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8/10/2019 Transformations Betwen Fractals
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Transformations between fractals
Michael F. Barnsley
Abstract. We observe that there exists a natural homeomorphism b etween theattractors of any two iterated function systems, with coding maps, that haveequivalent address structures. Then we show that a generalized Minkowski
metric may be used to establish conditions under which an affine iteratedfunction system is hyperbolic. We use these results to construct families offractal homeomorphisms on a triangular subset ofR2.We also give conditionsunder which certain bilinear iterated function systems are hyperbolic anduse them to generate families of homeomorphisms on the unit square. Thesefamilies are associated with tilings of the unit square by fractal curves, someof whose box-counting dimensions can be given explicitly.
1. Introduction
In this introduction we refer to various terms, some more or less commonplace tofractal geometers, such as iterated function system and attractor, and othersmore specialized, such as top of an attractor and address structures. Theseterms are explained in subsequent sections of the paper.
A fractal transformation is a special kind of transformation between the at-tractors of pairs of iterated function systems. Its graph is the top of the attractor ofan iterated function system that is defined by coupling the original pair of iteratedfunction systems. Approximations to fractal transformations can be calculated inlow dimensional cases by means of a modified chaos game algorithm. They haveapplications in digital imaging, see [7] for example.
This paper concerns several topics related to the construction of fractal trans-formations and conditions under which they are homeomorphisms. The first maintopic is fractal tops, introduced in [5] and [6]. We generalize the theory to encom-pass what Kigami [18] and Kameyama [14] call topological self-similar systems.Theorem 3.1 shows that a fractal top is a certain bijection between a shift invari-
ant subspace of code space, called the tops code space, and the attractor of aniterated function system. It is associated with a natural dynamical system, on theattractor, that can provide information about the tops code space.
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We use fractal tops to define fractal transformations and to provide condi-tions, related to address structures, under which they are homeomorphisms; thisprovides fractal homeomorphisms between the attractors of suitably matched pairsof iterated function systems. See Theorem 3.2.
In order to apply fractal homeomorphisms to digital imaging, for example, wefind that we need families of iterated function systems that satisfy two conditions.First, assuming that each member of the family has a well-defined coding mapand attractor, we require that the address structure of each member of the familyis the same, so that Theorem 3.2 can be applied. Second, we require that eachmember of the family indeed possesses a well-defined coding map and attractor.In particular, under what conditions does an affine iterated function system possessa unique attractor?
Consider for example the linear transformations f1, f2 : R2 R2 defined
by f1(x1, x2) = (x2, x1/2) and f2(x1, x2) = (x2/2, x1). The eigenvalues of eachtransformation are all real and of magnitude less than one. Each transformationpossesses a unique fixed point, the origin. But there are many different closed
bounded sets A R2
such that A = f1(A)f2(A). Consequently the affineiterated function system (R2, f1, f2) does not possess a unique attractor. Thereexists no metric, compatible with the natural topology ofR2, such that both f1and f2 are contractions. In this case there does not exist a well-defined codingmap.
Thus, our second main topic concerns this question: Under what conditionsdoes there exist a metric, compatible with the natural topology ofRM,such that agiven affine iterated function system on RM is contractive? We answer with the aidof the antipodal metric, introduced in Theorem 4.1. This leads us to the followingconstruction. LetK RM be a convex body. We will say that two distinct pointsl,m, both belonging to the boundary ofK, are antipodal when there are twodisjoint support hyperplanes ofK, one that contains l and one that contains m.We will also say that two distinct points p,q, both belonging to the boundary ofK, are chordal when their distance apart maximizes the distance between pairs ofdistinct pointsp, q inKsuch thatqp is parallel to qp. The key observation,Theorem 4.2, is that the set of antipodal pairs of points is the same as the set ofchordal pairs of points ofK. We say that a transformation f : RM RM is non-antipodal with respect to Kwhenf(K) Kand fmaps each antipodal pair into apair of points that are not antipodal. Then a corollary of Theorem 4.4 implies that,for any iterated function system (RM, f1, f2,...,fN) of affine transformations, eachof which is non-antipodal with respect toK, there exists a metric compatible withthe euclidean metric such that all of thefns are contractions. Such systems possessa well-defined coding map and attractor. The converse statement is provided byTheorem 4.6 and is the subject of a separate paper, [1].
In Section 5 we present families of affine iterated function systems that both
illustrate and apply the theory. We use Theorem 4.4 to prove that all the functionsin the families are contractive with respect to the antipodal metric and that, foreach family, the address structure is constant. We describe the resulting families of
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homeomorphisms, from a triangular region to itself, and relate them to Kameyamametrics, [14]. In particular, Theorem 5.3 states that there exists a metric, compat-ible with the euclidean metric, with respect to which certain affine IFSs are IFSsof similitudes.
