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A mandelbrot set for pairs of linear maps
Physica 150 (1985) 421-432 North-Holland. Amsterdam
A MANDELBROT SET FOR PAIRS OF LINEAR MAPSt M.F. BARNSLEY School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332, USA
and A.N. HARRINGTON Department of Mathematical Sciences, Loyola University, 6525 N. Sheridan Road, Chicago, Iflinois 60626, USA
Received 26 July 1984
The set of points A(s) = (± 1 ± s ± s2 ± S3 ± ... for all sequences of + and -}, where sEC and lsi < I, is generically a fractal. For example A(lj3) is the classical Cantor set and A(t + ij2) is a dragon curve. This set of fractals can be classified in terms of an associated Mandelbrot set D = (s E C: lsi < I, A(s) is disconnected}. The structure of D and its boundary are investigated.
1. Introduction
The discovery [1] of the Mandelbrot set M for the iterated complex polynomial Z2 - s has generated considerable research activity [2, 3], especially because of its relation to cascades of bifurcations and universal phenomena [4]. In this paper, we describe an analogous set associated with linear mappings. Our motivations are best explained starting from M, and so we characterize the Julia set [5] J(s) for Z2 - s. For almost any SEC, if we start at almost any point Z E C and calculate the limits of all sequences of compositions of the two maps
W-i-(Z)=/Z+s, w_(z)= -/z+s, then we obtain the attractor J(s)= {±y(s±y(s±y(s±..~
for all sequences of
+ and - }.
tSupported in part by NSF grant OMS-8401609.
In the nomenclature of Bamsley and Demko [6], {C U {oo}, W +(z), W _(z)} is an iterated function system (lFS). J(s) is anattractor for the IFS. M is the set of parameter values sEC such that J(s) is connected. Since J(s) is either connected or totally disconnected, M is the complement of the set of parameter values sEC such that J(s) is disconnected. Parameter space, with M marked on it, is a map of fractals [7] in the sense that each point in the space corresponds to a single Julia set J(s); and knowing where one is on this map provides good information about the structure of the corresponding J(s). The boundary aM of M is of special interest; for example, its intersection with the real line gives the parameter values at which bifurcations occur for the iterated real map x 2 - S, s real. aM appears to contain all of the values of s such that J(s) is a fractal tree (i.e., such that J(s) is connected and C-......J(s) has only one component); it is connected but may not be locally connected [12]; and it represents the set on which J(s) is "just connected." We illustrate the latter statement. For s> 2 J(s) is a Cantor set contained in the real line [8]. However, when s = 2,
0167-2789/85/$03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
M.F. Barnsley and A.N. Harrington Z A Mandelbrot set/or pairs 0/ linear maps
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since 2 = .;(2 + 1(2 + 1(2 + ..., any point represented by a code whose tail is - - + + + + + ... is also represented by the same code with the tail switched to + - + + + + + ... , and the Cantor set becomes "joined to itself' at these equivalent points resulting in J(s)=[-2,2] . also has this feature: magnifications of it reveal structures which themselves resemble parts of Julia sets J(s), for s-values near where one is magnifying. In this paper we consider the iterated function system [6] consisting of two complex linear maps with the same coefficient s on the z term, lsi < 1. Bya similarity transformation we may assume the functions are
aM
T+=sz
+ 1,
T_= sz-1. We remark that the IFS {C, T +(z), T _(z)} is one of the simplest one parameter families such that
the moment theory of p-balanced measures applies [6]. The associated fractals A(s) described below are basic entities in the approximation theory of fractals [9]. One may easily see that if we start with any point z E C and calculate the limits of all sequences of compositions of the two maps, then we obtain the attractor A(s)= {±1 ±s±s2±s3±... for all sequences of
+ and - }.
Figs . 1 to 3 show examples that were plotted using a microcomputer, by truncating the series of points for A(s) after the term s13 and scaling up the results to reach the borders of a 320 X 200 grid. The value of s is shown in the top left corner, in each figure. Some special cases of A(s) are these: (i) If 0< s < 1/2, A(s) is a Cantor set, obtained by
Fig. 1. The attractor A (s) for s = 0.1279188 + 0.6360408i.
M.F. Barlls1ey al ld A.N. Harrillg /oll/ A
Mallde1bro/ se/}o r pairs o/
Iillear maps
Fig: 2. The at tracto r A( s) .
Fig. 3. The at tracto r A (s ).
