Real Projective Iterated Function Systems

8/10/2019 Real Projective Iterated Function Systems

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REAL PROJECTIVE ITERATED FUNCTION SYSTEMS

MICHAEL F. BARNSLEY, ANDREW VINCE, AND DAVID C. WILSON

Abstract. This paper contains four main results associated with an attractorof a projective iterated function system (IFS). The first theorem characterizes

when a projective IFS has an attractor which avoids a hyperplane. The secondtheorem establishes that a projective IFS has at most one attractor. In thethird theorem the classical duality between points and hyperplanes in projec-

tive space leads to connections between attractors that avoid hyperplanes andrepellers that avoid points as well as hyperplane attractors that avoid points

and repellers that avoid hyperplanes. Finally, an index is defined for attractorswhich avoid a hyperplane. This index is shown to be a nontrivial projective

invariant.

1. Introduction

This paper provides the foundations of a surprisingly rich mathematical theoryassociated with the attractor of a real projective iterated function system (IFS).In addition to proving conditions which guarantee the existence and uniqueness ofan attractor for a projective IFS, we also present several related concepts. Thefirst connects an attractor which avoids a hyperplane with a hyperplane repeller.The second uses information about the hyperplane repeller to define a new indexfor an attractor. This index is both invariant under projective transformations

and nontrivial, which implies that it joins the cross ratio and Hausdorff dimensionas nontrivial invariants of the projective group. Thus, these attractors belong in anatural way to the collection of geometrical objects of classical projective geometry.

The definitions that support expressions such as iterated function system,attractor, contractive IFS, basin of attraction and avoids a hyperplane,used in this Introduction, are given in Section 3.

Iterated function systems are a standard framework for describing and analyzingself-referential sets such as deterministic fractals [2, 3, 23]and some types of randomfractals [8]. Attractors of affine IFSs have many applications, including imagecompression [4, 5, 21] and geometric modeling [16]. They relate to the theoryof the joint spectral radius [14] and to wavelets [15]. Projective IFSs have moredegrees of freedom than comparable affine IFSs [7] while the constituent functionsshare geometrical properties such as preservation of straight lines and cross ratios.

Projective IFSs have been used in digital imaging and computer graphics, see forexample [6], and they may have applications to image compression, as proposedin [9, p. 10]. Projective IFSs can be designed so that their attractors are smoothobjects such as arcs of circles and parabolas, and rough objects such as fractalinterpolation functions.

The behavior of attractors of projective IFSs appears to be complicated. Incomputer experiments conducted by the authors, attractors seem to come and goin a mysterious manner as parameters of the IFS are changed continuously. See

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arXiv:1003.34

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2 MICHAEL F. BARNSLEY, ANDREW VINCE, AND DAVID C. WILSON

Example 4 in Section 4 for an example that illustrates such phenomena. Theintuition developed for affine IFSs regarding the control of attractors seems to bewrong in the projective setting. Our theorems provide insight into such behavior.

One key issue is the relationship between the existence of an attractor and thecontractive properties of the functions of the IFS. In a previous paper [1] we in-vestigated the relationship between the existence of attractors and the existenceof contractive metrics for IFSs consisting of affine maps on Rn. We establishedthat an affine IFSFhas an attractor if and only ifF is contractive on all ofRn.In the present paper we focus on the setting where X =Pn is real n-dimensionalprojective space and each function inF is a projective transformations. In thiscaseF is called a projective IFS.

Our first main result, Theorem1,provides a set of equivalent characterizations ofa projective IFS that possesses an attractor that avoids a hyperplane. The adjointFt of a projective IFSF is defined in Section11, and convex body is defined inDefinition6. For a set Xin a topological space, Xdenotes its closure, and int(X)

denotes its interior.Theorem 1. IfF is a projective IFS onPn, then the following statements areequivalent.

(1)Fhas an attractorA that avoids a hyperplane.(2) There is a nonempty open setUthat avoids a hyperplane such thatF(U)

U.(3) There is a nonempty finite collection of disjoint convex bodies{Ci} such

thatF(iCi) int(iCi).(4) There is a nonempty open setU Pn such thatF is contractive onU.(5) The adjoint projective IFSFt has an attractorAt that avoids a hyperplane.

