MARIA PIRES DE CARVALHO
Chaotic Newton's
Sequenccs
s a route to ever m o r e exact knowledge, successive a p p r o x i m a t i o n has been
a m a j o r theme i n the development o f science. M a n y algorithms to f i n d ap-
p r o x i m a t i o n s of roots o f equations were devised. I n all such reasonings w e
begin w i t h an idea o f where the root lies, albeit less than accurate, and w e have
a s t r a t e g y to i m p r o v e the e s t i m a t e s . To l o o k up "whale" in a dictionary, t h e first s t e p is to o p e n the d i c t i o n a r y c l o s e to the end, b e c a u s e y o u h a v e a r o u g h i d e a w h e r e the w o r d is; next, y o u t u r n the p a g e s b a c k w a r d o r f o r w a r d till y o u f'md it, a n d this is the s t r a t e g y to i m p r o v e the first a p p r o x - imation. In t h e s e a r c h for z e r o s o f functions, y o u n e e d t o k n o w t h a t a zero e x i s t s a n d h o w t h e m a p b e h a v e s in t h e n e i g h b o r h o o d o f t h a t zero.
N e w t o n f o r m u l a t e d a g e n e r a l a n d s i m p l e m e t h o d to find a p p r o x i m a t i o n s o f z e r o s o f functions. F o r a r e a l (or c o m - p l e x ) f u n c t i o n f w i t h a z e r o at ~, a n d an initial c h o i c e x0, N e w t o n s u g g e s t e d the following r e c u r r e n c e f o r m u l a to ob- tain b e t t e r a p p r o x i m a t i o n s o f ~:
f ( X n ) X n + l = Xn f ' ( X n ) '
w h i c h is d e f i n e d if the derivative o f f v a n i s h e s at n o Xn, a n d
which, if convergent, will s u r e l y p i c k up a z e r o o f f as its limit. Given x0, the t e r m x~ is o b t a i n e d b y c o n s i d e r i n g t h e t a n g e n t line at (x0, f ( x o ) ) to the g r a p h o f f and i n t e r s e c t i n g it w i t h t h e r e a l axis; to get the w h o l e sequence, j u s t i t e r a t e this p r o c e s s . Sufficient c o n d i t i o n s for the m e t h o d to w o r k a r e e a s y t o state, b u t a m a j o r p r o b l e m raises: the c o m p e t i - tion a m o n g t h e s e v e r a l z e r o s o f t h e function. As a c o n s e - quence, t h e b a s i n o f a t t r a c t i o n o f e a c h zero (that is, t h e s e t o f initial c o n d i t i o n s x0 s u c h t h a t the c o r r e s p o n d i n g se- q u e n c e (x**)n E ~o c o n v e r g e s to the s p e c i f i e d zero) m a y h a v e a v e r y c o m p l i c a t e d b o u n d a r y , a n d the d y n a m i c s as- s o c i a t e d to t h e s e q u e n c e s (xn),~ E ~o m a y b e highly sensi- tive to p e r t u r b a t i o n s on initial c o n d i t i o n s . T h e s e b o u n d -
a r i e s have b e e n a favorite s h o w p i e c e in p o p u l a r i z i n g frac- tals ( s e e for i n s t a n c e [DS]).
But here I will f o c u s on a n o t h e r p r o b l e m . W h a t h a p p e n s if a m a p f : ~ - - ) ~ h a s no real zeros? N e w t o n ' s s e q u e n c e s
(Xn)n ~ ~o m a y b e defmed, although t h e y will n e v e r converge. H o w do t h e s e s e q u e n c e s b e h a v e ? I will exmnine here the particular c a s e o f the quadratic family x ~ ~ ~-->fc(X) = x 2 + c, w h e r e c is a real positive parameter. The n a t u r a l exten- sion to C o f e a c h m a p o f the family h a s the real line R • {0} as the b o u n d a r y of the b a s i n s o f a t t r a c t i o n o f its t w o (com- plex) roots, so its g e o m e t r y is trivial. However, the s e q u e n c e s
(Xn)n ~ % s h o w irregular and u n p r e d i c t a b l e behavior, w h i c h n e v e r t h e l e s s h a s an underlying o r d e r t h a t I will describe.