423
424
M.F. Barnsley and A.N. llarrington f A Mandelbrot set for pairs of linear maps
Fig. 4. The attractor A(s) for s = 0.5 - 0.5i. This is an example of a dragon curve. The horizontal and vertical lines arc artifa cts of the rescaling of the attractor, see text.
starting from an interval and taking out the middle (l - 2s )th parts at each stage of its construction. In particular, if s = 1/3 we obtain the classical ternary Cantor set. (ii) If s is pure imaginary the even and odd powers separate into real and imaginary parts; or, written in terms of s real,
Hence A(is) is a cartesian product of Cantor sets if 0 < s < 1/12, and it is a rectangle if s ~ 1/12. (iii) If we take s = (1 + i)/2, see fig. 4, we obtain the dragon considered by numerous people, see for example Davis and Knuth ~10), generated here by a new method, as an attractor for an IFS , rather than as a space filling curve generated geometrically - see the review by Mandeibrot (7).
Depending on the value of s, A(s) may be connected or disconnected. In the space of the parameter s we define
D= (SEC:
lsi <1, A(s) is disconnected}.
A computer approximation is shown in fig. 5 and a magnification of part of the boundary is shown in fig. 6. We will say more about the method of calculation later. Our focus is on the boundary aD of D , which is analogous to the boundary aM of the Mandelbrot set. A(s) tends to be most interesting visually when s E IlD, We illustrate this in figs. 7 to 13, all of which correspond approximately to s E aD and are such that the set "just touches itself'. The corresponding values of the real and imaginary parts of s are shown in the top left comers.
425
M.F. Barnsley and A.N. Harrington/ A Mandelbrot set lor pairs a/linear maps
-. ...
:
... .
Fig. 5. The Mandelbrot set D is approximated in black. See text.
2. Some analysis of A(s) and
aD
We formalize the notation for points in A(s). Let Q be the space of all sequences w = (w O,w t,w2 "") where wj = +1 or -1 (abbreviated +,-). Let gs: Q-)A(s) with
When s is fixed we will shorten our notation to g and A. With the usual cylinder set topology on Q, g: Q --+ A is continuous [6].
To better understand the geometry of A(s) it is instructive to consider the generation of the set in relation to the fixed points and cycles of T ±. The fixed points of T ± are ± 1/(1 - s). One can imagine shrinking the attractor A(s) by a factor s (denoting rescaling by lsi and rotation by arg(s)) about either fixed point. We denote the images by A += T +(A) and A _= T _(A). Then
Thus, A is generally a fractal, as defined by
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M.F. Barnsley and A.N. llarrington f A Mal/de/brat set for pairs of linear maps
We can now examine the set D, where A(s) is disconnected. There are simple bounds on the location of dl). Proposition 1. The attractor A(s) for the IFS {C, sz + 1, sz - 1} is totally disconnected if lsi <
1/2 and connected if 1> lsi> 1/12. Hence the boundary aD of D is contained in the annulus 1/2.:$; lsi.:$; lj12 . D is symmetric about the real and imaginary axes. Proof. We apply a theorem of Barnsley and
Demko [6]. For the special IFS considered here their theorem states that the Hausdorff dimension of A(s) is less than or equal to log(lj2)/loglsl, with equality if A(s) is totally disconnected. Combining the theorem with the facts that A(s) has Hausdorff dimension .:$; 2 and A(s) must be disconnected if its Hausdorff dimension is less than 1, we find that aD lies in the stated annulus. From the definition of A(s) it is clear that A(s) is symmetric both across the real axis and about the origin. 0 A practical method of determining whether A(s) is connected is provided by the following. Proposition 2. If A +n A _ = 0 then A is totally disconnected. If A + n A _ 0 then A is con-
*
nected. Fig. 6. Blow up of part of D, where 0.49 ~ Re s s 0.55 and 0.35 ~ Im s ~ 0.45.
Mandelbrot [7], since each of the pieces A + and A _ contain all of the structure of A.
From the sequences of + and - signs defining A(s) with our normalization, it is immediate that A ( - s) = A (s). The fixed points are not the same, but there is a complementarity when 2-cycle are considered: (a, b) is a 2-cycle for (T +' T _) if T +( a ) = b, T _( b) = a. The 2-cycles (-lj(1 + s) , + 1/(1 + s)) for the given system contains the fixed points for the system with s replaced by -s, and vice-versa.