Statement (4) is of particular importance because if an IFS is contractive, thenit possesses an attractor that depends continuously on the functions of the IFS,

see for example [3, Section 3.11]. Moreover, if an IFS is contractive, then variouscanonical measures, supported on its attractor, can be computed by means of thechaos game algorithm [2], and diverse applications, such as those mentionedabove, become feasible. Note that statement (4) of Theorem 1immediately impliesuniqueness of an attractor in the setU, but not uniqueness in Pn. See also Remarks2and 3 in Section13.

Our second main result establishes uniqueness of attractors, independently ofwhether or not Theorem 1 applies.

Theorem 2. A projective IFS has at most one attractor.

The classical projective duality between points and hyperplanes manifests itselfin interesting ways in the theory of projective IFSs. Theorem3below, which de-pends on statement (5) in Theorem1, is an example. It is a geometrical description

of the dynamics ofFas a set operator onPn

. The terminology used is provided inSection11.

Theorem 3. (1) A projective IFS has an attractor that avoids a hyperplane if andonly if it has a hyperplane repeller that avoids a point. The basin of attraction ofthe attractor is the complement of the union of the hyperplanes in the repeller.

(2) A projective IFS has a hyperplane attractor that avoids a point if and only ifit has a repeller that avoids a hyperplane. The basin of attraction of the hyperplaneattractor is the set of hyperplanes that do not intersect the repeller.


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PROJECTIVE IFS 3

Figure 1. The image on the left shows the attractor and hyper-

plane repeller of a projective IFS. The basin of attraction of theleaf-like attractor is the black convex region together with the leaf.The image on the right shows the attractor and repeller of theadjoint system.

Figure1illustates Theorem3. Here and in the other figures we use the disk modelof the projective plane. Diametrically opposite points on the boundary of the diskare identified in P2. In the left-hand panel of Figure1the leaf is the attractorAof

a certain projective IFSFconsisting of four projective transformations on P2

. Thesurrounding grainy region approximates the set R of points in the correspondinghyperplane repeller. The complement of R is the basin of attraction of A. Thecentral green, red, and yellow objects in the right panel comprise the attractor ofthe adjoint IFSFt, while the grainy orange scimitar-shaped region illustrates thecorresponding hyperplane repeller.

Theorem3 enables us to associate a geometrical index with an attractor thatavoids a hyperplane. More specifically, if an attractorA avoids a hyperplane thenAlies in the complement of (the union of the hyperplanes in) the repeller. Since theconnected components of this complement form an open cover ofA and since A iscompact,A is actually contained in a finite set of components of the complement.These observations lead to the definition of a geometric index of A, index(A),as is made precise in Definition 14. This index is an integer associated with an

attractor A, not any particular IFS that generates A. As shown in Section 12,as a consequence of Theorem 4, this index is nontrivial, in the sense that it cantake positive integer values other than one. Moreover, it is invariant under underP SL(n+ 1,R), the group of real, dimension n, projective transformations. Thatis, index(A) = index(g(A)) for all g P SL(n + 1,R).

See Remark 4 of Section 13 concerning attractors and repellers in the case ofaffine IFSs. See Remark 5 in Section 13 concerning the fact that the Hausdorffdimension of the attractor is also an invariant under the projective group.


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2. Organization

Since the proofs of our results are quite complicated, this section describes the

structure of this paper, including an overview of the proof of Theorem1.Section3 contains definitions and notation related to iterated function systems,

and background information on projective space, convex sets in projective space,and the Hilbert metric.

Section4provides examples that illustrate the intricacy of projective IFSs andthe value of our results. These examples also illustrate the role of the avoidedhyperplane in statements (1), (2) and (5) of Theorem 1.

The proof of Theorem1 is achieved by showing that

(1) (2) (3) (4) (1) (5).

Section5contains the proof that (1) (2),by means of a topological argument.Statement (2) states that the IFS

Fis a topological contraction in the sense that

it sends a nonempty compact set into its interior.Section 6 contains the proof of Proposition 4, which describes the action of a

projective transformation on the convex hull of a connected set in terms of its actionon the connected set. This is a key result that is used subsequently.