A f t e r a c l e v e r c h a n g e o f variable, a n a l y s i s o f the se- q u e n c e s (xn)n E % will b e s t r a i g h t f o r w a r d b y a p p e a l i n g to s o m e e a s y t e c h n i q u e s a n d r e s u l t s f r o m d y n a m i c a l s y s t e m s and e l e m e n t a r y n u m b e r theory. The m a i n r e s u l t is t h a t ra- tional initial c o n d i t i o n s p r o d u c e finite o r infinite p e r i o d i c sequences, w h e r e a s t h e i r r a t i o n a l o n e s y i e l d infinite b u t n o t p e r i o d i c s e q u e n c e s . This r e c a l l s w h a t h a p p e n s w i t h deci- mal or b i n a r y e x p a n s i o n s (luckily, e v e n t h e t e r m i n o l o g y is the same), a n d t h e sensitivity with r e s p e c t to the initial c h o i c e x0 is e v i n c e d at once. Moreover, t h e d y n a m i c s as- s o c i a t e d with t h e s e s e q u e n c e s is m o d e l e d b y a left shift on t h e b i n a r y r e p r e s e n t a t i o n o f x0 in the n e w variable.
Let m e s t a r t b y t a k i n g a b r i e f t o u r o f d i s c r e t e d y n a m i c a l systems. Given a m a p G : X--~ X, I m a y c o m p o s e G with it- s e l f as m a n y t i m e s a s I p l e a s e (the n - f o l d c o m p o s i t i o n o f G w i t h itself is d e n o t e d b y Gn). T h e r e f o r e for e a c h x in X the
b i t o f x b y G. The set of all orbits is a d y n a m i c a l s y s t e m . D y n a m i c a l systems form a category in which a n isomor-
p h i s m b e t w e e n two d y n a m i c a l systems f : X - - > X a n d
g : Y--~ Yis given by a h o m e o m o r p h i s m h : X---) Y s u c h that g o h = h o J} such a n h is called a c o n j u g a c y b e t w e e n f a n d g. Essentially, the aim of the t h e o r y is to know, u p to con- jugacy, the asymptotic behaviors of each orbit a n d h o w they vary with x. The fixed p o i n t s are the orbits easier to d e t e c t a n d t h e ones to look for first; m o r e generally, a n o r b i t is
p e r i o d i c w i t h p e r i o d p ~ ~ ff it is a f l i e d p o i n t o f G P ; if n o t h i n g is said to the con-
trary, p is u n d e r s t o o d to be the smallest period. An or-
bit is p r e - p e r i o d i c w i t h
p r e - p e r i o d n E No a n d pe- r i o d p E ~ ff G n ( x ) is a
fixed p o i n t for Gp.
F o r m a p s G defined o n s u b s e t s of ~, the c o m p o s i t i o n of G with itself m a y be p i c t u r e d o n the graph of G, a n d this is a good way of guessing h o w the orbits behave. F o r in- stance, c o n s i d e r G : [0, 1] ---> [0, 1] given by G ( x ) = 1 - x. T h e n G ( x ) = x if a n d only if x = 89 for this is the only in- t e r s e c t i o n of the graphs of G a n d the identity map. If x r
• t h e n G2(x) = G(1 - x) = x, so the orbit of x is periodic 2'
with p e r i o d 2: I suggest y o u c h e c k this o n the graph of G. The orbits m a y p r e s e n t m a n y differences with r e s p e c t to their topological properties, asymptotic behavior, or car- dinality of their range of values. There are d y n a m i c a l sys- t e m s that c o n t a i n essentially all the kinds of orbits t h a t n o n - injective m a p s m a y be e x p e c t e d to have. O n e such s y s t e m is b a s e d o n the space of s e q u e n c e s c o n s t r u c t e d with the
digits 0 a n d 1, say Z = {0, 1} ~ = { ( a l , a2, " 9 9 a n , " " " ) : aj E
{0, 1}}, with the metric
~. laj - ~'jl
D ( z , w ) = ~ 2J 'j = l
f o r z = (at, a2, 9 9 9 a n , 9 " " ) a n d w = (bl, b2, 9 9 9 b n , " " " ) .
