Spirals in the Mandelbrot set I

North-HollandPhysica A 205 (1994)634-645 ~ I S E I ~1

SSDI 0378-4371(93)E0412-8

Spirals in the Mandelbrot set I

J o h n S t e p h e n s o n Physics Department, University of Alberta, Edmonton, Alberta, Canada T6G 2J1

Received 11 August 1993

An explicit function is constructed to permit easy calculation of the asymptotic structure of the various spirals and branches around the main cardioid in the Mandelbrot set. Details of its application to "exterior" spirals are presented.

1. Introduction

The simplest non-linear iterative process is the quadratic map defined by the recurrence relation

Z m + l = R ( Z m ) = Z 2 + C , m = 0 , 1 , 2 . . . . . (1)

For real values of the parameter c, (1) is equivalent to the logistic map, and exhibits bifurcation leading to the onset of chaos [1,2]. Those complex values of c for which iterates starting at the origin remain bounded form the Mandelbrot set (M-set) [3,4]. Those parts of the Mandelbrot set originating in cycles of finite order n, so z n = z0, lie inside smooth figures such as cardioids and attached buds. I presume the reader is familiar with the general appear- ance of the Mandelbrot set, and has a copy of "Peitgen and Richter" (PR) [4] available for reference to the MAPs. My motivation was to obtain an arithmetic and analytical explanation of the "great spiral" and "giant tentacle" on the branch shown in MAP 36 in PR. More generally, one has to sort out the structure of the branches and embedded spirals which constitute the cyclic part of the Mandelbrot set (e.g. MAPs 27, 30, 38, 42 in PR).

The scaling properties of the Mandelbrot set, in the vicinity of preperiodic points and near Myrberg points, which terminate the Feigenbaum bifurcation sequences, have been studied previously by various authors: in particular Guckenheimer [5], Eckmann and Epstein [6], Douady and Hubbard [7], Tan Lei [8] and Milnor [9] have established the existence of asymptotic scaling for

0378-4371/94/$07.00 © 1994-Elsevier Science B.V. All rights reserved

J. Stephenson / Spirals in the Mandelbrot set I 635

rather general classes of quadratic maps. My aim is to obtain explicit " formulae" for the various spirals, so one can extract arithmetic results quite easily.

In this paper, I obtain a function (alpha function) which permits one to calculate the (asymptotic) properties of the various spirals and branches which terminate in a preperiodic point of cycle order one. First the arithmetic band structure of the (numerical) centres of cycles is examined in relation to the allocation of levels and sign sequences for centres along the "branches" emerging from the dominant "buds" attached to the left-hand side of the main cardioid. The "multiplier" and "ampli tude" (for a particular series of centres) of the "external" spiral along the top-most branch are obtained numerically. A derivation of the alpha function, which determines to leading order the asymptotic properties of the spiral, is presented, and the exact multiplier and amplitude are extracted. The zeros of the alpha function supply the amplitudes for the components of the spiral formed from the series of centres, of increasing cycle order, associated with various sets of levels. (Decimal numbers in this paper have been t r u n c a t e d - n o t rounded.)

2. Branch 3: MAP 27

It is simplest to start on branch 3 which emerges from the n = 3 cycle bud at the top of the main cardioid as in MAP 27 in PR. This branch extends to a " t ip" or limit point at c o = ( -0 .228, 1.115) in the complex c-plane. One calculates the values of c corresponding to the centres of cycles of increasing order n = 3, 4, 5, 6, 7, 8 , . . . , and selects those in the vicinity of this limit point, as in table I. The positions of the centres are located (PR) by the values of c for which the origin is a member of the cycle in question. Thus the centres are

n _ _ n - 1 n the zeros of the polynomials R c(0) = R c (c) = 0, where R c is the nth iterate, which can be constructed algebraically. The centres are arranged in conformity with a level scheme determined by the sign sequence {s,,} associated with the inverse cycle. At level h s m = sgn{cos[(l - ½)rr/2m]). In practice one calculates centres by iteration. For a particular choice of level l and some initial value of c, one iterates the entire inverse cycle. This method was used by Myrberg [10] for real centres. It seems to work generally quite well for getting rough values of c at complex centres. These values can then be refined by iteration of the (forward) cycle using a form of Newton's method.