Section7contains the proof that (2) (3) by means of a geometrical argument,in Lemmas2 and3. Statement (3) states that the compact set, in statement (2),that is sent into its interior can be chosen to be the disjoint union of finitely manyconvex bodies. What makes the proof somewhat subtle is that, in general, there isno single convex body that is mapped into its interior.

Sections8 and9 contain the proof that (3) (4). Statement (4) states that,with respect to an appropriate metric, each function inF is a contraction. Therequisite metric is constructed in two stages. On each of the convex bodies in

statement (3), the metric is basically the Hilbert metric as discussed in Section3.How to combine these metrics into a single metric on the union of the convex bodiesis what requires the two sections.

Section10 contains both the proof that (4) (1) and the proof of Theorem2.Section11contains the proof that (1) (5), namely thatFhas an attractor

if and only ifFt has an attractor. The adjoint IFSFt consists of those projec-tive transformations which, when expressed as matrices, are the transposes of thematrices that represent the functions ofF. The proof relies on properties of anoperation, called the complementary dual, that takes subsets ofPn to subsets ofPn.

Section11also contains the proof of Theorem3, which concerns the relationshipbetween attractors and repellers. The proof relies on classical duality betweenPn

and its dualPn, as well as the equivalence of statement (4) in Theorem1. Note

that, ifF has an attractor A then the orbit underFof any compact set in thebasin of attraction ofA will converge toA in the Hausdorff metric. Theorem3tellsus that if A avoids a hyperplane, then there is also a set R of hyperplanes thatrepel, under the action ofF, hyperplanes close toR. The hyperplane repellerR is such that the IFSF1, consisting of all inverses of functions in F, whenapplied to the dual space ofPn, hasR as an attractor. The relationship betweenthe hyperplane repeller of an IFSF and the attractor of the adjoint IFSFt isdescribed in Proposition10.


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PROJECTIVE IFS 5

Section 12considers properties of attractors that are invariant under the pro-jective groupP SL(n+ 1,R) . In particular, we define index(A) of an attractor Athat avoids a hyperplane, and establish Theorem4which shows that this index isa nontrivial group invariant.

Section 13contains various remarks that add germane information that couldinterrupt the flow on a first reading. In particular, the topic of non-contractiveprojective IFSs that, nevertheless, have attractors is mentioned. Other areas opento future research are also mentioned.

3. Iterated Function Systems, Projective Space, Convex Sets, andthe Hilbert Metric

3.1. Iterated Function Systems and their Attractors.

Definition 1. LetX be a complete metric space. Iffm: X X, m= 1, 2, . . . , M ,are continuous mappings, then

F = (X; f1, f2,...,fM) is called an iterated func-

tion system(IFS).

To define the attractor of an IFS, first define

F(B) =fF

f(B)

for any B X. By slight abuse of terminology we use the same symbolF for theIFS, the set of functions in the IFS, and for the above mapping. For B X, letFk(B) denote the k-fold composition ofF, the union offi1 fi2 fik(B) overall finite words i1i2 ik of length k. DefineF0(B) = B.Definition 2. A nonempty compact setAX is said to be an attractorof theIFS

F if

(i)F(A) = A and(ii) there is an open setU X such thatA U and limkFk(B) = A, for

all compact setsB U, where the limit is with respect to the Hausdorff metric.The largest open setU such that (ii) is true is called the basin of attraction

[for the attractorA of the IFSF].See Remark7 in Section13 concerning a different definition of attractor.

Definition 3. A functionf : X X is called a contraction with respect to a metricd if there is0


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3.2. Projective Space. Let Rn+1 denote (n+1)-dimensional Euclidean space andlet Pn denote realprojective space. Specifically,Pn is the quotient ofRn+1 \ {0}bythe equivalence relation which identifies (x

0, . . . , x

n) with (x

0, . . . , x

n) for any

nonzero R. Let: Rn+1 \ {0} Pn

denote the canonical quotient map. The set (x0, . . . , xn) of coordinates of anyx Rn+1 such that (x) = p is referred to as homogeneous coordinates of p. If

p, q Pn have homogeneous coordinates (p0, . . . , pn) and (q0, . . . , q n), respectively,and

ni=0

piqi = 0, then we say that p and q are orthogonal, and write pq. Ahyperplane inPn is a set the form

H= Hp = {q Pn :pq= 0} Pn,for some p Pn.