Proof. The first statement follows easily from the geometric similarity of A to its disconnected halves A+ and A_. For the second part, let pEA +n A _. There must be w, a E Q such that
p=g(w)=g(a), where Wo = + 1 and ao = -1. Now we argue by contradiction, and suppose A is disconnected. Since A is compact [6], it must be a disjoint union of compact sets E and F with some separation /) > O. Since all codes w that agree through the
M.F. Barnsley and A.N. Harrington ZA Mandelbrot set for pairs of linear maps
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1M:
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.12791.88
Fig. 7. The attractor A(s) for s close to aD. The actual values of the real and imaginary parts of s are posted in the top left comer. The two tones represent the two components A + and A _. See also figs. 8-13, and the text.
Re:
1M:
.593491.2 .2522843
Fig. 8.
428
AI.F. Barnsley and A.N. llurrington ZA Mandelbrot set f or pairs of linear maps
He: I fit:
• 58274J.3
.36599
Fig. 9.
He: I fit:
.5152286 .4299493...
Fig. 10.
M.F. Barnsley and A .N. llarrington f A Mandelbrot set/or pairs 0/ linear maps
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.3695433
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Fig. 11.
.2~3~98
.6573696
Fig. 12.
429
M.F. Barnsley and A.N. Harrington f A Mandelbrot set for pairs of linear maps
430
Fig. 13.
term
wn
are separated by at most
there must be a maximum number of initial signs that the codes for points x E E and y E F can agree on. Suppose that codes x and y do agree to the maximum number of signs, X=g(Y1'Y2'···'Yn ' +, ... ), y=g(Y1'Y2'···'Yn '
- , ••• ).
Consider the point
z = g( Y1' Y2,···, Yn , + ,WI' W 2, W 3,···,), =
g( Y1' Y2,···' Yn ,
- ,<11,
(12' <13, ... ).
Whichever of the sets E or F contains z, we can choose a representation of z to contradict our
assumption about the maximum number of agree0 ing signs. Hence A is connected. We remark that if A + n A _ consists of a single point, then A(s) is a fractal tree. The picture of the set D in fig. 5 was generated on a microcomputer [11] as follows. Calculations were made only in the first quadrant of the s-plane which was represented with 200 points in both the real and imaginary directions from magnitude 0 to Ijli. Where lsi < 0.5 or lsi> Vii the picture was immediately filled in, black and white representing disconnected and connected respectively. For each point in the remaining quarter annulus a calculation was made. The powers of s through S13 were calculated and scaled up using 32 bit floating point arithmetic. If lsi was small enough the terms were scaled up as much as possible to have the attractor lie in the region with real and imaginary parts
M.F. Barnsley and A .N. Harrington f A Mandelbrot set f or pairs of linear maps
between ± 2 15• For points that might be in the overlap between A +' and A _, 16 bit integer additions were done to sum up through the s13 term. The results were rounded to keep the eight most significant bits and the corresponding points in A + were plotted in a 256 X 256 grid. If any point in A _ (= - A +) then matched a plotted point, the attractor was considered connected. .1f lsi was near 1/1i, however, the attractor was not scaled up so much so each point of the attractor would lie within half a pixel of some point calculated. Looking at the picture of the Mandelbrot set D suggests many conjectures. One we have proved is that a segment on the real axis sticks out on the boundary. Proposition 3. The interval 1= {0.5 ~ s s 0.53} lies in dI). Proof. First we show that if SE! then A(s) is connected. Clearly, A(s) is real for 0.5 ~ s < 1, and hence its Hausdorff dimension is bounded by 1. Using this bound and the theorem of Barnsley and Demko mentioned before we findA(s) is connected if s>0.5.1f s=0.5 then A(s)=[-2,2] in binary representation. In the remainder of the proof we show that as s leaves the real axis A(s) becomes disconnected. We do this with inequalities for the derivative of gs( w) with respect to s for real s. We consider real and imaginary parts separately. For s E J let
0: = Max(A_) = -1 +S+S2+ ...
431
taining I we can only have
if
for some sequence (w 4 , ws, ... ) E il. Then d
dsgs(w) = 1 + 2s
> 1 + 2s = 2(1
+ 3s 2 + 4w4 S 3 + ... + 3s 2 -
(3s 2 + 4s 3 + 5s 4 +
'" )
+ 2s + 3s 2) -1/(1 - S)2 >- 1
for all s in an open interval containing I, The opposite behavior occurs in A + = -A_. Since
uniformly in w, we can make a first order Taylor approximation with Lls imaginary and find that uniformly in a neighborhood of the only part of A + and A _ where the real parts can match, the imaginary parts separate as s leaves the real axis.
o The segment extending from 0.5 in our picture aD appears to go much further than the end of I. By considering separate cases for different combinations of the first few signs in w, one should be able to extend the proposition. of
= (2s-1)/(l-s).