In Theorem 6.1 in Section 6 we give sufficient conditions under which certainbilinear iterated function systems are hyperbolic. Then we use such IFSs to con-struct a family of homeomorphisms on the unit square in R2.This example involvesa tiling of the unit square by 1-variable fractal interpolation functions. A closedform expression for some related box-counting dimensions is provided. In this waywe obtain some information about the smoothness of fractal homeomorphisms.
2. Some Kinds of Iterated Function Systems and Attractors
2.1. Iterated function system with a coding map
Let N 1 be a fixed integer. Let (X, dX) be a nonempty complete metric space.Let H denote the nonempty compact subsets ofX and letdHdenote the Hausdorffmetric; then (H, dH) is a complete metric space.
Let denote the set of all infinite sequences of symbols{k}k=1 belongingto the alphabet{1,...,N}. We write = 123... to denote an element of ,and we write k to denote the kth component of . We define a metric don by d(, ) = 0 when = and d(, ) = 2
k when k is the least indexfor which k= k. Then (, d) is a compact metric space that we refer to ascode space. The natural topology on , induced by the metric d, is the same asthe product topology that is obtained by treating as the infinite product space{1,...,N}.
Letfn : X X, n= 1, 2,...,Nbe mappings. We refer to (X, {fn}Nn=1) as aniterated function system. Let fn : X X, n = 1, 2,...,N be continuous and let:
Xbe a continuous mapping such that
() = f1((S())) (2.1)
for all where S : is the shift operator, defined by S() = wherek = k+1 fork = 1, 2,... Then we define
F= (X, {fn}Nn=1, )to be an iterated function system with coding map . Throughout we use theabbreviation IFS to mean an iterated function system with coding map.
If() = X thenF is also called a topological self-similar system, as intro-duced by Kigami [18] and by Kameyama [15], see [14].
2.2. Point-fibred, contractive, and hyperbolic IFSs
We say that the IFSF is point-fibredwhen it possesses a coding map given by() = lim
kf1 f2 fk(x) (2.2)
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where it is assumed that the limit exists for all , is independent ofx X,depends continuously on , and the convergence to the limit is uniform in x, for(, x) (, B) , for any fixed B H. It is straightfoward to prove that ifF ispoint-fibred then its coding map is unique.
The notion of a point-fibred iterated function system was introduced byKieninger [17], p.97, Definition 4.3.6; however we work in a complete metric spacewhereas Kieninger frames his definition in a compact Hausdorff space.
We say that the IFSF is contractivewhen each fn is a contraction, namelythere is a number Ln [0, 1) such that dX(fn(x), fn(y)) LndX(x, y) for allx, y X, for all n. Then L = max {ln} is called a contractivity factor forF. Wesay that a metric dXon X is compatible withdXwhen both metrics induce the sametopology on X. We say thatF is hyperbolic if there exists a metric, compatiblewithdX, with respect to whichFis contractive.
When Fis hyperbolic its coding map is given by equation (2.2). It is straight-foward to prove that any hyperbolic iterated function system is point-fibred, seefor example [2] (Theorem 3), but the converse is not true: Kameyama [14] has
shown that there exists an abstract point-fibred IFS, wherein X= , that is nothyperbolic. If the IFSF is such that () = X then it is point-fibred and itscoding map is given by{()} = limk f1 f2 fk(X); this is provedin section 2.4. It follows that the restrictionF|A = (A, {fn}Nn=1, ) of the IFSFto its attractor A, see below, is point-fibred. Since it is possible to construct twodistinct IFSs, each with the same set of functions{fn}Nn=1, but different codingmaps, there exists an IFS that is not point-fibred. Thus, the set of IFSs strictlycontains the set of point-fibred IFSs which, in turn, strictly contains the set ofhyperbolic IFSs.
2.3. Attractors
LetF= (X, {fn}Nn=1, ) be an IFS. We define the attractor ofF to be
A={() : } X.ClearlyAH, because is compact and nonempty, and : X is continuous.Kameyama [14] refers to A as a topological self-similar set. It follows from thecommutation condition (2.1) thatA obeys
A= f1(A) f2(A) fN(A) with AH. (2.3)WhenF is point-fibred, we have
A= limk
Fk(B), (2.4)with respect to the Hausdorff metric, for all B H. Consequently, ifF is point-fibred then its attractorA can be characterized as the unique solution of (2.3). Anelegant proof of this, in the hyperbolic case, is given by Hutchinson [13], Section
3.2. He observes that a hyperbolic IFSFinduces a contractionF : H H (we usethe same symbolFboth for the IFS and the mapping) defined byF(S) =fn(S)for allS H. See also [12] and [25]. To prove that equation (2.4) holds whenF is
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point-fibred, we prove that the convergence in (2.2) is uniform in , andxB,for any fixed B H. Suppose the contrary. Then for some >0,for someB H,for each k, we can find (k) , and bk B so that dX(f(k),k(bk),
(k)
)
for allk , wheref(k),k = f(k)1
f(k)
2 ...
f(k)
k
. Using compactness of bothB and
, we can find subsequences (kl) and{bkl} that converge to and bBrespectively. Since the convergence in equation (2.2) is uniform in (, x) B,it follows that f(kl),kl(bkl) converges to (). Using the continuity of and ofd
in both its arguments, we obtain dX((), ()) which is a contradiction.Note that if A is the attractor of the IFSF then the following diagram
commutes, forn = 1, 2,...,N,
sn
A
fnA
(2.5)
where sn : denotes the inverse shift defined by sn() = where 1 = nand k+1 = k for k = 1, 2, ....This set of assertions is equivalent to Equation(2.1) holds for all .