Acting o n ~, the one-sided full shift m a p (r takes each se- q u e n c e (al, a2, 9 9 9 an, 9 9 9 ) to (a2, 9 9 ", an, 9 " " ). T h i s m a p is c o n t i n u o u s with r e s p e c t to the above metric; it h a s peri- odic p o i n t s of all periods, because, for each p @ ~J,
GP(at, a2, " 9 ", ap, at, a2, " 9 ", ap, " . . . ) =
(at, a2, 9 9 ", ap, at, a2, 9 9 ", ap, 9 . . . );
a n d it has dense orbits (e.g., that of the e l e m e n t of Z that is o b t a i n e d by writing d o w n c o n s e c u t i v e l y all p o s s i b l e fi- n i t e b l o c k s of digits 0 or 1 o r d e r e d b y their l e n g t h - - s e e [D] for m o r e details). I will c o n s i d e r each e l e m e n t of Z as a bi- n a r y e x p a n s i o n of a n u m b e r in [0, 1]; in this process, the fi- nite b i n a r y r e p r e s e n t a t i o n (of each dyadic rational) is t h o u g h t of as having a n infinite tail of zeros: thus, 0.01(2) is the e l e m e n t of ~ given b y 01000000 9 9 9 a n d is distinct, in Z, f r o m 00111111 9 9 9 although they are e x p a n s i o n s of the s a m e n u m b e r .
In e x p a n s i o n of the real n u m b e r s in a given base b, each n u m b e r is replaced by a s e q u e n c e a N ' ' ' a o ' e t c 2 c 3 ' ' '
Ck . . . (b) with aj, Ck in {0, 1, 9 9 9 b - 1}, m e a n i n g t h a t
the n u m b e r is given by the s u m
Rational numbers have
finite or infinite periodic
representations in any base.
a N ( b ) N - l + " ' ' + a l b + a o + C ( 1 ) +
9 . . § § . . .
It will be f o u n d useful to discard the i n t e g e r part a n d keep i n f o r m a t i o n only a b o u t the digits Ck. Rational n u m b e r s have finite or infinite periodic r e p r e s e n t a t i o n s in a n y base, in general n o t unique; irrationals a p p e a r as u n i q u e non-peri- odic infinite r e p r e s e n t a t i o n s . To simplify the notation, a pe-
riodic s e q u e n c e of Z, say (at, a 2 , ' ' ' , ap, al, a 2 , ' " ,
a p , . . . . ), will be de- n o t e d b y a l a 2 " " a p , and similarly a pre-periodic b i n a r y r e p r e s e n t a t i o n 0 . a l a 2 9 a n a n + l a n + 2 " a n + p a n + l
an+2 " " " an+p 9 9 9 .. 9 will be abbreviated to 0.ala2 9 9 "an an + l an+2 9 " " an + p.
When a r a t i o n a l n u m b e r is w r i t t e n in irreducible form, i n f o r m a t i o n o n its e x p a n s i o n in a given b a s e c a n be read from the d e n o m i n a t o r only. In the case b = 2 it is k n o w n that (see [RT]):
(I) A r a t i o n a l ro E ]0, 1[ h a s f i n i t e b i n a r y r e p r e s e n t a -
t i o n i f a n d o n l y i f i t i s d y a d i c ; t h a t is, i t m a y be w r i t t e n a s ro = k / 2 n w h e r e k, n E ~ a n d k i s odd.
In this case the (finite) r e p r e s e n t a t i o n o f t 0 has precisely n digits.
(II) A r a t i o n a l ro ~ ]0, 1[ h a s i n f i n i t e b i n a r y r e p r e s e n -
t a t i o n w i t h a p e r i o d t h a t s t a r t s j u s t a f t e r the deci- m a l p o i n t i f a n d o n l y i f it i s a n i r r e d u c i b l e f r a c - t i o n t/q w h e r e q i s odd.
F u r t h e r m o r e , the length of the p e r i o d does n o t e x c e e d 4)(q), w h e r e 4) is the mulet t o t i e n t f u n c t i o n (for each q E N, 4)(q) is the n u m b e r of positive integers less t h a n q a n d co-prime to q); in fact, it divides #)(denominator) a n d is in- d e p e n d e n t of the n u m e r a t o r . (For instance, 1/5 = 0.0011(2) has period 4 = 4)(5) a n d 1/13 = 0.000100111011(2) has pe- riod 12 = 4)(13).) (III) The d e n o m i n a t o r i s e v e n b u t n o t a p o w e r o f 2 - - t h a t is, ro = t/2nQ, a n i r r e d u c i b l e f r a c t i o n w h e r e Q i s odd a n d n i s a p o s i t i v e i n t e g e r - - i f a n d o n l y i f the bi- n a r y r e p r e s e n t a t i o n i s i n f i n i t e p r e - p e r i o d i c w i t h a p r e - p e r i o d n. F o r example, 1/(2 9 5) = 0.00011(2) h a s period 4 as 1/5 a n d pre--period 1.