By symmetry A _ can intersect A + only in points in [-0:,0:]. Now one can easily show that , in an open interval containing I, 0: < S3 which mean s
Hence, for any point s in an open interval con-
3. Conjectures and discussion It appears from fig. 6 that aD is disconnected, in contrast to aM. To prove such a result would require a broad analytical tool for confirming when A(s) is connected. Douady and Hubbard [2], in establishing the connectedness of aM, make central use of the fact that the forward orbit of the origin, under iterative application of Z2 - s, re-
432
At.F. Barnsley and A.N. Harrington f A Mandelbrot set f or pairs of linear maps
mains bounded if and only if J(s) is connected. We do not know an analogous result for piecewise linear maps. We expect some relationship between the local structure of aD and the structure of A(s), for this reason. A(s) varies uniformly with changing s; that is, the Haudorff distance between A(s) and A(t) goes continuously to zero as s approaches t. Hence, if we let s be close to t, such that t E aD, then a reasonable first approximation to A(s) is A(t) itself, and an improved approximation to A(s) is T +( A ( t » U T _(A(t». In order that these two approximate components, sA(t) + 1 and -sA(t) + 1, of A(s) "just touch," as s varies close to t, it must be that s follows a path dictated by this order of approximation by the geometry of A(t). In this way we can interpret a few of the features of aD such as the spiral peninsula about 30° off the x-axis, and the straight line segment on the x-axis, A similar Mandelbrot set can be associated with mildly nonlinear hyperbolic IFS obtained by perturbing T ± by a terms proportional to e, so that A(s, e) -.. A(s) as e -.. O. One might expect aDE to have geometrical features in common with aD. D can be characterized in terms of the existence of a class of dynamical systems. It follows from Barnsley and Demko (theorem 6) that sED if and only if there exists a compact subset K c C, numbers a and b e c, and a two-to-one onto mapping F: K -.. K such that F(z) E {(l/s)z + a,(I/s)z+b} for each zEK. Thus aD corresponds to the boundary of a class of dynamical systems.
References [1] B. Mand elbrot, Fractal aspects of the iteration of Z --> hZ • (1 - z ), Annals of N.Y. Acad. Sci. 357 (1980) 249-259; On the quadratic mapping Z --> z2 - JL for complex /1 and z : the fractal structure of its M set and scaling. Physica 7D (1983) 224-239; On the dynamics of iterated maps VIII , Ch aos and Statistical Methods. Y. Kuramoto, ed . (Sprin ger, Berlin, 1~ 84) . pp. 32-42. [2] A Douady and J. Hubbard, Comptes Readus (Paris) 294 (1982) 123-126. Also "On the dynamics of polynomial like mapping," Preprint (1984). W. Thur ston, " On the dynamics of iterated rational maps," Prcprint. [3] A Douady, "Systemes dynamique holornorphe s,' Seminaire Bourbaki, 35 e annee, 1982/83, no 599. (4) M. Feigenb aum, Quantitative universality for a class of nonline ar transformations, J. Stat. Phys. 19 (1978) 25-52. P.J. Myrberg, Sur !'iteration des polynomes reels quadratiques, J. de Math. Pures et Appliquees Ser 9, 41 (1962) 339-351. [5] A good review of Julia sets is given by P. Blanchard, Complex analytic dynamics on the Riemann sphere, Bulletin of the A.M.S. 11 (1984) 85-141. [6] M.F. Barnsley and S.G. Demko, Iterated function systems and global construction of fractals, Georgia Institute of Technology Preprint, July 1984. (7) B. Mandelbrot, The Fractal Geometry of Nature (Freeman, San Francisco, 1983). [8] See for example M.F. Bamsley, J.S. Ger onimo and AN. Harrington, Geom etry and combinatori cs of Julia sets for real quadratic maps, J. Stal. Phys. 37 (1984) 51-92. [9J M.F. Barnsley and S.G. Demko, Rational approximation of fractals, to appear in Springer Lecture Notes' in Mathemat ics, Proc. Tampa Conf. on Rational Approximation. E.B. SalTand P.R. Graves-Morris, cds, IIOJ C. Davis and D.E. Knuth. Number representat ions and dra gon curves, J. of Recreational Mathemati cs 3 (1970) 66-81, 133-149. [11] We have available copies of the program, for IBM PC with 128K of memory and color graphics capability. (12) A Douady recently informed us that he believes M to be locally connected, with a very poor modulus of local connectivity.