2.4. When is an IFS point-fibred?
LetF = (X, {fn}Nn=1, ) be an IFS such that () = X. ThenF is point-fibred.This fact was pointed out to me by Jun Kigami after my lecture at the conference,and is contained in a remark in [19]. To prove it here, we simply note that
limk
f1 f2 fk(X)= limk
f1 f2 fk(())= lim
k(s1 s2 ... sk()) (by (2.5))
={()} for all .
The last equality follows from the observation that the IFS (, s1, s2,...,sn) ispoint-fibred with attractor and coding map : given by ()= forall .
2.5. When is a p oint-fibred IFS hyperbolic?
Kameyama [14] has shown that there exists an IFS, wherein X = , for whichthere is no metric, compatible with the original topology, with respect to which itis contractive. Inspection of this abstract IFS shows that it is point-fibred.
3. Fractal Transformations
3.1. The top of a topological self-similar systemThe notion of fractal tops, for hyperbolic IFSs, was introduced in [4] and developedin [5] and [6].
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LetF = (X, {fn}Nn=1, F) denote an IFS, and let AF denote its attractor.Then we define
1F ({x}) ={ :F() = x}
to be the set ofaddressesof the point xAF.The following definitions and observations, which are implied by the continu-
ity ofF : AFand the commutative diagrams (2.5), generalize correspondingstatements for hyperbolic IFSs. See [5], Chapter 4, and [6] for examples and dis-cussion, in the hyperbolic case, of tops functions, top addresses, and tops codespaces.
We order the elements of according to < iffk > k,where k is theleast index for which k= k. Let F(x) = max{ : F() = x} for allx AF. Then F :={F(x) : x AF} is called the tops code space and :AF
onto F is called the tops function, for the IFSF. The valueF(x) is calledthetop address ofxAF. The tops function :AF is well-defined, one-to-one and onto, [6]. It provides a right inverse to the coding map; that is, FF isthe identity onAF. The inverse function,
1
F : FAF,is one-to-one, onto, andcontinuous. However, Fmay not be continuous, [6]. Let
1F
: F AF denotethe restriction of F to F (the closure of F) or, equivalently, the continuous
extension of1F to F. Then1F is continuous and onto. The ranges of both
1F
and 1F
are equal toAF becauseAFis closed.Notice that (2.5) implies
fn(x) =F sn F(x) for all xAF. (3.1)
3.2. Symbolic dynamics
The structure of the tops code space is related to symbolic dynamics as the fol-lowing theorem shows.
Theorem 3.1. LetF = (X, {fn}Nn=1, F) be an IFS with attractor AF, and letF be the associated tops code space. Then (i) S(F) F, and (ii) if f1 isone-to-one onAF thenS(F) = F.
Proof. Suppose thatF.(i) To see that S() F, suppose that there is some > S() such that
F() =F S(). ThenF(1) = f1 F() =f1 F S() =F (1S()) = F().
But 1 > , so this contradicts the fact that F. Therefore S() is thelargest address ofF(S()), soS()F.
(ii) We show that, when f1 is invertible on AF, we have 1F. Supposethat 1 /
F. Then there is some >1 such thatF() = F(1),and = 1where > . Then
f1F() = F(1) = F() =F(1) = f1 F(),
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so since f1 is one-to-one, F() = F(), which leads to a contradiction. Hence1F.
3.3. The tops dynamical system
Theorem 3.1 tells us that we can define what we call the tops dynamical systemTF :AFAF(associated with the IFSF) by
TF =1F
S F.When Fis continuous the topological entropy ofTF : AF AF is the same asthat of the shift operator acting on the tops code space F. This follows fromthe invariance of topological entropy under topological conjugation, see Corollary3.1.4 on p.109 of [16].
We can use the orbits of a tops dynamical system to calculate the tops codespace: for each xAF the value ofF(x) =12... is given by
k = min{n {1, 2,...,N}: T(k1)F (x)fn(AF)},
whereT0F (x) = x,T1F (x) = TF(x), T
2F (x) =TFTF(x),and so on. This formula
is useful when, as in the examples illustrated in Figure 3, the sets fn(AF) havestraight edges and a simple formula forTF(x) is available.
3.4. Transformations between attractors
LetF = (X, {fn}Nn=1, F) denote an IFS with attractor AF = F(X). SimilarlyletG = (Y, {gn}Nn=1, G) be an IFS with attractor AG. Then the correspondingfractal transformation TFG :AFAG is defined to be
TFG =G F.