We define the round metric dP onPn as follows. Each point p ofPn is repre-

sented by a line inRn+1 through the origin, or by the two points ap and bp wherethis line intersects the unit sphere centered at the origin. Then, in the obvious no-tation,dP(p, q) = min {ap aq , ap bq}wherex y denotes the Euclideandistance betweenxandy in Rn+1. In terms of homogeneous coordinates, the metricis given by

dP(p, q) =

2 2|p, q|pq ,

where, is the usual Euclidean inner product. The metric space (Pn, dP) iscompact.

A function f :PnPn is a projective transformation if there is a non-singularlinear map Lf :Rn+1 \ {0} Rn+1 \ {0} such that Lf = f , i.e. the followingdiagram commutes:

LfRn+1 Rn+1

Pn Pn.

f

The mapLfcan be represented by a real (n + 1) (n +1) non-singular matrix. AnIFSF= (Pn; f1, f2,...,fM) is called a projective IFS if each f F is a projectivetransformation onPn.

3.3. Convex subsets of Pn. We now define the notions of convex set, convexbody, and convex hull of a set with respect to a hyperplane. In Proposition4westate an invariance property that plays a key role in the proof of Theorem 1.

Definition 6. A setS Pn is said to beconvex with respect to a hyperplaneH if S is a convex subset of the affine subspacePn\H. A closed convex set thatavoids a hyperplane and has nonempty interior is called a convex body.

It is important to distinguish this definition of convex from projective convex,which is the term often used to describe a set S Pn with the property that ifl is a line in Pn then S l is connected. (See [18, 22] for a discussion of relatedmatters.)


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PROJECTIVE IFS 7

Definition 7. Given a hyperplaneH Pn and two pointsx, y Pn\H, the uniquelinexy throughx andy is divided into two closed line segments byx andy. Theone that does does not intersectHwill be called the line segment with respectto Hand denoted xyH.

Note that C is convex with respect to a hyperplane H if and only if xyH Cfor all x, y C.Definition 8. LetS Pn and letHbe a hyperplane such thatS H =. Theconvex hull of S with respect to H is

convH(S) = conv(S),

whereconv(S) is the usual convex hull ofS, treated as a subset of the affine spacePn\H.

We can also describe convH(S) as the smallest convex subset of Pn\H thatcontains S, i.e., the intersection of all convex sets ofPn

\H containing S. The

key result concerning convexity and projective transformations is Proposition4 inSection6.

3.4. The Hilbert metric. In this section we define the Hilbert metric associatedwith a convex body.

Let p, q Pn, with p= qand with homogeneous coordinates p = (p0, . . . , pn)and q = (q0, . . . , q n). Any point r on the line pqhas homogeneous coordinatesri = 1pi+2qi, i= 0, 1, . . . , n. The pair (1, 2) is referred to as thehomogeneousparametersofrwith respect topandq. Since the homogeneous coordinates ofpandqare determined only up to a scalar multiple, the same is true of the homogeneousparameters (1, 2).

Leta = (1, 2), b= (1, 2), c= (1, 2), d= (1, 2) be any four points on sucha line in terms of homogeneous parameters. Theircross ratio R(a,b,c,d), in terms

of homogeneous parameters on the projective line, is defined to be

(3.1) R(a,b,c,d) =

1 12 21 12 2

1 12 21 12 2 .

The key property of the cross ratio is that it is invariant under any projectivetransformation and under any change of basis{p, q} for the line. If none of thefour points is the first base pointp, then the homogeneous parameters of the pointsare (, 1), (, 1), (, 1), (, 1) and the cross ratio can be expressed as the ratio of(signed) distances:

R(a,b,c,d) =( )( )(

)(

)

.

Definition 9. Let K Pn be a convex body. Let H Pn be a hyperplane suchthat H K = . Let x and y be distinct points in int(K). Let a and b be twodistinct points in the boundary ofKsuch thatxyH abH, where the order of thepoints along the line segmentabH isa, x,y, b. The Hilbert metric dK onint(K)is defined by

dK(x, y) = log R(a,b,x,y) = log

|ay| |bx||ax| |by|

.