One observes that at each cycle order n there are up to 2 n- 1 centres of which roughly ~th are clustered in a "band" terminating at the limit point c 0. This composite band contains up to [2" 2 (_)n] /3 centres and consists of "sub- bands" of centres located on the various branches emerging from the odd order

6 3 6 J. Stephenson / Spirals in the Mandelbrot set I

T a b l e I

C e n t r e s , l e v e l s a n d s i g n s e q u e n c e s i n b a n d s o n t h e l e f t s i d e o f t h e m a i n c a r d i o i d f o r c y c l e s o f o r d e r s

n = 3 t o 8 . ( B # = b r a n c h ; m c = m a i n c a r d i o i d . )

n = 3

2 - + + - 0 . 1 2 2 5 6 1 1 6 6 , 0 . 7 4 4 8 6 1 7 6 6 B 3 b u d o n m c

n = 4

4 + - + + - 0 . 1 5 6 5 2 0 1 6 6 , 1 . 0 3 2 2 4 7 1 0 8 B 3 c a r d i o i d

n = 5

6 - - - + + - 0 . 5 0 4 3 4 0 1 7 5 , 0 . 5 6 2 7 6 5 7 6 1 B 5 b u d o n m c

7 - + - + + - 0 . 0 4 4 2 1 2 3 5 7 , 0 . 9 8 6 5 8 0 9 7 6 B 3

8 + + - + + - 0 . 1 9 8 0 4 2 0 9 9 , 1 . 1 0 0 2 6 9 5 3 7

n = 6

1 2 + - - - + + - 0 . 5 9 6 8 9 1 6 4 4 , 0 . 6 6 2 9 8 0 7 4 4 B 5

13 + - + - + + - 0 . 0 1 5 5 7 0 3 8 6 , 1 . 0 2 0 4 9 7 3 6 6 B 3

1 4 - - + - + + - 0 . 1 1 3 4 1 8 6 5 5 , 0 . 8 6 0 5 6 9 4 7 2

15 - + + - + + - 0 . 1 6 3 5 9 8 2 6 1 , 1 . 0 9 7 7 8 0 6 4 2

1 6 + + + - + + - 0 . 2 1 7 5 2 6 7 4 7 , 1 . 1 1 4 4 5 4 2 6 5

n = 7

2 2 . . . . . + + - 0 . 6 2 2 4 3 6 2 9 5 , 0 . 4 2 4 8 7 8 4 3 6 B 7 b u d o n m c

2 3 - + - - - + + - 0 . 5 3 0 8 2 7 8 0 4 , 0 . 6 6 8 2 8 8 7 2 5 B 5

2 4 + + - - - + + - 0 . 6 2 3 5 3 2 4 8 5 , 0 . 6 8 1 0 6 4 4 1 4

2 5 + + - + - + + - 0 . 0 1 4 2 3 3 4 8 1 , 1 . 0 3 2 9 1 4 7 7 5 B 3

2 6 - + - + - + + - 0 . 0 0 6 9 8 3 5 6 8 , 1 . 0 0 3 6 0 3 8 6 2

2 7 - - - + - + + - 0 . 1 2 7 4 9 9 9 7 3 , 0 . 9 8 7 4 6 0 9 0 9

2 8 + - - + - + + - 0 . 2 7 2 1 0 2 4 6 1 , 0 . 8 4 2 3 6 4 6 9 0

2 9 + - + + - + + - 0 . 1 5 7 5 1 6 0 5 3 , 1 . 1 0 9 0 0 6 5 1 4 B 3 m a p f r o m n = 6

3 0 - - + + - + + - 0 . 1 7 4 5 7 8 2 2 1 , 1 . 0 7 1 4 2 7 6 7 1

3 1 - + + + - + + - 0 . 2 0 7 2 8 3 8 3 5 , 1 . 1 1 7 4 8 0 7 7 2

3 2 + + + + - + + - 0 . 2 2 4 9 1 5 9 5 1 , 1 . 1 1 6 2 6 0 1 5 7

n = 8

4 3 f- + - 0 . 3 5 9 1 0 2 3 9 0 1 1 2 , 0 . 6 1 7 3 5 3 4 5 3 3 9 8 b u d o n m c

4 4 + . . . . . + + - 0 . 6 9 0 9 4 2 8 9 7 6 5 2 , 0 . 4 6 5 3 4 9 5 3 8 5 8 1 B 7 e d g e a t 4 4