The transformation TFG depends upon the ordering of the functions inFandG. For example, ifF= (X, f1, f2),G= (X, f2, f1), then in general TFG is notthe identity map on AF.
The transformation TFG may be characterized with the aid of the IFS withcoding map
K= (AF AG ; {kn}Nn=1 , K)wherekn(x,y,) = (fn(x), gn(y), sn()) and the coding map is defined by K() =(F(), G(), ). The graph ofTFG is the same as
{(x, y) : (x,y,)AK, for all (x,y, )AK}.
This characterization may be used to facilitate the computation of values ofTFGwhen AF and AG are subsets ofR2, bothF andG are hyperbolic, and a chaosgame type algorithm is used, see [6].
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4.1. Notation
We treat Rm as a vector space, an affine space, and a metric space. We iden-tify a point x = (x1, x2,...,xm) Rm with the vector whose coordinates arex1, x2,...,xm. We write 0
Rm for the point in Rm whose coordinates are all zero.
We writexy to denote the closed line segment with endpoints at x and y. We writey x to denote the vector, in the direction from x to y, whose magnitude is thelength ofxy. The inner product between x, y Rm is denoted by < x,y > . The2-norm of a point xRm isx=< x, x >. IfxRm\{0},then the unit vectorwhich points in the direction ofx is denoted x. That is, x= x/ x. The euclideanmetricdE : R
m Rm [0, ) is defined bydE(x, y) =x y for all x, y Rm.
LetK Rm denote a compact convex set with boundary K. We assumethroughout that there exists a ballBr(x) ={y Rm : dE(x, y) r}, of radiusr > 0 and center at x K, such thatBr(x) K. Let uRm\{0}. We defineLu =Lu(K) to be the unique support hyperplane ofK with outer normal in thedirection ofu. See [20], p.14. Then{Lu, Lu} denotes the unique pair of distincthyperplanes, perpendicular to u, that intersect K but contain no points of theinterior ofK. [11] refers to{Lu, Lu} as the two supporting hyperplanes ofKorthogonal to u. ForuRm\{0} we define
Au={(l, m)(LuK) (LuK)} andA=Au.We say that (l, m) Au is an antipodal pairof points corresponding to the direc-tion ofu, and thatA=A(K) is the set of antipodal pairsof points ofK.4.2. The antipodal metric
We define the width ofK in the direction of u to bew(u) = min{l m: l Lu(K) , m Lu(K)}, for all uRm\{0},
and we define the diameter ofK to be|K|= max{w(u) : uRm\{0}};
see for example [20], p.15, and [11]. Note that w(u) is a continuous function ofuSm1 :={uRm :||u||= 1},see [26], p.368. Since Sm1 is compact it followsthat|K|= w(u) for someu Sm1.
The following metric was discovered by Ross Atkins, a student at the Aus-tralian National University. It is related to the Minkowski metric, see Corollary4.3 below, [9], p.21, ex.5, and [10], p.100; for instance, the two metrics are thesame whenK is symmetric about 0, that is, when x K x K. We definetheantipodal metric dK: R
m Rm [0, ) by
dK(x, y) = max
{
(y x),uw(u)
:u
Rm
\{0
}}The maximum here is achieved at some uRm\{0} because(y x),u /w(u) iscontinuous in uSm1.
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Theorem 4.1. (i) dK is a metric onRm.
(ii) The metricsdK anddE onRm are equivalent, with
dE(x, y)/
|K| dK(x, y)
dE(x, y)/r for allx, y
Rm.
(iii) For allx, y KdK(x, y)1 with equality iff(x, y) A(K).
Proof. First we prove that dK is a metric on Rm. (a) dK is clearly symmetric. (b)
Ifx = y then dK(x, y) = 0. Ifx=y then
dK(x, y) = max{(y x),uw(u)
: uRm\{0}}
(y x), (y x)w(y x) y x =
y xw(y x) >0.
We have shown that dK(x, y) 0, with equality if and only ifx = y . (c) For allx,y,z Rm we have
dK(x, y) = max{(y z) + (z x),uw(u)
: uRm\{0}}
max{(y z),uw(u)
: uRm\{0}} + max{(z x),uw(u)
: uRm\{0}}=dK(x, z) +dK(z, y).
This establishes the triangle inequality and completes the proof thatdKis a metricon Rm.
To prove (ii) we simply note that
x y|K| dK(x, y)x yr .
To prove (iii) we suppose first that (x, y) A. Then xyconv(Lv Lv) ,the convex hull ofLvLv, for all vRm\{0}. It follows that(y x), v w(v)for allvRm\{0}. Hence(y x), v /w(v)1 for all vRm\{0}.Also (x, y) Aimplies there is uRm\{0}such that (x, y)(LuK) (LuK) . It follows that(y x),u= w(u). So
dK(x, y) = max{(y x), vw(v)
: vRm\0}=(y x),uw(u)
= 1.