A MANDELBROT SET FOR PAIRS OF LINEAR MAPSt M.F. BARNSLEY School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332, USA
and A.N. HARRINGTON Department of Mathematical Sciences, Loyola University, 6525 N. Sheridan Road, Chicago, Iflinois 60626, USA
Received 26 July 1984
The set of points A(s) = (± 1 ± s ± s2 ± S3 ± ... for all sequences of + and -}, where sEC and lsi < I, is generically a fractal. For example A(lj3) is the classical Cantor set and A(t + ij2) is a dragon curve. This set of fractals can be classified in terms of an associated Mandelbrot set D = (s E C: lsi < I, A(s) is disconnected}. The structure of D and its boundary are investigated.
1. Introduction
The discovery [1] of the Mandelbrot set M for the iterated complex polynomial Z2 - s has generated considerable research activity [2, 3], especially because of its relation to cascades of bifurcations and universal phenomena [4]. In this paper, we describe an analogous set associated with linear mappings. Our motivations are best explained starting from M, and so we characterize the Julia set [5] J(s) for Z2 - s. For almost any SEC, if we start at almost any point Z E C and calculate the limits of all sequences of compositions of the two maps
W-i-(Z)=/Z+s, w_(z)= -/z+s, then we obtain the attractor J(s)= {±y(s±y(s±y(s±..~
for all sequences of
+ and - }.
tSupported in part by NSF grant OMS-8401609.
In the nomenclature of Bamsley and Demko [6], {C U {oo}, W +(z), W _(z)} is an iterated function system (lFS). J(s) is anattractor for the IFS. M is the set of parameter values sEC such that J(s) is connected. Since J(s) is either connected or totally disconnected, M is the complement of the set of parameter values sEC such that J(s) is disconnected. Parameter space, with M marked on it, is a map of fractals [7] in the sense that each point in the space corresponds to a single Julia set J(s); and knowing where one is on this map provides good information about the structure of the corresponding J(s). The boundary aM of M is of special interest; for example, its intersection with the real line gives the parameter values at which bifurcations occur for the iterated real map x 2 - S, s real. aM appears to contain all of the values of s such that J(s) is a fractal tree (i.e., such that J(s) is connected and C-......J(s) has only one component); it is connected but may not be locally connected [12]; and it represents the set on which J(s) is "just connected." We illustrate the latter statement. For s> 2 J(s) is a Cantor set contained in the real line [8]. However, when s = 2,
0167-2789/85/$03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
M.F. Barnsley and A.N. Harrington Z A Mandelbrot set/or pairs 0/ linear maps
422
since 2 = .;(2 + 1(2 + 1(2 + ..., any point represented by a code whose tail is - - + + + + + ... is also represented by the same code with the tail switched to + - + + + + + ... , and the Cantor set becomes "joined to itself' at these equivalent points resulting in J(s)=[-2,2] . also has this feature: magnifications of it reveal structures which themselves resemble parts of Julia sets J(s), for s-values near where one is magnifying. In this paper we consider the iterated function system [6] consisting of two complex linear maps with the same coefficient s on the z term, lsi < 1. Bya similarity transformation we may assume the functions are
aM
T+=sz
+ 1,
T_= sz-1. We remark that the IFS {C, T +(z), T _(z)} is one of the simplest one parameter families such that
the moment theory of p-balanced measures applies [6]. The associated fractals A(s) described below are basic entities in the approximation theory of fractals [9]. One may easily see that if we start with any point z E C and calculate the limits of all sequences of compositions of the two maps, then we obtain the attractor A(s)= {±1 ±s±s2±s3±... for all sequences of
+ and - }.
Figs . 1 to 3 show examples that were plotted using a microcomputer, by truncating the series of points for A(s) after the term s13 and scaling up the results to reach the borders of a 320 X 200 grid. The value of s is shown in the top left corner, in each figure. Some special cases of A(s) are these: (i) If 0< s < 1/2, A(s) is a Cantor set, obtained by
Fig. 1. The attractor A (s) for s = 0.1279188 + 0.6360408i.
M.F. Barlls1ey al ld A.N. Harrillg /oll/ A
Mallde1bro/ se/}o r pairs o/
Iillear maps
Fig: 2. The at tracto r A( s) .
Fig. 3. The at tracto r A (s ).