4 5 + - + - - - + + s e e 5 2

4 6 - - + - - - + + - 0 . 5 9 2 4 6 5 9 0 2 7 4 6 , 0 . 6 2 1 3 4 8 6 8 9 2 6 1

4 7 - + + - - - + + - 0 . 6 0 6 1 8 5 5 5 8 8 9 4 , 0 . 6 8 4 0 3 1 6 1 6 4 7 6

4 8 + + + - - - + + - 0 . 6 3 2 3 8 4 1 2 1 9 6 9 , 0 . 6 8 4 7 0 1 7 5 7 6 2 3 B 5 e d g e a t 4 8

4 9 + + + - + - + + - 0 . 0 1 5 6 0 8 5 3 3 2 2 3 , 1 . 0 3 6 6 4 5 6 6 5 8 4 7 B 3

5 0 - + + - + - + + - 0 . 0 0 9 8 6 2 4 3 6 9 8 6 , 1 . 0 2 9 6 3 0 1 1 8 3 6 1

5 1 - - + - + - + + - 0 . 0 2 3 2 4 2 2 9 1 9 6 1 , 0 . 9 9 8 9 9 2 6 4 4 6 8 4

5 2 + - + - + - + + - 0 . 5 2 5 9 7 1 0 8 2 5 3 0 , 0 . 6 9 6 9 4 3 6 4 8 5 5 2 B 5 a n o m a l o u s ( 4 5 )


Table I (Continued)

n = 8 53 + - - - + - + + 53 missing; 52 is replaced by 77 on B3

54 . . . . . ~ - + + -0.074470451380, 0.970542136899 B3 55 - + - - + - + + -0.240416194737, 0.870487421314 56 + + - - + - + + -0 .296350364863, 0.845154528884

57 + + - + + - + + -0.157781855891, 1.112590613593 B3 map f r o m n = 7 58 - + - + + - + + -0 .154454617643, 1.104395461077 59 - - - + + - + + -0.183173449982, 1.091560429186 60 + - - + + - + + -0.162867707067, 1.037313240810 61 + - + + + - + + -0.206598609869, 1.121383303100 62 - - + + + - + + -0 .207991431776, 1.109132944418 63 - + + + + - + + -0 .222235654179, 1.118375889201

64 + + + + + - + + -0 .227331323047, 1.115883598594 B3 band edge

77 + - + - + - - + 0.000464217447, 1.004501022924 B3 anomalous (52)

88 + + . . . . . + 0.025970522960, 0.864697720038 B3 rhs of mc

90 - + - + - - - + 0.022860533259, 0.813669918234 end of B3

93 + - + + - - - + 0.135782238839, 0.670855910094 near n = 7 bud

b u d s w h i c h a p p e a r d o w n t h e l e f t - h a n d s i d e o f t h e m a i n c a r d i o i d . T h e w i d t h o f

e a c h s u b - b a n d d o u b l e s f o r e v e r y i n c r e a s e i n c y c l e o r d e r . T h e s u b - b a n d c l o s e s t

t o t h e l i m i t p o i n t c o r r e s p o n d s t o b r a n c h 3 a t t h e t o p o f t h e c a r d i o i d . F o r b r a n c h

3 t h e r e a r e 2 n - 4 c e n t r e s i n e a c h b a n d . F o r a g i v e n c y c l e o r d e r , t h e o t h e r

s u b - b a n d s c o n t a i n f e w e r c e n t r e s b y a f a c t o r o f 2 a t e a c h s t a g e a s o n e m o v e s

f r o m b r a n c h t o b r a n c h . F o r b r a n c h p , p o d d , t h e r e a r e 2 n - p - 1 c e n t r e s i n e a c h

s u b - b a n d . U s i n g t a b l e I , a n d i t s e x t e n s i o n f o r l a r g e r n , o n e c a n e s t i m a t e t h e

n u m b e r o f c e n t r e s o f a g i v e n c y c l e o r d e r w h i c h a p p e a r a l o n g e a c h b r a n c h . F o r

c y c l e o r d e r n a n d b r a n c h n u m b e r p , t h e n u m b e r s o f c e n t r e s i n t h e v a r i o u s

s u b - b a n d s a r e

p•n: 6 7 8 9 10 . . . 3 4 5

3 1 1 2 4 8 16 32 64 . - -

5 1 1 2 4 8 16 . . .