Now supposex, y K but (x, y) / A.Then, for eachvRm\0, xyconv(LvLv), butxydoes not intersect both Lvand Lv.It follows that (y x), v /w(v)
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4.3. The setAequals the setD, the chordal pairs of points ofKLetuRm\{0}. We define the diameter ofK in the direction of u to be
d(u) = max{x y: x, yK, x y = u, R}.The maximum is achieved at some pair of points belonging toKbecause K Kisconvex and compact, andx yis continuous for (x, y) K K. ForuRm\{0}we define
Du={(p, q)K K: d(u) =qp}andD=Du.We say that (p, q) Du is a chordal pairof points in the direction ofu, and thatD is the set of chordal pairsof points ofK.
Theorem 4.2 is probably present in the convex geometry literature, but it isnot well-known. For example, it is not mentioned in [20] or [23]. It is crucial tothis work because it provides the heart of Theorem 4.4.
Theorem 4.2. The set of antipodal pairs of points ofK is the same as the set ofchordal pairs of points ofK. That is,
A=D.Proof. See [1]. The tools used are (a) that a convex body is the intersection of allstrictly convex bodies that contain it and (b) that, whenK is strictly convex, thefunction f(u) = xu xu/ xuxu is a continuous function f : Sn1 Sn1with the property that < f(u), u >> 0 for all u. Hence f does not map x toxfor any x, from which it follows by an elementary exercise in topology (see, forexample, Munkres, problem 10, page 367) that fhas degree 1 and, in particular,is surjective.
Letdbe aMinkowski metricon Rm.That is,dis a metric with the propertiesd(x + z, y + z) = dK(x, y) anddK(x, (1 )x + y) = dK(x, y) for all x, y,z Rmand all [0, 1]. Then there exists a compact convex setC, symmetric about theorigin, such that d = dC. The setC is given by
C ={xRm :k(x)1}wherek is the unique norm such that k(x y) =d(x, y) for all x, y Rm. See [22]pp.31-32.
Corollary 4.3. Letx, y K withx= y . Then there exists (l, m) A(K) be suchthat lm andxy are parallel and
dK(x, y) =y xm l =
dE(x, y)
d(y x) .In particular, dK is a Minkowski metric.
Proof. We can find (l, m) D such that lmand xy are parallel. By Theorem 4.2A=D, so there is a nonzero vector v with (l, m) Av. Now, by definition,
dK(x, y) = max{(y x),uw(u)
:uRm\{0}}.
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We claim that the maximum occurs when u= v, because if u= v then
(y x), vw(v)
= (y x), v =
y xm l ,
and if u = v then(y x),u /w(u) y x / m l . Hence, dK(x, y) =y x / m l=y x /d(m l) = dE(x, y)/d(y x).
For example, ifK is triangle then dK = dC whereC is hexagon, symmetricabout the origin.
4.4. IFSs of non-antipodal affine transformations
We say that f : Rm Rm is non-antipodal with respect toK iff(K) K and(x, y) A (K) implies (f(x), f(y)) / A (K).Theorem 4.4. Let f : Rm Rm be affine and non-antipodal with respect to aconvex bodyK. Thenf is a contraction with respect to dK.Proof. Let (x, y) K Kbe given with x=y. Then by Corollary 4.3 we can find(l, m) A(K) such thatlm is parallel toxy, and
dK(x, y) =y xm l .
Now consider (f(x), f(y)) K. Iff(x) =f(y) then d(f(x), f(y) = 0< d(x, y). Iff(x)= f(y) then f(l)= f(m) and the line segments f(x)f(y) and f(l)f(m) areparallel, because fis affine. Also(f(l), f(m)) K K is not an antipodal pair so,by Theorem 4.1 (iii),
dK(f(l), f(m))< 1.
In fact, let
L, M A(K) be such that L M is parallel to both f(l)f(m) and
f(x)f(y), again using Corollary 4.3; then we must have
dK(f(l), f(m)) =f(m) f(l) ML
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In the penultimate line we have used the facts that f is affine and xy is paralleltolm. Hence
g(u) :=dK(f(y), f(x))
dK(y, x)
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Figure 1. The triangles used to define the affine transformationsof the IFSF,,={R2; f1, f2, f3, f4}
5.1. Non-antipodal subtriangles
We chooseK=T, a filled triangle in R2,with strictly positive area and vertices atthe points A, B, and C. LetTdenote a triangle with vertices at the points P, Q,andR. Suppose that T T. Suppose also that the statement both T {P} =andT QR= is not true, for all cyclic permutations P QRofABC. Then wesay thatT is a non-antipodal subtriangle ofT.Corollary 5.1. Let the affine IFSF ={T; f1, f2,...,fN} be such thatfn(T) is anon-antipodal subtriangle ofT for each n. ThenF is contractive with respect todT.
This result is useful for applications because it provides a convenient geomet-
rical condition under which an affine IFS is point-fibred. We use it next to yieldfamilies of affine IFSs, such that each family has a constant address structure.