423
424
M.F. Barnsley and A.N. llarrington f A Mandelbrot set for pairs of linear maps
Fig. 4. The attractor A(s) for s = 0.5 - 0.5i. This is an example of a dragon curve. The horizontal and vertical lines arc artifa cts of the rescaling of the attractor, see text.
starting from an interval and taking out the middle (l - 2s )th parts at each stage of its construction. In particular, if s = 1/3 we obtain the classical ternary Cantor set. (ii) If s is pure imaginary the even and odd powers separate into real and imaginary parts; or, written in terms of s real,
Hence A(is) is a cartesian product of Cantor sets if 0 < s < 1/12, and it is a rectangle if s ~ 1/12. (iii) If we take s = (1 + i)/2, see fig. 4, we obtain the dragon considered by numerous people, see for example Davis and Knuth ~10), generated here by a new method, as an attractor for an IFS , rather than as a space filling curve generated geometrically - see the review by Mandeibrot (7).
Depending on the value of s, A(s) may be connected or disconnected. In the space of the parameter s we define
D= (SEC:
lsi <1, A(s) is disconnected}.
A computer approximation is shown in fig. 5 and a magnification of part of the boundary is shown in fig. 6. We will say more about the method of calculation later. Our focus is on the boundary aD of D , which is analogous to the boundary aM of the Mandelbrot set. A(s) tends to be most interesting visually when s E IlD, We illustrate this in figs. 7 to 13, all of which correspond approximately to s E aD and are such that the set "just touches itself'. The corresponding values of the real and imaginary parts of s are shown in the top left comers.
425
M.F. Barnsley and A.N. Harrington/ A Mandelbrot set lor pairs a/linear maps
-. ...
:
... .
Fig. 5. The Mandelbrot set D is approximated in black. See text.
2. Some analysis of A(s) and
aD
We formalize the notation for points in A(s). Let Q be the space of all sequences w = (w O,w t,w2 "") where wj = +1 or -1 (abbreviated +,-). Let gs: Q-)A(s) with
When s is fixed we will shorten our notation to g and A. With the usual cylinder set topology on Q, g: Q --+ A is continuous [6].
To better understand the geometry of A(s) it is instructive to consider the generation of the set in relation to the fixed points and cycles of T ±. The fixed points of T ± are ± 1/(1 - s). One can imagine shrinking the attractor A(s) by a factor s (denoting rescaling by lsi and rotation by arg(s)) about either fixed point. We denote the images by A += T +(A) and A _= T _(A). Then
Thus, A is generally a fractal, as defined by
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M.F. Barnsley and A.N. llarrington f A Mal/de/brat set for pairs of linear maps
We can now examine the set D, where A(s) is disconnected. There are simple bounds on the location of dl). Proposition 1. The attractor A(s) for the IFS {C, sz + 1, sz - 1} is totally disconnected if lsi <
1/2 and connected if 1> lsi> 1/12. Hence the boundary aD of D is contained in the annulus 1/2.:$; lsi.:$; lj12 . D is symmetric about the real and imaginary axes. Proof. We apply a theorem of Barnsley and
Demko [6]. For the special IFS considered here their theorem states that the Hausdorff dimension of A(s) is less than or equal to log(lj2)/loglsl, with equality if A(s) is totally disconnected. Combining the theorem with the facts that A(s) has Hausdorff dimension .:$; 2 and A(s) must be disconnected if its Hausdorff dimension is less than 1, we find that aD lies in the stated annulus. From the definition of A(s) it is clear that A(s) is symmetric both across the real axis and about the origin. 0 A practical method of determining whether A(s) is connected is provided by the following. Proposition 2. If A +n A _ = 0 then A is totally disconnected. If A + n A _ 0 then A is con-
*
nected. Fig. 6. Blow up of part of D, where 0.49 ~ Re s s 0.55 and 0.35 ~ Im s ~ 0.45.
Mandelbrot [7], since each of the pieces A + and A _ contain all of the structure of A.
From the sequences of + and - signs defining A(s) with our normalization, it is immediate that A ( - s) = A (s). The fixed points are not the same, but there is a complementarity when 2-cycle are considered: (a, b) is a 2-cycle for (T +' T _) if T +( a ) = b, T _( b) = a. The 2-cycles (-lj(1 + s) , + 1/(1 + s)) for the given system contains the fixed points for the system with s replaced by -s, and vice-versa.