Cases (II) a n d (III) merit closer i n s p e c t i o n :
( I V ) I f a n i r r e d u c i b l e f r a c t i o n o f p o s i t i v e i n t e g e r s t/q E
]0, 1[ h a s a n odd d e n o m i n a t o r , it m a y be expressed
i n the f o r m s / ( 2 P - 1) w h e r e s a n d p are p o s i t i v e i n -
tegers a n d a r e m i n i m a l . O n c e t h i s i s achieved, p g i v e s the length o f the p e r i o d o f i t s b i n a r y r e p r e s e n t a t i o n .
F o r example,
1 3 1 5 x 6 3
5 - 24 - 1 - 0.0011(2); 13 - 2 t2 - 1 = 0.000100111011(2).
(V) I f the f r a c t i o n t/q has a n even d e n o m i n a t o r w h i c h is n o t a p o w e r o f 2 - - t h a t is, t/q = t/2nQ w i t h n E a n d Q o d d - - i t m a y be expressed i n the f o r m s/2n(22
- 1) w h e r e n, s, a n d p are p o s i t i v e integers, m i n i -
mal, a n d p is greater than 1. The integer p is the length o f the period o f the b i n a r y representation o f t/q, a n d n is the pre-period.
F o r e x a m p l e 1/12 = 1/(2 2 (2 2 - 1)) = 0.0001(2).
Let m e s k e t c h a p r o o f of these t w o properties. (V) im- plies (IV) if n is also allowed to b e zero; to prove (V), con- sider the fraction 1/Q a n d the e q u a t i o n s that p r o d u c e its bi- n a r y e x p a n s i o n :
1 = Q x 0 + I
2 X 1 = Q x d l + r l 0 < r i < Q
2 X r l = Q x d 2 + r 2 0 < r ~ < Q
As the r e m a i n d e r s rj are positive integers less t h a n a n d co-
p r i m e to Q, they r e p e a t t h e m s e l v e s after r steps, at the
most. The first r e m a i n d e r to r e a p p e a r is precisely 1 be- cause, by (II), the b i n a r y r e p r e s e n t a t i o n of 1/Q has a p e r i o d that starts j u s t after the decimal point. Therefore there ex-
ists a positive i n d e x p s u c h that
rp
= 1, a n d so the last ofthe above equations, before they start repeating, is 2 x
rp 1 " ~ Q x dp + rp = Q x dp + 1. Multiply the s e c o n d
e q u a t i o n b y 22-1, the third o n e b y 22 -2 a n d so on, a n d a d d t h e m all to get 22 = Q [22 1 dl + 22-2 d2 + . . . + 2dp-1 + dp] + 1. Therefore 1 [22-i dl + 22-2 d2 + " 9 9 + 2 d p - i +dp] A Q 2 2 - 1 2 2 - 1 ' SO t A t Q 2 2 - 1 ' t A t s 2nQ 2 n (22 - - 1) 2 n (22 -- 1)"
Further, the type of the binary representation of s/(2 n (22 - 1)) is the s a m e as that of 1/(2 n (22 - 1)), a n d the latter m a y b e o b t a i n e d from the following calculation:
1 1 1/22
2 n ( 2 2 - 1) 2 n 1 - 1/22
: o . o . . , o o o . . .
