b (n,l x ) = F^+l, (4.14) l<a,b<. Proof: We will prove this theore by induction. We start fro a line segent L a calculate the nuber of Z^-type line segents generated after n iterations of L a. Consider first L a = L. To calculate Z^w, Z ), we note that after one iteration we have L -> L x. The L x in 122 [MAY

6 OBTAINING DIVIDING FORMULAS n \Q(ri) FROM ITERATED MAPS the right-h side, after n-\ iterations, generates all the Z^-type line segents in B^n\/u) in this range, the nuber of which is L b {n - 1, Z x ). Using (4.14), we conclude that -a L b (n,lj = L b (n-l,l l ) = F^ = Y. F n ( -L, w h e r e a s. (4.15).9=0 Consider next L a = L _ l. After one iteration, L _ x -> Ly + L. The L x L in the right-h side, after n-1 iterations, generate all the 4 -type line segents in B [n \/u) in this range, the nuber of which is I^in -1, Z^ + L b (n -1, L ). Using (4.14) (4.15), we conclude that L b (n,lj = l b (n-l,l 1 ) + L b (n-l,l ) -a (A 15) = 4 ( -l } +4 ( -ti = l 4 ( ; - t s, wherea = / W -l. ^ We can now prove the general case by induction. Consider L a = Z^., suppose that s=0 -a L b (n, 4 ) = I4 ( -"L, wherea = -/. (4.17) Now consider L a = Z ^ ^. After one iteration, Z ^ ^ -> Ly + L^. The Zj Z w _? in the righth side, after n-\ iterations, generates all the Z^-type line segents in B^n\ju) in this range, the nuber of which is 1^ (n -1, Zj) + Z^ (n - 1, Z w _ y ). Using (4.14) (4.16), we conclude that L b (n, L a ) = 4(/i - 1, Zj) + 4(/i - 1, V / ) = 4 ( -t + l 4 ( - t, - i = X4 ( -"L, wherea = i «- / - L Q.E.D. With Theore 4.2 established, we can now discuss the intersections of line segents of these w-types in B^n\fj) with the diagonal line. We consider line segents of type Z^ in B^n\fi). All these line segents are parallel can be divided into two parts; one part is in the range 0 < x < x b, the other is in the range x b < x < 1. We easily see that each line segent of those in the range 0 < x < x b intersects the diagonal line once others in the range x b < x < 1 cannot intersect the diagonal line. We divide the range 0<x<x b into 0 < x < l / 2 l/2<x<x b. Consider first the range 0 < x < 1/2. The original line segent in this range is an L x. The nuber of line segents of type L b in B [n] (ju; x) in this range is L h (n, Ly) = F^+l. Consider next the range 1 / 2 < x < x b. The original line segent in the range 1 / 2 < x < x b is Z^+1. The nuber of Z^-type line segents in B [n] (ju) in this range is L b (n, L b+1 ) - H =o~ l Fn-b-s- Therefore, the total nuber of intersections of these w-type line segents with the diagonal line is b=\ 1 n-b+l T -b-l Z**J s=0 n-b-s -b-l b=\ b=\ s=0 -\ -\ -\ = M *#S = (* + l) (4.19) b=0 k=l fe=0 sz$">. (4.20) 1998] 123

7 OBTAINING DIVIDING FORMULAS n \Q(ri) FROM ITERATED MAPS /, defined above, ay be called the Lucas nubers of degree, whose definition is 4 W) = X 4-y > w i t h t h e first eleents defined by 4 w ) = 2 i - l, \<i<. (4.21) We then have the final results: N z (n) = L n (), (4.22) w tf(«), withiv(^)-x^/rf )4 W) ' ( 4-23 ) w (4 w) -1), for n a prie. (4.24) 5. CONCLUSIONS Many N(n) such that n\n(n) can be obtained in this way for other iterated aps. In principle, infinite N(n) can be obtained, since each iterated ap contributes an N(n). It sees that the existence of dividing forulas is not so rare not so ysterious. ACKNOWLEDGMENTS I would like to thank the anonyous referee for a very thorough reading of the anuscript for pointing out soe errors. I also thank Professors Leroy Jean-Pierre, Yuan-Tsun Liu, Chih-Chy Fwu for stiulating discussions offering any valuable suggestions. I would also like to express y gratitude to the National Science Council of the Republic of China for support under NSC Grant Nos. 82~ REFERENCES 1. D. Gulick. Encounters with Chaos, pp Singapore: McGraw-Hill, Inc., Loo-Keng Hua. Introduction to Nuber Theory, Ch. 6. Berlin-Heidelberg-New York: Springer- Verlag, Chyi-Lung Lin. "On Triangular Baker's Maps with Golden Mean as the Paraeter Value." The Fibonacci Quarterly 34.5 (1996): Chyi-Lung Lin. "Generalized Fibonacci Sequences the Triangular Map." Chin. J. Phys. 32 (1994): E. Ott. Chaos in Dynaical Systes, p. 29. New York: Cabridge University Press, A. Rotkiewicz. "Probles on Fibonacci Nubers Their Generalizations." In Fibonacci Nubers Their Applications 1: Ed. A. N. Philippou, G. E. Bergu, & A. F. Horada. Dordrecht: D. Reidel, W. Sierpinski. Eleentary Theory of Nubers, Ch. 4. Warsaw: PWN-Polish Scientific Publishers, AMS Classification Nubers: 11B83, 11B39, 11A [MAY