Proof. The first statement follows easily from the geometric similarity of A to its disconnected halves A+ and A_. For the second part, let pEA +n A _. There must be w, a E Q such that
p=g(w)=g(a), where Wo = + 1 and ao = -1. Now we argue by contradiction, and suppose A is disconnected. Since A is compact [6], it must be a disjoint union of compact sets E and F with some separation /) > O. Since all codes w that agree through the
M.F. Barnsley and A.N. Harrington ZA Mandelbrot set for pairs of linear maps
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1M:
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.6644672
.12791.88
Fig. 7. The attractor A(s) for s close to aD. The actual values of the real and imaginary parts of s are posted in the top left comer. The two tones represent the two components A + and A _. See also figs. 8-13, and the text.
Re:
1M:
.593491.2 .2522843
Fig. 8.
428
AI.F. Barnsley and A.N. llurrington ZA Mandelbrot set f or pairs of linear maps
He: I fit:
• 58274J.3
.36599
Fig. 9.
He: I fit:
.5152286 .4299493...
Fig. 10.
M.F. Barnsley and A .N. llarrington f A Mandelbrot set/or pairs 0/ linear maps
He:
1M:
.3695433
.5~52286
Fig. 11.
.2~3~98
.6573696
Fig. 12.
429
M.F. Barnsley and A.N. Harrington f A Mandelbrot set for pairs of linear maps
430
Fig. 13.
term
wn
are separated by at most
there must be a maximum number of initial signs that the codes for points x E E and y E F can agree on. Suppose that codes x and y do agree to the maximum number of signs, X=g(Y1'Y2'···'Yn ' +, ... ), y=g(Y1'Y2'···'Yn '
- , ••• ).
Consider the point
z = g( Y1' Y2,···, Yn , + ,WI' W 2, W 3,···,), =
g( Y1' Y2,···' Yn ,
- ,<11,
(12' <13, ... ).
Whichever of the sets E or F contains z, we can choose a representation of z to contradict our
assumption about the maximum number of agree0 ing signs. Hence A is connected. We remark that if A + n A _ consists of a single point, then A(s) is a fractal tree. The picture of the set D in fig. 5 was generated on a microcomputer [11] as follows. Calculations were made only in the first quadrant of the s-plane which was represented with 200 points in both the real and imaginary directions from magnitude 0 to Ijli. Where lsi < 0.5 or lsi> Vii the picture was immediately filled in, black and white representing disconnected and connected respectively. For each point in the remaining quarter annulus a calculation was made. The powers of s through S13 were calculated and scaled up using 32 bit floating point arithmetic. If lsi was small enough the terms were scaled up as much as possible to have the attractor lie in the region with real and imaginary parts
M.F. Barnsley and A .N. Harrington f A Mandelbrot set f or pairs of linear maps
between ± 2 15• For points that might be in the overlap between A +' and A _, 16 bit integer additions were done to sum up through the s13 term. The results were rounded to keep the eight most significant bits and the corresponding points in A + were plotted in a 256 X 256 grid. If any point in A _ (= - A +) then matched a plotted point, the attractor was considered connected. .1f lsi was near 1/1i, however, the attractor was not scaled up so much so each point of the attractor would lie within half a pixel of some point calculated. Looking at the picture of the Mandelbrot set D suggests many conjectures. One we have proved is that a segment on the real axis sticks out on the boundary. Proposition 3. The interval 1= {0.5 ~ s s 0.53} lies in dI). Proof. First we show that if SE! then A(s) is connected. Clearly, A(s) is real for 0.5 ~ s < 1, and hence its Hausdorff dimension is bounded by 1. Using this bound and the theorem of Barnsley and Demko mentioned before we findA(s) is connected if s>0.5.1f s=0.5 then A(s)=[-2,2] in binary representation. In the remainder of the proof we show that as s leaves the real axis A(s) becomes disconnected. We do this with inequalities for the derivative of gs( w) with respect to s for real s. We consider real and imaginary parts separately. For s E J let
0: = Max(A_) = -1 +S+S2+ ...
431
taining I we can only have
if
for some sequence (w 4 , ws, ... ) E il. Then d
dsgs(w) = 1 + 2s
> 1 + 2s = 2(1
+ 3s 2 + 4w4 S 3 + ... + 3s 2 -
(3s 2 + 4s 3 + 5s 4 +
'" )
+ 2s + 3s 2) -1/(1 - S)2 >- 1
for all s in an open interval containing I, The opposite behavior occurs in A + = -A_. Since
uniformly in w, we can make a first order Taylor approximation with Lls imaginary and find that uniformly in a neighborhood of the only part of A + and A _ where the real parts can match, the imaginary parts separate as s leaves the real axis.
o The segment extending from 0.5 in our picture aD appears to go much further than the end of I. By considering separate cases for different combinations of the first few signs in w, one should be able to extend the proposition. of
= (2s-1)/(l-s).