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PROJECTIVE IFS 9

Figure 2. Projective attractor which includes a hyperplane, anda zoom. See Example 3.

and / is irrational. In terms of homogeneous coordinates (x,y,z), the attractorofF is the line x = 0. Another example is illustrated in Figure 2, where

Lf1 =

41 19 1919 41 1919 19 41

and Lf2 =10 1 1910 21 1

10 10 10

,Neither functionf1 norf2 has an attractor, but the IFS consisting of both of themdoes. The unionA of the points in the red and green lines is the attractor. Sinceany two lines in P2 have nonempty intersection, the attractor A has nonemptyintersection with every hyperplane. Consequently by Theorem 1, there exist nometric with respect to which both functions are contractive. In the right panel a

zoom is shown which displays the fractal structure of the set of lines that comprisethe attractor. The color red is used to indicate the image of the attractor underf1, while green indicates its image under f2.

EXAMPLE 4 [Attractor discontinuity ]: This example consists of a family F ={F(t) : tR} of projective IFSs that depend continuously on a real parameter t.The example demonstrates how behaviour of a projective family F may be morecomplicated than in the affine case. LetF(t) = (P2; f1, f2, f3) where

Lf1=

198t + 199 198t + 198 198t2 297t 990 1 0198 198 198t 98

,Lf2

= 397 396 594

0 1 0198 198 296 , and Lf3 =

595 594 14850 1 0

198 198 494 .This family iterpolates quadratically between three IFSs,F(0),F(1), andF(2),each of which has an attractor that avoids a hyperplane. But the IFSsF(0.5) andF(1.5) do not have an attractor. This contrasts with the affine case, where similarinterpolations yield IFSs that have an attractor at all intermediate values of theparameter. For example, if hyperbolic affine IFSsF andGeach have an attractor,then so does the average IFS, (tF+ (1 t)G) for all t [0, 1].


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5. Proof that (1) (2) in Theorem 1Lemma 1. [Theorem 1 (1)

(2)] If the projective IFS

F has an attractor A

and there is hyperplaneH such thatH A= , then there is a nonempty open setUsuch thatU H= andF(U) U.Proof. Let O denote an open set containing A such that O is compact and OPn \ H. Since A is an attractor, there is an integer k0 such thatFk(O) O fork k0. DefineV by

A V :=

k=k0

Fk(O) O Pn \ H.

Since the setVis open, each function inFis an open map, and Vis compact, thesetV possesses the following properties:

1. V H= ,2. V is compact,3.F(V) V,F(V) V4.Fm(V) V for some integer m.We next show that there is a (nonempty) set V that satisfies the above four

properties withm = 1. By way of contradiction, assume thatm > 1 is the leastinteger for which there exists aVsatisfying the above four conditions. Let k < m bean integer for which 2k m. We will find an open set Wsatisfying the above fourconditions withk replacingm in the fourth condition, contradicting the minimalityofm. The set Wwill be obtained by a slight fattening ofV .

DefineG:= {f1 f2 fk) | i {1, 2, . . . , M }}B:= {x V| g(x) V :=V\ V for someg G}D:= G(B).

Since 2k m, we have G(D) = G2(B) V. Hence there is an open set ODcontaining D such that OD is compact and G(OD) V. Likewise there is anopen set OB containing B such that OB is compact and G(OB) OD. Now letW := V OB OD. IfxV\ B, then by the definition ofB we have g(x)Vfor any g G. Ifx OB , then by the definition ofOB we have g(x) OD Vfor any g G. Also, ifx OD, then by the definition ofOD, we have g(x) Vfor any g G. Lastly, we can ensure that W H =. This is straightforward:sinceO Pn\H and VO, it follows thatV Pn\H. Since B , DV the opensetsOB , OD can be chosen so that they are subsets ofPn\H. In fact, they can bechosen so that OB, OD Pn\H. Since W = V OB OD we have W Pn\H.Thus, W is an open set that satisfies the four conditions with m = k in condition(4). Since we have the desired contradiction, we conclude that there is a set V

which obeys the four conditions with m = 1.SinceF(V) =F(V)V, it follows thatF(V) H=. IfU =F(V), then U

obeys all four conditions withm = 1. In addition U H= .