5.2. Families of homeomorphisms
LetT be a triangle with vertices A, B,C as above. Let c denote a point on theline segment AB, let a denote a point on the line segment BC, and let b denotea point on the line segment CA, such that{a,b,c} {A,B,C} =. Then eachof the triangles caB, Cab, cAb, and cab is a non-antipodal subtriangle ofT, seeFigure 1.
Letf1: R2 R2 denote the unique affine transformation such that
f1(ABC) = caB,
by which we mean that f1
maps A to c, B to a, and C to B. Using the samenotation, let affine transformations f2, f3, andf4 be the ones uniquely defined by
f2(ABC) = C ab, f3(ABC) =cAb, and f4(ABC) =cab.
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Transformations between fractals 15
Let us write F ={R2; f1, f2, f3, f4}, where= (|Bc|/|AB|, |Ca|/|BC|, |Ab|/|CA|).Then, for all (0, 1)3,F is contractive with respect to the metric dT, hasconstant attractorT, and has constant address structureC :=CF . The latterassertion is proved in [6], Example 1. Consequently Theorem 3.2 provides a fractal
homeomorphism T :T T defined by T = where := F and := F ,for all, (0, 1)3. Note thatTT =Tfor all ,, (0, 1)3and thatT preserves area when = (c,c,c) and= (1 c, 1 c, 1 c) for anyc(0, 1).
SinceF is hyperbolic with respect to the antipodal metric dK, we can finda contractivity factor L(0, 1) such that dT(fn (x), fn (y))LdT(x, y) for all,n, x.Clearly, L is not a constant function of. However, we will use the nextlemma to prove that there is a metric, compatible with the euclidean metric, withrespect to which all of the fn s are similitudes.
Lemma 5.2.
T(fn (x)) = f
n (T(x)) for allx T, for all,,n.
Proof. Equation (3.1) implies fn (x) = sn (x) for all x T. Hence, sincethe tops code space F is independent of, we have
T(fn (x)) = T sn (x) = sn (x)
= sn (x) = sn (x)= sn T(x) =fn (T(x)),
for all x T, for all , ,n. We define a family of metrics d onT by
d(x, y) = dE(T(x), T(y)).
For each, (0, 1)3, this is indeed a metric, compatible with the euclidean met-ric, becauseT
:T T
is a homeomorphism.
Theorem 5.3. The maps fn of the IFSF, restricted toT, are similitudes withscaling factor0.5 with respect to the metricd , where
:=(0.5, 0.5, 0.5).
Proof. Using Lemma 5.2 and the definition d we have
d(fn (x), f
n (y)) = dE(T(f
n (x)), T(f
n (y))
=dE(fn (T(x)), f
n (T(y))
= (0.5)dE(T(x), T(y))
= (0.5)d(x, y),
for all x, y T, n = 1, 2, 3, 4, and (0, 1)3.
An example of T applied to a picture is illustrated in Figure 2, for =(0.65, 0.65, 0.65) and= where = (0.5, 0.5, 0.5).The meaning of a picture oftransformation on the euclidean plane applied to a picture is intuitively obvious;
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16 Michael F. Barnsley
Figure 2. Euclidean geodesics within a triangle, on the left, aretransformed byT into nondifferentiable paths, on the right, that
are geodesics for the Kameyama metric D(F).
it is discussed objectively in Section 2.2 of [5]. The picture on the left in Figure2, a Cartesian grid masked by the triangle T, is the before image, P, whilethe picture on the right is the after image, T(P). Notice how straight linesegments on the left are transformed into fractal paths on the right. These pathsrepresent geodesics ofd. In fact, using the nomenclature of Kameyama [14], it
can be demonstrated thatd
=D
(F
|T),the standard pseudodistanceD
with
metric polyratio , for the topological self-similar systemF|T := (T, {fn } , ).We notice that the shift map S: FFrespects the relationship C C,as
discussed in Remark 3.4. It follows that the dynamical system T :T T,definedbyT = S,is continuous. It is readily seen thatTmaps Tonto itself, withmost points having four distinct preimages. The entropy ofT is ln 4, the same asthat of the shift map acting on the code space of four symbols. Note however, thatT(x) goes continuously clockwise three times round T when x goes clockwiseonce round T. The two dynamical systems T, T are topologically conjugate,with T = T T T for all ,. The action ofT on some of the points ofT is illustrated in the top left panel of Figure 3. The other panels illustrate thedynamics of the five other possible families of affine IFSs, that can be constructed
similarly toF.Each family has a constant address structure. Thus we obtain sixfamilies of homeomorphisms onT. Of these, only three families are distinct in thesense that no pair is conjugate via a euclidean transformation.
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Transformations between fractals 17
Figure 3. Three distinct families of fractal homeomorphisms,which are not conjugate under any affine transformation, are gen-erated by orienting the four subtriangles fn(ABC) =abc, in oneof six ways. In each case the corresponding tops dynamical systemis four-to-one, at almost all points, and continuous: its action onthe points A, ...G is illustrated.