7 1 1 2 4 - - .

T h i s b a n d s t r u c t u r e h a s b e e n c h e c k e d t o o r d e r 12. O n b r a n c h 3 a t e a c h

i n c r e a s e i n c y c l e o r d e r t h e c e n t r e s i n t h e s u b - b a n d a t o r d e r n a r e " m a p p e d " a t

o r d e r n + 1 t o a s e t o f l e v e l s i n t h e ( b o t t o m ) h a l f o f t h e n e x t s u b - b a n d c l o s e s t

t o t h e l i m i t p o i n t c 0. G e o m e t r i c a l l y a l l t h e 2 n - 4 p o i n t s o n b r a n c h 3 a t c y c l e

o r d e r n m o v e c l o s e r t o t h e l i m i t p o i n t , w h i l e a n e w s e t o f 2 n - 4 p o i n t s a p p e a r i n

638 J. Stephenson / Spirals in the Mandelbrot set I

Table II Magnitudes and angles of the "radi i" do = c ' n - c o from the limit point c o to the centres at the band

edges for n = 3 - 20, for one revolution.

n Magnitude Angle n Magnitude Angle

3 0.3850428 -74.083 12 0.00001220403 134.344 4 0.1095594 -49 .167 13 0.000003952982 157.470

5 0.03358603 -26 .284 14 0.000001280379 -179.403 6 0.01065100 -3 .704 15 0.0000004147151 -156.277 7 0.003426919 19.034 16 0.0000001343260 -133.151

8 0.001108364 41.961 17 0.00000004350813 - 110.025 9 0.0003589965 65.004 18 0.00000001409225 -86 .899

10 0.0001163030 88.101 19 0.000000004564474 -63 .773 11 0.00003767590 111.219 20 0.000000001478430 -40 .647

their place, thereby doubling the number of points. The branch is therefore composed of segments or "cells", which map into one another with each increase in cycle order. The "mot ion" of each segment is along a spiral terminating at the limit point. (Similarly for the other branches.) To see this

arithmetically, one calculates the successive positions of the centres c ' n for a particular sequence of "associated" levels of interest, say the band-edge levels 2 n 2, which are closest to the limit point at each stage. Table II shows the

magnitudes and angles of the "radii" dn = c'~ - c o f rom the limit point to the centres. The magnitudes are in approximately constant ratios approaching 3.0873780, while the angle differences are also approximately constant, approaching -23.125998 ° . So successive iterates for the band-edge levels lie along an equiangular exponential spiral. It is convenient to write this spiral in

the complex form

d , , = c ' n - c o = A ' / a n , with a = la[ e i" , (2)

so the complex spiral multiplier a comprises a "magni tude" lal ~ 3 , and an "angle" a ~ - 23 °. Further calculation for other sets of associated levels

reveals that they give rise to spirals with the same multiplier.

3. Levels and sign sequences

The level structure and corresponding sign sequences in the vicinity of the band edge on branch 3 are readily constructed. For cycle order n on branch 3 the band edge is at level 2 n-2. The 2 n-4 levels adjacent to the band edge give

rise to centres. The signature of branch 3 is { - + + } , as for the n = 3 bud attached to the main cardioid, and the centres have signature { + - + +} (as at


cycle order n = 4), "fol lowed" (reading from right to left for the " forward" iteration) by the 2 n-4 possible choices of the - sign. The sign sequences are quoted in table I in the same order (left to right) as they are used in calculating the "reverse" (or " inverse") cycle. (The signs are "used" from left to right but " read" from right to left). Similarly for the succeeding branches p (odd), with q = p - 2, one has a signature { q - , + + } for the bud on the main cardioid, and an initial sequence {+, q - , + + } for the centres, with -+ signs thereafter. For the "principal" series of centres on branch 3 associated with the band edge levels themselves the sign sequences are {m +, - + + }, m = 1, 2 . . . . . for cycle order n = m + 3 in cell m, m = 0 being the bud. The next closest set of levels ends with a - sign, and the corresponding series of centres has sign sequences { - , m + , - + + } . In general for a "tail" of - signs {t}, the series of centres

have signs {{ t} ,m+ , - + + } . Some of the signs sequences are "irregular", and not what one would expect

from the overall sign structure, so the corresponding centre has an "anomalous" level. This occurs occasionally (unexpectedly) when the cycle element whose sign is to be allocated (during the construction of the inverse cycle, after the complex square-root has been calculated) has a "small" real part. For example at cycle order n = 8, on branch 3, level 52 is replaced by level 77 for which the seventh sign (as listed) is opposite. Similarly on branch 5, level 45 is replaced by level 52 for which the fifth sign is opposite. Level 53 is missing altogether! And there are two "stray" centres both with positive real parts at

levels 88 and 90.