6. Projective transformations of convex sets

This section describes the action of a projective transformation on a convex set.We develop the key result, Proposition 4, that is used subsequently.


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PROJECTIVE IFS 11

Proposition1 states that the property of being a convex subset (with respect toa hyperplane) of a projective space is preserved under a projective transformation.

Proposition 1. Let f : Pn Pn be a projective transformation. For any twohyperplanesH, H withS H=andf(S) H =, the setS Pn is a convexset with respect to H if and only iff(S) is convex with respect to H.

Proof. Assume that S is convex with respect to H. To show thatf(S) is convexwith respect toH it is sufficient to show, given any two points x, y f(S), thatxyH f(S). Ifx = f1(x) and y = f1(y), then by the convexity ofS andthe fact that S H =, we know that xyH S. Hence f(xyH) f(S). Sincef(S) H = , and ftakes lines to lines, xyH =f(xyH) f(S).

The converse follows by repeating the above proof forf1.

Proposition2states that convH(S) behaves well under projective transformation.

Proposition 2. LetS Pn and letHbe a hyperplane such thatS H =. Iff : Pn Pn be a projective transformation, then

convf(H)f(S) = f(convH(S)).

Proof. Since S convH(S), we know that f(S) f(convH(S)). Moreover, byProposition1, we know thatf(convH(S)) is convex with respect tof(H). To showthatconvf(H)f(S) =f(convH(S)) it is sufficient to show that f(convH(S)) is thesmallest subset containing f(S), i.e., there is no set C such that Cis convex withrespect to f(H) and f(S) C f(convH(S)). However, if such a set exists,then by applying the inverse f1 to the above inclusion, we have S f1(C)convH(S). Sincef

1(C) is convex by Proposition 1, we arrive at a contradictionto the fact thatconvH(S) is the smallest convex set containing S.

In general,convH(S) depends on the avoided hyperplane H. But, as Proposition3shows, it is independent of the avoided hyperplane when S is connected.

Proposition 3. IfS Pn is a connected set such thatS H = S H = forhyperplanesH, H ofPn, then

convH(S) = convH(S).

Proof. Since S H=, the set convH(S) is the ordinary convex hull ofS in theaffine space EH := Pn \ H. Since S H =, the set convH(S) also equals theordinary convex hull ofSin EH\H =EH \H, which, in turn, equals the ordinaryconvex hull ofS in EH := Pn \ H, i.e. equals convH(S).

The key result, that will be needed for example in Section 7, is the following.

Proposition 4. Let S Pn be a connected set and let H be a hyperplane. IfS H= andf : Pn Pn is a projective transformation such thatf(S) H= ,then

convHf(S) = f(convH(S)).

Proof. This follows at once from Propositions 2 and 3.


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7. Proof that (2)(3) in Theorem1The implication (2)

(3) in Theorem 1 is proved in two steps. We show that

(2)(2.5)(3) where (2.5) is the following statement.(2.5) There is a hyperplane H and nonempty finite collection of nonempty dis-

joint connected open sets{Oi} such thatF(iOi) iOi andiOi H= .Lemma 2. [(2)(2.5)] If there is a nonempty open setU and a hyperplaneHwith U H = such thatF(U) U, then there is a nonempty finite collectionof nonempty disjoint connected open sets{Oi} such thatF(iOi) iOi andiOi H= .Proof. Let U =U, where the U are the connected components of U. LetA= kFk(U) and let{Oi} be the set ofU that have nonempty intersection withA. This set is finite because the sets in {Oi} are pairwise disjoint and Ais compact.SinceF(A) A andF(U) U, we find thatF(Oi) iOi. SinceiOi UandU H= , we haveiOi H= . Lemma 3. [(2.5)(3)]: If there is a nonempty finite collection of nonemptydisjoint connected open sets{Oi} and a hyperplaneH such thatF(iOi) iOiandiOiH =, then there is a nonempty finite collection of disjoint convexbodies{Ci} such thatF(iCi) int(iCi).Proof. Assume that there is a nonempty finite collection of nonempty disjoint con-nected open sets{Oi} such thatF(iOi) iOi andiOi avoids a hyperplane.Let O =iOi. SinceF(O) O, it must be the case that, for each f F andeach i, there is an index that we denote by f(i), such that f(Oi) Of(i). SinceOi is connected and both Oi and f(Oi) avoid the hyperplane H it follows fromProposition4 that