where the first b l o c k of zeros h a s size n a n d the repeating b l o c k has p - 1 zeros followed b y a single 1. The integer s m a y c h a n g e the digits b u t n o t the m e a n i n g of n a n d p. No- tice that if the d e n o m i n a t o r is even b u t n o t a p o w e r of 2, t h e n p m u s t b e bigger t h a n or equal to 2. The effect of the p o w e r 2 n in the d e n o m i n a t o r is to p u s h the period to the right, creating a pre-period of length n. I suggest y o u c h e c k this o n s o m e examples, such as
1 1 9 9
14 - 2(2 3 - 1) - 0.0001(2); 14 - 2(2 3 - 1) - 0.1010(2);
1 1
28 - 2 2 (2 ,3 - 1) - 0.00001(2). Let m e s u m m a r i z e for later use:
r0 E Q ~ r0 has a unique representation, infinite, non- periodic
3k, n E ~ : ro = k / 2 ~' ~ h a s a finite b i n a r y
r e p r e s e n t a t i o n that t e r m i n a t e s at 0 after n
r 0 E Q ~ digits
3k, n E N 3p E N 0 : r 0 = k/(2P(2n - 1)) u n i q u e binary representation with pre-period p a n d period n
It is time to go b a c k to N e w t o n ' s m e t h o d a n d t h e m a p f l . If I start with a n initial c o n d i t i o n x0 E ~, t h e n the cor- r e s p o n d i n g N e w t o n ' s s e q u e n c e (x~)nE~0, if well defined, is real a n d thus c a n n o t converge: if it did, t h e r e c u r r e n c e for-
m u l a Xn+t = (Xn 2 - 1)/2xn w o u l d imply t h a t t h e limit L E
0~ verifies the i m p o s s i b l e e q u a t i o n 2 L 2 - L 2 - 1. The dy-
n a m i c a l s y s t e m a s s o c i a t e d with this r e c u r r e n c e f o r m u l a m a y be d e s c r i b e d b y the iterates of the m a p q3 : R ---) E, q3(t r 0) = (t 2 - 1)/2t, ~J(0) = 0. If well defined, the se- q u e n c e (xn)n E ~0 is the orbit b y q3 of x0; however, o n c e a n orbit of ~ lands o n the fixed p o i n t 0, it s t o p s b e i n g a New- t o n ' s sequence. The m a p u5 is a n odd f u n c t i o n , i n c r e a s i n g in ]-0% 0[ a n d in ]0, +oo[, a n d is a s y m p t o t i c to the line y =
x/2. It is easy to identify s o m e orbits b y o b s e r v i n g the graph
of q3:
(1) Consider x0 = l; t h e n q3(x0) = 0, so q3n(x0) = 0 for n --> 1; Xn is n o t d e f i n e d for n -> 2. I describe this b y saying that the orbit of 1 is f i n i t e a n d t e r m i n a t e s at 0 after
one iterate.
(2) If x0 = 1 + ~/2, t h e n q3(x0) = I a n d q32(x0) = 0, so
~n(xo) = 0 for n --> 2 although xn is n o t d e f i n e d for n ->
3. This orbit is also finite a n d t e r m i n a t e s at 0 after two iterates.
(3) Take n o w x0 = 1P~/3; t h e n q 3 ( x 0 ) = - 1 / ~ / 3 a n d q32(x0) = 1/~/-3. This is a periodic orbit of p e r i o d two.
The equality q~2(X) = X leads to a p o l y n o m i a l equation
o f degree 4 with o n l y even e x p o n e n t s ; it h a s n o solu- tions other t h a n 1/~/-3 a n d - 1 / ~ / 3 .
(4) If x0 = V ~ , t h e n qJ(x0) = 1/~/3 a n d q32(qJ(x0)) = qJ(x0). So x0 is a pre-periodic orbit of p e r i o d two a n d pre-pe- riod one.
More s o p h i s t i c a t e d tools are n e e d e d to d e t e c t other kinds of orbit. The r e c u r r e n c e f o r m u l a Xn+l = ((Xn) 2 - 1)/ (2xn) is similar to the t r i g o n o m e t r i c f o r m u l a cotan(20) =
( c o t a n 2 ( O ) - - 1)/(2 cotan(0)) for 0 E ]0, ~-[ / {7r/2}. Let x0 =
cotan(1rr0) for r0 E ]0, 1[: this is p e r m i s s i b l e since cotan: ]0, ~r[ ~ R is a h o m e o m o r p h i s m , a n d so the topological p r o p e r t i e s of the o r b i t s of q3 are p r e s e r v e d u n d e r this c h a n g e of variable. Moreover, in this n o t a t i o n , w e have q~n (X0) = c o t a n ( T r 2 n r 0 ) for each n, p r o v i d e d that 2n~rr0 is n o t a n integer multiple of ~r. The n u m b e r s in ]0, 1[ that fail
to satisfy this r e q u i r e m e n t for s o m e integer n are j u s t the dyadic rationals; m o r e precisely:
1st Conclusion: ro = k / 2 n, w i t h k, n E ~ a n d k a n o d d i n -
t e g e r , i f a n d o n l y i f t h e o r b i t b y ~ t e r m i n a t e s a t 0 a f t e r n i t e r a t e s .