4. Preperiodic points

The limit point at c o at the end of branch 3 is an example of a preperiodic point which arises by setting R 3= - R 2, so all subsequent iterates have the same "constant" value, c o is the complex zero (with positive imaginary part) of c3+ 2(c2+ c + 1). The real zero is the band-merging point at - 1 . 5 4 3 6 8 9 . . .

For the tip of the p = 3 branch the precycle is

c o = ( -0 .228, 1.115), c 1 = ( -1 .419, 0.606), c 2 = (1.419, - 0 . 6 0 6 ) .

At a general preperiodic point (ppp), after a certain number of iterations q', say, one finds an iterate R q' with the property that some later iterate R q'+p' =

- R q'. So the iterates settle into a cycle of order p' . I will refer to these special sequences as "precycles" {q', p ' } , with precursor order q', cycle order p ' , and (total) preperiodic order P = q ' +p ' . For example the "t ips" of the branches (above) are ppp's with sign sequences { q - , + + } and preperiodic order P

6 4 0 J. Stephenson / Spirals in the Mandelbrot set 1

equal to the branch number p = q + 2. The sign sequence for the preperiodic

point at the tip is the same as that of the corresponding bud where the branch is attached to the main cardioid.

For a precycle of cycle order p ' = 1 the Pth precycle element satisfies R e = (Re) 2 + c, so there are two possible values for R e :

/ z = ( l + X / ] - - 4 c ) / 2 or ~ , ' = l - / z = ( 1 - X / 1 - 4 c ) / 2 . (3)

For "exterior" preperiodic points at the "tips" of branches the former value is

required. I will refer to ~ as the "modulus" and ~ ' as the "complementary modulus". As I will describe below: a = 2~, so the multiplier, and hence the

magnitude and angle, are determined exactly, once c o is known. In order to

obtain the actual spiral, one requires the amplitude, A', which depends on the level sequence in question.

5. Asymptotic method

I will present an outline of the theory for the preperiodic case of branch tips with cycle order p ' = 1, and precursor order q', and preperiodic (pp) order p = p ' + q'. So for m ~ p , R m = R p = - R p - 1 . On branch 3, p ' = 1, q ' = 2 and

p = 3. (In general p must be replaced by P.) The method has been used

previously to calculate the first and second order contributions to the asymp-

totic behaviour of iterates R m near the "spike" at c = - 2 .

The idea is to construct a Taylor series expansion of R m, the mth iterate of

the function R, about the preperiodic limit point Co. The coefficients in this

expansion are related by a set of difference equations. These equations can be solved sequentially, and the asymptotic form of the Taylor series coefficients determined for large m. The Taylor series can then be resummed, to leading

asymptotic order, as a generating function which satisfies a simple non-linear

functional equation, permitting immediate solution to provide an asymptotic representation of R m. Higher order terms can be obtained [11].

The first few successive iterates are obtained exactly from R m+l = (Rm) 2 + c, so R 1 = c, R 2 = c 2 + c, R 3 = (c 2 + c) 2 + c, et seq. Then the first derivatives

satisfy D R m+l = 2 R m • D R m + 1 (D --- O/Oc) , from which one may calculate DR n

exactly (numerically or algebraically) by recursion. When m/> p, R m assumes a constant value, which one identifies as/~ (=c 2 -= R 3 = R 4 et seq. on branch 3). Denoting the first derivatives of R m evaluated at c o by t~, one sees that for

m ~>p, they satisfy a linear first order difference equation:

1 i tm+ ~ -- (2~)t m = 1, m >~p, (4)


1 1 O n branch 3, t 3 = D R 3 = 1 + 2Ca(1 + start ing at the exact k n o w n value of tp.