f(convH(Oi)) = convH(f(Oi)) convH(Of(i)) int(convH(Of(i))).For eachi, let Ci= convH(Oi), so that each Ci is a convex body. Then we have

f(Ci) int(Cf(i)).However, it may occur, for some i= j , that Ci Cj=. In this case Ci Cj

is a connected set that avoids the hyperplane H, and is such that f(Ci Cj) alsoavoids H. It follows again by Proposition4 that

convH(f(Ci Cj)) = f(convH(Ci Cj)) int(conv(Cf(i) Cf(j)).Define Ci and Cj to be related ifCi Cj=, and let denote the transitive

closure of this relation. (That is, ifCi is related to Cj andCj is related toCk, thenCi is related to Ck.) From the set{Ci}define a new setU whose elements are

U = convCZ

C : Zis an equivalence class with respect to .By abuse of language, let{Ci}be the set of convex sets in U. It may again occur,for some i= j , that Ci Cj=. In this case we repeat the equivalence process.In a finite number of such steps we arrive at a finite set of disjoint convex bodies{Ci}such thatF(Ci) int(Ci).

Lemma2 and Lemma3 taken together imply that (2) (3) in Theorem1.


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PROJECTIVE IFS 13

8. Part 1 of the proof that (3) (4) in Theorem1The standing assumption in this section is that statement (3) of Theorem 1 is

true. We begin to develop a metric with respect to whichF is contractive. Thefinal metric is defined in the next section.

LetU :={C1, C2,...,Cq} be the set of nonempty convex connected componentsin statement (3) of Theorem 1. Define a directed graph (digraph) G as follows.The nodes ofG are the elements ofU. For each f F, there is an edge coloredfdirected from node U to node V iff(U)int(V). Note that, for each node Uin G, there is exactly one edge of each color emanating from U. Note also thatGmay have multiple edgesfrom one node to another and may have loops. (A loop isan edge from a node to itself.)

Adirected pathin a digraph is a sequence of nodesU0, U1, . . . , U k such that thereis an edge directed from Ui1 to Ui for i = 1, 2 . . . , k. Note that a directed pathis allowed to have repeated nodes and edges. Letp = U0, U1, . . . , U k be a directedpath. Iff1, f2, . . . , f k are the colors of the successive edges, then we will say that p

has typef1 f2 fk.Lemma 4. The graphG cannot have two directed cycles of the same type startingat different nodes.

Proof. By way of contradiction assume that U=U are the starting nodes of twopathsp and p of the same typef1 f2 fk. Recall that the colors are functions ofthe IFS F. Ifg = fk fk1 f1 f0, then the compositiong takes the convex setU into int(U) and the convex set U into int(U). By the Krein-Rutman theorem[19]this is impossible.

Each functionf Facts on the set of nodes ofG in this way: f(U) = V where(U, V) is the unique edge of color f starting at U.

Lemma 5. There exists a metricdG on the set of nodes ofG such that

(1) dG(U, V) 2 for allU=V and(2) eachf F is a contraction with respect to dG.

Proof. Starting from the graphG, construct a directed graphG2whose set of nodesconsists of all unorder pairs{U, V} of distinct nodes ofG. In G2 there is an edgefrom{U, V}to{f(U), f(V)} for all nodes{U, V} in G2 and for each f F. SinceGhas no two directed cycles of the same type starting at different nodes, we knowby Lemma4 thatG2 has no directed cycle. Because of this, a partial ordercanbe defined on the node set ofG2 by declaring that{U, V} {U, V}if there is anedge from{U, V} to{U, V} and then taking the transitive closure. Every finitepartially ordered set has a linear extension (see [17] for example), i.e. there is anordering< of the nodes ofG2:

{U1, V1} < {U2, V2} < < {Um, Vm}such that if{U, V} {U, V} then{U, V}


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Properties (1), (2) and (3) guarantee that dG is a metric on N(G). The fact2 dG(Ui, Vi) 4 for all i guarantees the triangle inequality. If

s= min1i0, where is chosenso small that (i) Hi

Ci= ; and (ii)f(

Ci) int(

Cf(i))f F, i {1, 2,...q}.