Because k is odd, we have X n - 1 = cotan(Tr2 n-1 r0) =
cotan(~rk/2) = 0 a n d therefore xm is n o t d e f i n e d for m - n; so the orbit of x0 = cotan(Trr0) by ~ t e r m i n a t e s at the fLxed p o i n t 0 after n iterates. This is the case of r0 = 1/4 =
0.01(2), X0 = cotan(~-r0) = c o t a n 0 r / 4 ) = 1 a n d x l = 0. Con-
versely, if an orbit of u3 t e r m i n a t e s at 0, say ~ n ( X o ) = 0, t h e n
cotan(~r2nr0) = 0 a n d therefore there exists m E 2~ s u c h
that 2n~rro = m~r + ~T/2. So 2nro = m + 1/2, t h a t is, r0 =
( 2 m + 1)/2 n+ ~.
What real n u m b e r s r0 p r o d u c e periodic or pre-periodic or- bits by ~3? r0 c a n n o t be dyadic, and there m u s t be N a n d P
such that ~eoN+P (X0) = (ejV (X0); this implies that r0 r k / 2 n for
all integers k and n a n d c o t a n ( T r 2 g + P r o ) = cotan(~2Nr0).
Solving this equation, it is f o u n d that ro = k/2 N (2 p - 1) with
k ~ ~, N ~ ~0, P ~ N a n d P -> 2. These are the remaining ra- tionals of ]0, 1[ (see (IV) a n d (V) above): they have infinite periodic or pre-periodic b i n a r y expansions with period P.
2 n d Conclusion: The o r b i t o f Xo b y ~ i s f i n i t e o r i n f i n i t e
p e r i o d i c / p r e - p e r i o d i c i f a n d o n l y i f ro i s r a t i o n a l ; i f s u c h i s t h e c a s e , t h e n t h e o r b i t t y p e o f Xo i s c o m p l e t e l y d e t e r - m i n e d b y t h e d e n o m i n a t o r o f ro. I n p a r t i c u l a r , i f ro i s i r - r a t i o n a l , t h e n Xn i s d e f i n e d f o r all n ~ ~ .
Let m e review in this n e w setting s o m e of the a b o v e ex- amples.
(a) r0 = 1/3 = 1/(2 -2 - 1) = 0.01(2): then N = 0, P = 2, x0 = cotan(~-/3) = 1/~/3, a n d Xl = cotan(2~r/3) = - 1 / ~ / 3 . The orbit b y ~3 of x0 is periodic with p e r i o d P.
(b) r 0 = 1 / 6 = 1 / 2 ( 2 2 - 1 ) = 0 . 0 0 1 ( 2 ) : N = I , P = 2 , a n d
x0 = cotan(~r/6) = V 3 , x~ = cotan(2~r/6) = cotan(~-/3) = 1/~/3. The orbit of x0 is pre-periodic with pre-period N = 1 a n d p e r i o d P = 2.
(c) r0 = 1/5 = 3/(2 -4 - 1) = 0.0011(2):N = 0 , P = 4, a n d x 0 = cotan(rr/5), x~ = cotan(2~r/5), x2 = cotan(@r/5), xs = cotan(8~-/5), x4 = cotan(16~-/5) = x0. The orbit of x0 is periodic with p e r i o d P = 4.
I suggest y o u n o w c o m p a r e the following diagram with the s i m i l a r one above.
r0 ~ Q ~ its orbit b y ~3 is infinite n o n - p e r i o d i c
3k, n E ~ : ro = k / 2 n ~ its orbit b y ~3
ro E Q ~ t e r m i n a t e s at 0 after n iterations
3k, n E N 3p E No : r0 = k/2P(2~ - 1) ~ its orbit by u3 h a s pre-period p a n d p e r i o d n Thus the orbit of x0 by q3 is completely determined by the b i n a r y representation of r0. This also shows that the discrete dynamical system generated by q3 is highly sensitive to initial conditions: the distinction b e t w e e n rational and irrational r0 is e n o u g h to produce wide disparities b e t w e e n orbits.
Other m o r e p a r t i c u l a r traits of the orbits for irrational values of r0 c a n b e studied by p i c k i n g up two clues I left behind:
(1) the f u n c t i o n z ~ cotan(~rz) is periodic of period 1; (2) iterating x0 by ~3 corresponds, in the n e w variable, to simply d o u b l i n g the a r g u m e n t of the c o t a n function.