2c0). The solution has the fo rm

tml = 6~/(2/~) m + N = M(2IZ) m for large m (5a)

where one has

~3 = 1/(1 - 2 / x ) and M = (tip - ~ ) / ( 2 / x ) p = - t e ~ / ( 2 t z ) P , (5b)

say. For higher derivatives, using Leibni tz ' t heo rem, one can express the r th

derivative of R "÷1 with respect to the pa r ame te r c in terms of derivatives of all

the lower o rder i terates:

~=o r! D~Rm . D r _ ~ R m D~Rm+I=6~I + = s ! ( r - s ) ! r > 0 . (6)

N o w one can construct the Taylor expansion of R m about the limit point c o in

powers of d = c - Co:

oe

~ _ D r R m R m = t r d r / r ! , where t,, = at c = c o . (7) r = 0

The Taylor series coefficients are just the r th derivative of R m with respect to

the pa rame te r c, evaluated at c o where d = 0. These coefficients satisfy the same hierarchy of difference equat ions:

t r + l = ~;rl + s ! ( r - - S) t t s t rm-s ' s = O °

r > O . (8)

One can solve successively for higher o rder coefficients. For very high order

iterates it is apparen t that the Taylor series coefficients at c o have the asymptot ic fo rm t r = Ar[M(2/~)m]L

Substi tute this leading asymptot ic fo rm into the full recur rence relat ion (8), to obta in for the leading coefficients

r! s ! ( r - s ) ! or a~(2/x) ~= a,a~_ s , r > l , (9) $ = 0 s ~ O

with a r = A r / r ! for convenience.

642 J. Stephenson / Spirals in the Mandelbrot set I

6. The alpha function

Next construct a generating function, the "A lpha" function, A, by intro- ducing a dummy variable u and summing,

A ( u ) : ~ artt r . (10) r--O

Multiply the equations in (9) by u r for r = 1 . . . . , m, sum, and express the result in terms of the generating function, to obtain

A(2/xu) = A(u) 2 +/z(1 - / z ) , (11)

where I have set a 0 =/x, in order that A ( 0 ) - A ( 0 ) 2 = / x ( 1 - / x ) = c . The coefficient a I is undetermined by the above equation, so I set it equal to unity: a~ = 1, by choice. Also for algebraic reasons it is bet ter to work in terms of the spiral multiplier a = 2/z, so

A(at l ) : A(u) 2 + ½a(1 - ½a). (12)

The alpha function is defined by this functional equation. A power series solution is easily generated, and, in conjunction with the functional equation, can be used to evaluate the alpha function for complex values of a and u. For certain values of a the alpha function can be expressed in terms of known functions. The case a = 2 is trivial. The case a = 4 corresponds to the real axis "spike" at - 2 [11]. The case a = - 2 is also soluble. The alpha function is essentially the Schr6der function for R (PR). The first few terms in the power series solution are supplied in appendix A. In practice, some 15 to 20 terms were used.

7. Amplitudes

In order to calculate amplitudes from the asymptotic theory, we need to identify u in terms of d and/~, and to indicate how the appropriate value of u is to be extracted from the alpha function.

By matching the expansions of R m and the alpha function, one sees that u = d~la m = dtp~(21~) m-p. It is straightforward to calculate the latter combina- tion at some fairly large cycle order n (= 20 say) to find an "exper imenta l" estimate for u. For "principal" cardioid centres on branch 3, u ~ ( - 1.8689826, 0.3860138).

Now R" = 0 at the centre of a cycle of order n, and d, = c,] - Co, so at c = c ' ,


A ( u ) = 0. Tha t is, we seek the ze ro of the a lpha func t ion c loses t to the or ig in:

u 1 = ( -1 .8689826986 . . . . 0 . 3 8 6 0 1 3 6 9 8 9 . . . ) . Thus A ' = ul(21x)3/t3~. O n c e the

a m p l i t u d e is known , one can e s t ima te the pos i t ions o f the cen t re s for la rge

cycle o r d e r n. These e s t ima tes a re r e m a r k a b l y accura t e , and can be used to

m a t c h cen t res to levels when the sign sequences are i r regu la r .