6. Fractal transformations generated by bilinear functions
LetR = [0, 1]2 R2 denote the unit square, with vertices A = (0, 0), B =(1, 0), C = (1, 1), D = (0, 1). Let P,Q,R, S denote, in cyclic order, the succes-sive vertices of a possibly degenerate quadrilateral, as illustrated for example inFigure 4.
Then we uniquely define a bilinear function B:R R such that B(ABCD) =PQRSby
B(x, y) = P+x(Q P) +y(S P) +xy(R+P Q S).
This transformation acts affinely on any straight line that is parallel to either thex-axis or they-axis. That is, ifB|AB :ABP Qis the restriction toABofB and if
A: R2
R2 is the affine function defined by
A(x, y) =P+ x(Q
P) + y(S
P),
thenA|AB =B|AB. As we illustrate, this property makes it easy to constructelaborate parameterized families of bilinear IFSs with constant address structures.But first we need conditions under which bilinear IFSs are point-fibred.
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18 Michael F. Barnsley
Figure 4. Possibly degenerate quadrilaterals with verticesP,Q,R,Sin cyclic order.
Figure 5. The four quadrilaterals IEAH, IEBF, IGCD,IGDH, define four bilinear transformations that are contractivewith respect to an appropriately chosen metric that is compatiblewith the euclidean metric.
6.1. Contractivity of bilinear transformations
The following theorem provides practical sufficient conditions for a bilinear IFS tobe hyperbolic.
Theorem 6.1. The bilinear transformationB :R R defined byB(x, y) = P +(Q P)x+ (S P)y+ (P Q+R S)xy whereP,Q,R, S R, is contractivewith respect to the distance functiond,((x1, y1), (x2, y2)) = |x1x2|+ |y1y2|
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Transformations between fractals 19
for some choice of, >0 if
1 (x, y) +(x, y)> 0 (6.1)for allx, y
[0, 1] where
(x, y) =|(R1S1)y+ (Q1P1)(1 y)| + |(R2Q2)x + (S2P2)(1 x)|(6.2)
and
(x, y) =|((R S)y+ (Q P)(1 y)) ((R Q)x + (SP)(1 x))| . (6.3)The condition (6.1)is satisfied if
1 + 2 min {area (QRS) ,area (RSP) ,area (SP Q) ,area (P QR)} (6.4)>max {|R1 S1| , |Q1 P1|} + max {|R2Q2| , |S2P2|} .
Note that d, is a metric on R2 compatible with the euclidean metric pro-
vided that >0, >0.
Proof. We can writeB(x, y) = (P1+ a1(y)x +c1(0)y, P2+a2(0)x +c2(x)y)where ai(y) = (RiSi)y + (QiPi)(1y), and ci(x) = (RiQi)x+ (SiPi)(1 x), for i= 1, 2. Thus, we seek >0, >0, and 0
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20 Michael F. Barnsley
IfP,Q,R, Sare the vertices of a trapezium with sides P S and QR parallelto they-axis, then 1 (x, y) + (x, y) = (1 |Q1 P1|)(1|QR|x |P S|(1 x))is strictly positive for all x, y[0, 1], provided that|Q1 P1|< 1, |QR|< 1, and
|P S
|< 1.From this it follows that the parameterized family of IFSs
Fdefined in
equation (6.6) is hyperbolic. In a similar manner it is straightfoward to constructother families of hyperbolic bilinear IFSs whose attractors areR, as suggested forexample by Figure 5.
An example for which Theorem 6.1 does not imply contractivity is obtainedby choosing Q = (0.2, 0.9), R = (0.9, 0.1), S= (0.1, 0.9). Then, regardless of thelocation ofP inR, we have (1, 1) = 1.6> 1 + (1, 1) = 1.08.
6.2. Box-counting dimensions
LetNbe a positive integer. Let
0 =x0< x1< ... < xN= 1.
Let Ln : [0, 1]
[xn1, xn] be the unique affine transformation, of the formLn(x) = anx+bn, such that Ln(0) = xn1 and Ln(1) = xn, for n = 1, 2,...,N.Let 0 lj uj < 1 and sj = ujlj for j = 0, 1, . . . , N . Let Qn denote thetrapezium with vertices (xn1, ln1), (xn, ln), (xn, un), and (xn1, un1). Thenwe define fn :R Qn by
fn(x, y) = (Ln(x), cnx + [sn1+ (snsn1)x] y+ ln1),where cn = lnln1. It is readily verified that each fn is bilinear and, usingTheorem 6.1, that the IFSF := (R, f1, f2,...,fN) is hyperbolic. Using standardmethods, [3], it is readily verified that the attractor ofF is the graphG of acontinuous function g : [0, 1][0, 1].
For present purposes we define the box-counting dimension ofG to be
dimFG := lim0+
logN(G)log 1 (6.5)
whereN(G) is the minumum number of square boxes, with sides parallel to theaxes, whose union containsG. By the statement dimFG= D we mean that thelimit in equation (6.5) exists and equals D. .