The first o n e implies that, w h e n y o u c o m p u t e the suc- cessive v a l u e s of cotan(~r2~r0), w h a t m a t t e r s is the frac- tional p a r t of 2n~b (denoted by {2nr0}). If the irrational r0 is
w r i t t e n in b a s e 2 a s ro = 0 . a l a 2 a 3 9 9 9 a k " " " (2), this rep-
r e s e n t a t i o n is unique, a n d 2ro = a l 9 a2a3 9 9 9 ak 9 9 9 (2). Dis-
missing the integer part, we are left with {2r0} =
0.a2a3 9 9 9 ak " " " (2) and, by (2),
( c o t a n ( 1 r 2 n r o ) )n E ~0 = (cotan([ rr2nro}) )n 9 ~o
= (cotan(~r 9 0 . a n + l an+2 9 9 " (2)))n 9 t~ 0,
which c o r r e s p o n d s , up to the a c t i o n of c o t a n o (Iv x 9 ), ex- actly to the i t e r a t i o n ~ n of the shift o n the s e q u e n c e
a l a 2 a 3 9 " 9 ak " " 9 9 More precisely, the m a p
]0, 1[ / [dyadic numbers} ~ --~ ]0, 1[ / [dyadic numbers}
O . a l a 2 . . . ak . . . (2) ~ O . a2a3 . ak . . . (2)
(that is, 0-(t) = 2t if 0 -< t < 89 ~ ( t ) = 2t - 1 if~" 1 < t < 1) is
c o n j u g a t e d b y z ~ cotan(rrz) to the a c t i o n of ~3 o n the set of x0 w h o s e orbits by c6 do n o t t e r m i n a t e at the fLied p o i n t 0 after a finite n u m b e r of iterates; a n d ff is the same as the shift m a p ~ restricted to the s e q u e n c e s of zeros or o n e s that are n o t e v e n t u a l l y constant, for t h e m a p
h(O . a l a 2 9 9 9 ak " 9 9 (2)) = a l a 2 a 3 9 9 9 ak . . .
is a c o n j u g a c y b e t w e e n the c h o s e n r e s t r i c t i o n s of ff a n d q. Let m e illustrate the use of these o b s e r v a t i o n s in t w o examples:
(i) If r0 = 0.10100100010000 . . . (2), where each digit
1 is followed b y a block of zeros of i n c r e a s i n g length, t h e n r0 is irrational a n d the s e q u e n c e (Xn)nE~ =
(cotan(~-2nro))ne~ = (cotan({ ~ 2 n r o } ) ) n 9 ~ is b o u n d e d
away from zero, because {2nr0} <0.1010010010 - 9 9 (2) = ~4 for all n. But, since [2nro} gets arbitrarily close to 0, this orbit is n o t b o u n d e d from above.
( i i ) If r0 is a n irrational n u m b e r w h o s e b i n a r y r e p r e s e n t a - t i o n is given by a s e q u e n c e in ~ with dense ~-orbit,
t h e n the c o r r e s p o n d i n g s e q u e n c e (Xn)n 9 ~o is d e n s e
in ~.
If for each dyadic n u m b e r of ]0, 1[ I select the binary rep-
resentation with ending zeros (e.g., writing 1/2 =
0 . 1 0 0 0 0 . - - ( 2 ) i n s t e a d of 0 . 0 1 1 1 - - . ( 2 ) ) , then the corre- sponding e x t e n s i o n of h is n o t continuous. However, if
I let ~ ( x ) = h((1/Tr)cotan- x), t h e n the equation ~r o Y~(x) = 1
o ~3(x) is still valid for all x r 0. This yields the following:
3rd Conclusion: The d y n a m i c s o f t h e N e w t o n ' s s e q u e n c e s
(Xn)n 9 %, f o r a l l o w e d r e a l i n i t i a l c o n d i t i o n s x0, i s d e t e r - m i n e d b y t h e b i n a r y r e p r e s e n t a t i o n s o f t h e i n i t i a l c o n - d i t i o n s i n t h e n e w v a r i a b l e ro.
I n o w p r o c e e d to c h e c k h o w t h e p a r a m e t e r c affects t h e p r e v i o u s calculations. I will s h o w t h a t the d y n a m i c s o f t h e c o r r e s p o n d i n g N e w t o n ' s s e q u e n c e s for p a r a m e t e r c is t h e s a m e as for c = 1 w h e n c > 0, a n d c h a n g e s d r a s t i c a l l y at c = 0 .