8. Zeros of the alpha function

T h e amp l i t udes for the ser ies of cen t res a s soc ia t ed wi th the va r ious poss ib l e

levels , a re ca lcu la t ed f rom the zeros of the a lpha func t ion , wi th the sp i ra l

mu l t ip l i e r f ixed at a = 2/x = ( 2 . 8 3 9 2 8 6 , - 1 . 2 1 2 5 8 1 ) , a p p r o p r i a t e to the t ip of

b r anch 3. ( t ab le I I I ) . T h e zeros themse lves fall in to the s ame " p o w e r s - o f - t w o "

b a n d s t ruc ture as the successive sub -bands for the cen t res do. T h e b a n d s of

ze ros e x p a n d o u t w a r d by mul t ip les of the spiral scal ing fac tor . W h e n the zeros

(which are p r o p o r t i o n a l to the ampl i tudes ) a re " c o n t r a c t e d " back in, by

divis ion by a p p r o p r i a t e power s of the spi ra l mul t ip l i e r , the cyclic c o n t r i b u t i o n

to the M a n d e l b r o t set is o b t a i n e d , as it wou ld a p p e a r in an a sympto t i ca l l y high

o r d e r "ce l l " . In this (divis ion m o d u l o a) sense , the a lpha func t ion con ta ins a

" c o m p l e t e " cell , for b ranch 3.

I have l i s ted in t ab le I I I the first 32 zeros of the a lpha func t ion at c o in the

s ame o r d e r as they ar ise na tu ra l ly t h rough the level scheme . O n e can use these

to ob ta in r e m a r k a b l y accura te e s t ima tes of the 32 cen t res in the b r anch 3

s u b - b a n d at cycle o r d e r n = 9.

Table III Zeros of the alpha function at the tip of branch 3.

-1.868982698658 + 0.386013698900 -10.486865921864 + 4.996511921368 -11.268947900132 + 33.452197881659 -29.960806692852 + 24.219376641294

32.675299548869 + 129.025180110383 -11.354933172539 ÷ 83.813535465578 -51.071655334356 ÷ 113.103705403288 -60.756425795996 + 101.955747056093

295.176940812102 + 359.915734185482 225.115207076498 + 280.428088594771 75.033871343079 + 310.661142270951 47.175097849152 + 212.685256586682

-10.126694365466 + 394.378638072128 -3.342911850114 + 360.221655330561

-45.086691588097 + 370.374037139849 -53.624794251106 + 360.071379136659

1364.939636924028 + 710.230579014159 1222.466358287563 + 587.973401056974 946.408389433907 + 626.003632035865 984.958369453613 + 470.880165251689

560.661616434988 + 878.327375237002 655.016033963762 + 677.598542125031 400.062340225741 + 612.242871712307 348.686431774271 + 497.085321161334

445.685324818568 + 1141.465093865721 461.521781527876 + 1121.575797733614 399.131121251208 + 1069.064701095853 443.304060056395 + 977.734885915083

319.009112541173 + 1115.620196911414 326.512570066998 + 1087.176002083442 288.019086552188 + 1094.314341433437 279.687508166322 + 1084.339502229410

644 J. Stephenson / Spirals in the Mandelbrot set 1

9. Concluding remarks

1 h a v e p r e s e n t e d a n e l e m e n t a r y a r i t h m e t i c a n d a n a l y t i c a l a p p r o a c h to t h e

cycl ic a s p e c t s o f t h e p = 3 a n d o t h e r b r a n c h e s d o w n t h e l e f t - h a n d s ide o f t h e

m a i n c a r d i o i d in t h e M a n d e l b r o t se t . T h e m e t h o d s d e s c r i b e d h e r e h a v e b e e n

a p p l i e d w i t h e q u a l s ucce s s t o a c c o u n t f o r b r a n c h e s a n d e m b e d d e d s p i r a l s in

o t h e r r e g i o n s o f t h e M - s e t , a r o u n d t h e m a i n c a r d i o i d , as we l l as t h e a p p r o a c h

to t h e " s p i k e " a t c = - 2 [11].

Appendix A

T h e f i rs t 5 t e r m s in t h e e x p a n s i o n o f t h e a l p h a f u n c t i o n a r e

A ( u ) =1 + u + u2/[a(a 1)] + 2 u 3 / [ a 2 ( a - 1) (a 2 1)] ~ a - -

+ (a + 5)u4/[a3(a - 1) ( a 2 - 1 ) ( a 3 - 1 ) ] .