By symmetry A _ can intersect A + only in points in [-0:,0:]. Now one can easily show that , in an open interval containing I, 0: < S3 which mean s
Hence, for any point s in an open interval con-
3. Conjectures and discussion It appears from fig. 6 that aD is disconnected, in contrast to aM. To prove such a result would require a broad analytical tool for confirming when A(s) is connected. Douady and Hubbard [2], in establishing the connectedness of aM, make central use of the fact that the forward orbit of the origin, under iterative application of Z2 - s, re-
432
At.F. Barnsley and A.N. Harrington f A Mandelbrot set f or pairs of linear maps
mains bounded if and only if J(s) is connected. We do not know an analogous result for piecewise linear maps. We expect some relationship between the local structure of aD and the structure of A(s), for this reason. A(s) varies uniformly with changing s; that is, the Haudorff distance between A(s) and A(t) goes continuously to zero as s approaches t. Hence, if we let s be close to t, such that t E aD, then a reasonable first approximation to A(s) is A(t) itself, and an improved approximation to A(s) is T +( A ( t » U T _(A(t». In order that these two approximate components, sA(t) + 1 and -sA(t) + 1, of A(s) "just touch," as s varies close to t, it must be that s follows a path dictated by this order of approximation by the geometry of A(t). In this way we can interpret a few of the features of aD such as the spiral peninsula about 30° off the x-axis, and the straight line segment on the x-axis, A similar Mandelbrot set can be associated with mildly nonlinear hyperbolic IFS obtained by perturbing T ± by a terms proportional to e, so that A(s, e) -.. A(s) as e -.. O. One might expect aDE to have geometrical features in common with aD. D can be characterized in terms of the existence of a class of dynamical systems. It follows from Barnsley and Demko (theorem 6) that sED if and only if there exists a compact subset K c C, numbers a and b e c, and a two-to-one onto mapping F: K -.. K such that F(z) E {(l/s)z + a,(I/s)z+b} for each zEK. Thus aD corresponds to the boundary of a class of dynamical systems.
References [1] B. Mand elbrot, Fractal aspects of the iteration of Z --> hZ • (1 - z ), Annals of N.Y. Acad. Sci. 357 (1980) 249-259; On the quadratic mapping Z --> z2 - JL for complex /1 and z : the fractal structure of its M set and scaling. Physica 7D (1983) 224-239; On the dynamics of iterated maps VIII , Ch aos and Statistical Methods. Y. Kuramoto, ed . (Sprin ger, Berlin, 1~ 84) . pp. 32-42. [2] A Douady and J. Hubbard, Comptes Readus (Paris) 294 (1982) 123-126. Also "On the dynamics of polynomial like mapping," Preprint (1984). W. Thur ston, " On the dynamics of iterated rational maps," Prcprint. [3] A Douady, "Systemes dynamique holornorphe s,' Seminaire Bourbaki, 35 e annee, 1982/83, no 599. (4) M. Feigenb aum, Quantitative universality for a class of nonline ar transformations, J. Stat. Phys. 19 (1978) 25-52. P.J. Myrberg, Sur !'iteration des polynomes reels quadratiques, J. de Math. Pures et Appliquees Ser 9, 41 (1962) 339-351. [5] A good review of Julia sets is given by P. Blanchard, Complex analytic dynamics on the Riemann sphere, Bulletin of the A.M.S. 11 (1984) 85-141. [6] M.F. Barnsley and S.G. Demko, Iterated function systems and global construction of fractals, Georgia Institute of Technology Preprint, July 1984. (7) B. Mandelbrot, The Fractal Geometry of Nature (Freeman, San Francisco, 1983). [8] See for example M.F. Bamsley, J.S. Ger onimo and AN. Harrington, Geom etry and combinatori cs of Julia sets for real quadratic maps, J. Stal. Phys. 37 (1984) 51-92. [9J M.F. Barnsley and S.G. Demko, Rational approximation of fractals, to appear in Springer Lecture Notes' in Mathemat ics, Proc. Tampa Conf. on Rational Approximation. E.B. SalTand P.R. Graves-Morris, cds, IIOJ C. Davis and D.E. Knuth. Number representat ions and dra gon curves, J. of Recreational Mathemati cs 3 (1970) 66-81, 133-149. [11] We have available copies of the program, for IBM PC with 128K of memory and color graphics capability. (12) A Douady recently informed us that he believes M to be locally connected, with a very poor modulus of local connectivity.