Theorem 6.2. [8] LetFdenote the bilinear IFS defined above, and letG denote itsattractor. Letan = 1/N forn= 1, 2,...,N and let
Nn=1
sn1+sn2
>1. IfG is nota straight line segment then
dimFG= 1 +log
Nn=1
sn1+sn2
log N
Information about dimFG provides information about the smoothness ofgbecause dimFGis related to Holder exponents associated with g ,see [24], Section12.5, for example.
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22 Michael F. Barnsley
for all admissible,,.In particular, we can obtain information about the struc-ture and smoothness ofT by studying T :R R, for all admissible , in thecase where denotes the parameter set p = r = 2/3, q= s = 1/3.
We observe that T(Lx) = Lx for all x
[0, 1] where Lx is the line segment
{(x, y) : 0y1}. Consequently, ifG(f) Rdenotes the graph of a continuousfunctionf : [0, 1][0, 1],thenT(G(f)) is also the graph of a continuous functionfrom [0, 1] to itself. So let C[0, 1] denote the set of continuous functions from[0, 1] into itself, with metric dC[0,1](f, g) = max{|f(x) g(x)| : x [0, 1]}. ThenT :R R induces a continuous transformation T :C[0, 1]C[0, 1], definedby T(f) = g where gC[0, 1] is uniquely defined by T(G(f)) =G(g).
LetfcC[0, 1] be defined by fc(x) =c for c[0, 1]. Information about thesmoothness ofTis obtained by looking at the functionsgc := T(fc) for variousvalues ofc. In [8] it is proved that
gc1(x)< gc2(x) whenever 0c1< c21,for allx
[0, 1] and all admissible.Since
R=
{T(
G(fc)) : c
[0, 1]
},for each
admissible, ,it now follows that the graphs of the set of functions {gc : c[0, 1]}tileR, for each admissible . For example, when = , we have gc =fc and thegraphs of the set of functions{fc: c[0, 1]} tileR.
In [8] it is proved that
f0= g0< g1/2< g1= f1.
where G (g0) is the attractor of the IFS F(1) := (R, B1, B2), G
g1/2
is the attractor
of the IFSF(2) := (R, B3, B4), andG (g2) is the attractor of the IFSF(3) :=(R, B5, B6) . Furthermore, by Theorem 6.2, ifG
g1/2
is not a line segment and
(p q+r s)> 1 then
dim(G (g0)) = 1, dim G g1/2= 1 +log (p
q+ r
s)
log2 , dim(G (g1)) = 1.So for example if p = 5/8, q = 1/8, r = 7/8, s = 2/8 then dim
G g1/2 =(log 9 log 2)/ log 2 = 1.1699... So the image under Tof the three line segmentsG (f0) ,G
f1/2
,G (f1) is a sandwich of three curves, the upper and lower having
dimension one and the middle curve having box-counting dimension greater thanone and less than two. This sandwich is repeated at finer and finer scales, as canbe seen by applying compositions of finite sequences of operators from the setF(1) , F(2) , F(3)
to the sandwich. This notion is implicit in Figure 7.
ACKNOWLEDGEMENTS. I thank Louisa Barnsley for the illustrations. Ithank Maria Moszynska for helpful discussions concerning Theorem 4.2. I thankUta Frieberg for interesting observations about point-fibred affine IFSs on trian-
gles, at an early stage of the work reported here. I also thank David C. Wilson,Peter Massopust, and Andrew Vince, for observations and comments related toaspects of this work.
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Transformations between fractals 23
Figure 7. The image on the left, which is supported on R, istransformed to become the image on the right under the fractalhomeomorphism T discussed at the end of Section 6.3. Hori-zontal lines on the left are transformed to become the graphs offractal interpolation functions. For example the horizontal linethrough the center of the image on the left becomes a curve withfractal dimension 1.1699..., illustrated in black in the image onthe right.
References
[1] Ross Atkins, M. F. Barnsley, David C. Wilson, Andrew Vince, A characterizationof point-fibred affine iterated function systems,Preprint, submitted for publication,(2009).
[2] M. F. Barnsley and S. G. Demko, Iterated function systems and the global construc-tion of fractals, Proc. Roy. Soc. London Ser. A 399 (1985) 243275.
[3] M. F. Barnsley, Fractal functions and interpolation. Constr. Approx. 2 (1986), no.4, 303329.
[4] , Theory and application of fractal tops. 320,Fractals in Engineering: NewTrends in Theory and Applications. Levy-Vehel J.; Lutton, E. (eds.) Springer-Verlag,London Limited, 2005.
[5] , Superfractals, Cambridge University Press, Cambridge, 2006.
[6] , Transformations between self-referential sets,Amer. Math. Monthly 116(2009) 291-304.
[7] M. F. Barnsley, J. E. Hutchinson, New methods in fractal imaging,Proceedings of theInternational Conference on Computer Graphics, Imaging and Visualization, (July
26-28, 2006), IEEE Society, Washington D.C. 296-301.[8] M. F. Barnsley, P. Massopust, M. Porter, Fractal interpolation and superfractals
using bilinear transformations, Preprint, 2009.
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