I t is c l e a r f r o m t h e d i f f e r e n c e e q u a t i o n s t h a t t h e a l g e b r a i c d e n o m i n a t o r s in t h e te l 1 [ S

c o e f f i c i e n t o f u m a r e d m = 7rs= ~ ata - 1). T h e r e q u i r e d e x p a n s i o n h a s t h e f o r m

A ( u ) = ½a + Em_ 1 rlmum /drn. T h e c o r r e s p o n d i n g n u m e r a t o r s a r e q u i c k l y g e n e r -

a t e d b y r e c u r s i o n . T h e f i rs t 11 a r e

n 1 = 1 , n 2 = 1 , n 3 = 2 , n 4 = a + 5 ,

n 6 = 6 a 4 + 16a 3 + 28a 2 + 28a + 4 2 ,

n 5 = 4a 2 + 6a + 1 4 ,

n 7 = 4a 7 + 32a 6 + 48a 5 + 1 0 8 a 4 + 128a 3 + 148a 2 + 120a + 1 3 2 ,

n 8 = a 11 + 27a 1° + 89a 9 + 204a 8 + 327a 7 + 5 3 6 a 6 + 660a 5 + 8 1 1 a 4 + 7 5 6 a 3

+ 7 0 5 a 2 + 495a + 4 2 9 ,

n 9 = 8a 15 + 8 4 a 14 + 2 5 2 a 13 + 652a 12 + 1074a 11 + 1926a 1° + 2 7 1 8 a 9 + 3 7 0 8 a 8

+ 4 3 1 8 a 7 + 5140a 6 + 4 9 8 8 a 5 + 4 8 7 0 a 4 + 3 9 6 0 a 3 + 3 1 9 0 a 2 + 2 0 0 2 a + 143~

n l0 = 28a 2° + 184a 19 + 706a 18 + 1640a 17 + 3 4 8 6 a 16 + 6 1 1 8 a 15 + 9 7 8 8 a 14

+ 14452a 13 + 2 0 2 7 6 a 12 + 2 5 2 8 6 a 11 + 3 1 1 0 8 a 1° + 3 4 7 8 4 a 9 + 3 7 5 8 8 a 8

+ 37406a 7 + 3 6 2 5 6 a 6 + 3 1 0 2 0 a 5 + 2 6 4 0 0 a 4 + 19470a 3

+ 14014a 2 + 8008a + 4 8 6 2 ,


n l l = 56a 26 + 364a 25 + 1528a 24 + 3968a 23 + 9260a 22 + 17560a 2a + 31380a 2°

+ 49636a a9 + 76180a 18 + 106612a 17 + 144684a 16 + 183148a is

+ 224708a 14 + 260036a 13 + 292832a a2 + 308956a a~ + 319724a 1°

+ 310720a 9 + 293892a 8 + 259320a 7 + 223064a 6 + 175044a 5

+ 134992a 4 + 92092a 3 + 60424a 2 + 31824a + 16796 .

References

[1] R.M. May, Nature 261 (1976) 459. [2] M.J. Feigenbaum, J. Stat. Phys. 19 (1978) 25; 21 (1979) 669. [3] B.B. Mandelbrot, Ann. NY Acad. Sci. 357 (1980) 249. [4] H.-O. Peitgen and P.H. Richter, The Beauty of Fractals (Springer, New York, 1986). [5] J. Guckenheimer, Inventiones Math. 39 (1977) 165. [6] J.-P. Eckmann and H. Epstein, Commun. Math. Phys. 101 (1985) 283. [7] A. Douady and J.H. Hubbard, Ann. Sci. Ec. Norm. Sup. (4) 18 (1985) 287. [8] Tan Lei, Commun. Math. Phys. 134 (1990) 587. [9] J. Milnor, Computers in Geometry and Topology (Chicago, 1986), Lecture Notes in Pure and

Applied Mathematics, vol. 114 (Dekker, New York, 1989) pp. 211-257; Dynamics in one complex variable, SUNY preprint (1990/5).

[10] P.J. Myrberg, Ann. Acad. Sci. Fennicae A I 336/3 (1963) 10. [11] J. Stephenson, J. Stat. Phys. 58 (1990) 579; Proc. Conf. on Non-Linear and Chaotic

Phenomena (Edmonton, 1990), W. Rozmus and J.A. Tuszynsky, eds. (World Scientific, New Jersey, 1991) pp. 19-25; in: Proc. Symp. on Field Theory and Collective Phenomena (Perugia, 1992) (World Scientific, Singapore, 1993).

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Spirals in the Mandelbrot set I