Let m e r e w r i t e c a s _+a 2, w i t h a E [0, § D e n o t e b y ~Sa t h e m a p a s s o c i a t e d to N e w t o n ' s m e t h o d a p p l i e d to f~, w h e r e ___ = sign(c): t h u s ~Sa(0) = 0, ~a+(X) = (x 2 - a2)/2x, ( ~ ( x ) = ( x 2 + a2)/2x. F o r a f i x e d sign _+, the family o f m a p s (~J~)a ~ 10, +~[ c o n v e r g e s p o i n t w i s e , b u t n o t uniformly, to
~o(X) = x / 2 as a --) O. The limiting d y n a m i c s is u n i n t e r e s t - ing: f o r all x0 E R, the s e q u e n c e ((~30)n(x0))~e~ has limit 0, the unique f i x e d - p o i n t o f ~50. If a > 0, t h e n for x r 0 w e h a v e X 2 - - a 2 ~ ( X ) - - - - - - a 2x t h a t is,
(x;,
a This s u g g e s t s t h e c h a n g e o f v a r i a b l e w h i c h l e a d s to and, in general, to X 0 t o = - - , a x i _ (t0) 2 - 1 t l - a 2t0(x;_,
(tn) 2 - 1 tn+l -- 2 t nThis m e a n s that, up to a c h a n g e o f variable, the m a p ~ a + a c t s as ~ = ~ , a n d no f u r t h e r w o r k is n e e d e d in this case.
If a > 0 a n d c = - a 2, t h e n f c h a s t w o real zeros, a a n d - a , w i t h b a s i n s o f a t t r a c t i o n given b y ]0, +oo[ a n d ] - % 0[, r e s p e c t i v e l y . In fact, the m i n i m u m value o f ~ a ( X ) =
x 2 + a2/2x for x > 0 is a, w h i c h is also the u n i q u e f i x e d p o i n t of~ga in ]0, +oo[; and, s i n c e ~Jalla, +~176 is a c o n t r a c t i o n , it f o l l o w s that, for all initial c h o i c e s x0 > 0, t h e s e q u e n c e
(Xn)n c o n v e r g e s to a. S i m i l a r r e a s o n i n g s h o w s t h a t (Xn)n
c o n v e r g e s to - a for all c h o i c e s x0 < 0. It is along the imag- i n a r y a x i s t h a t the d y n a m i c s o f (6a is chaotic: for, if x0 = ip0 for s o m e P0 E R / {0}, t h e n N e w t o n ' s r e c u r r e n c e f o r m u l a
Xn+i = (Xn 2 + a2)/2 x n b e c o m e s
(i pn) 2 + a 2 _ i ( P n ) 2 - a 2 ipn+ l I
2 ipn 2pn
This m e a n s that, in t h e r e a l v a r i a b l e p, the d y n a m i c s is given b y P . + I = u3+ (Pn), w h i c h h a s a l r e a d y b e e n analyzed.
It is w o r t h r e m a r k i n g t h a t t h e c o n c l u s i o n s o b t a i n e d for
the q u a d r a t i c f a m i l y (fc)c e x t e n d e a s i l y to all q u a d r a t i c poly- nomials. Given a p o l y n o m i a l p ( x ) = d2 x2 + d i x + do, with
dj E R a n d d2 • 0, the equation p ( x ) = 0 is equivalent to
p ( x ) / d 2 = 0, a n d s o I m a y a s s u m e t h a t d2 = 1. By a s i m p l e t r a n s l a t i o n in t h e v a r i a b l e x, given b y x = t § di/2, p be- c o m e s
p ( t ) = t 2 + [do - d2/4],
w h i c h b e l o n g s to t h e family (fc)c. H e n c e all the p r e v i o u s r e s u l t s h o l d for this l a r g e r family.
Acknowledgments
My t h a n k s to P a u l o Arafijo for his h e l p in i m p r o v i n g t h e text.
REFERENCES
[D] Devaney, Robert L. An Introduction to Chaotic Dynamical Systems, 1989, Addison Wesley.
[DS] Devaney, Robert L., Keen, Linda (Editors). Chaos and Fractals:
The Mathematics Behind the Computer Graphics, Proceedings of Symposia in Applied Mathematics, Vol 39 (1989), American Mathe- matical Society.
[P] P61ya, George. MathematicatMethods in Science, 1977, The Math- ematical Association of America
[R-I-J Rademacher, Hans, and Toeplitz, Otto (H. Zuckerman, translator).