Given arbitraryx, y

intCi, leta, bbe the points where the linexy intersectsCi and let af, bfbe the points where the line f(x)f(y) intersectsCf(i). (Clearly

a,b,af, bf depend on i but we have not specified this to avoid clutter.) Let didenote the Hilbert metric on the interior ofCi for eachi, and define

f,i = min{|xy| :x Cf(i), y f(Ci)} >0, forf F,i {1, 2,...q} .We claim that


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PROJECTIVE IFS 15

df(i)(f(x), f(y)) = ln|aff(y)| |f(x) bf||aff(x)| |f(y) bf|(9.1)

1f,i+ 1

ln

|f(a) f(y)| |f(x) f(b)||f(a) f(x)| |f(y) f(b)|

=

1

f,i+ 1 ln

|a y| |x b||a x| |y b|

=

1

f,i+ 1di(x, y),

for allx, y intCi, for allf F, and alli = 1, 2,.... Here| |denotes Euclidean

distance as discussed in Section3. The second to last equality is the invariance of thecross ratio under a projective transformation. To prove the inequality, let, withoutloss of generality,|f(a) f(b)| = 1 and let h :=|f(a) f(y)| and s :=|f(a) f(x)|.Moreover letr := |aff(x)| and t := |f(y) bf|. Finally lets = 1 sand h = 1 h.Note that s h < 1. With this notation, the inequality in the claim is

R:= ln (r+ h)(t + s)

(r+ s)(t + h)

1f,i+ 1

lnhs

sh

.

It is easy to check thatR, as a function of the single positive variabler,is decreasing.Hence R is maximized when r = f,i; similarly when t = f,i. Therefore it issufficient that

ln

(f,i+h)(f,i+s)

(f,i+s)(f,i+h)

lnhs

sh

= ln

(1h)/(f,i+1)(1s)/(f,i+1)

+ ln

(1s)/(f,i+1)(1h)/(f,i+1)

ln

(1h)(1s

+ ln

(1s)(1h)

=

n=1

hnsn

n(f,i+1)n+ s

nhn

n(f,i+1)n

n=1 hnsnn + snhnn 1

f,i + 1

.

The last inequality holds because it holds for corresponding terms in the numeratorand denominator. This completes the proof of Equation (9.1).

Now let = max{ 11+f,i :f F,i = 1, 2,...q}


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Proof. LetU= iint(Ci). Define d : U U by

d(x, y) = di(x, y) if (x, y) Ci Ci for some i,dG(Ci, Cj) if (x, y) Ci Cj for some i =j,where the metrics di and dG are defined in Lemma 5 and Proposition5.

First we show that d is a metric on U. We only need to check the triangleinequality. Ifx, y and z lie in the same connected component ofCi, the triangleinequality follows from Proposition5. Ifx, yand z lie in three distinct components,the triangle inequality follows from Lemma 5. If x, y Ci and z Cj for somei =j , then

d(x, y) + d(y, z) = di(x, y) + dG(uy, uz) dG(uy, uz) = dG(ux, uz) = d(x, z),d(x, z) + d(z, y) = dG(ux, uz) + dG(uz, uy) 2 di(x, y) = d(x, y).

Second we show thatF is contractive with respect to d. By Proposition5 thereis 0

< 1 such that, ifx and y lie in the same connected component ofU and

f F, thend(f(x), f(y)) d(x, y).

Ifx and y lie in different connected components ofU, then there are two cases. Iff(x) and f(y) lie in different connected components, then by Lemma5,

d(f(x), f(y)) = dG(f(x), f(y)) G dG(x, y) = d(x, y),whereG is the constant guaranteed by Lemma5. Iff(x) andf(y) lie in the sameconnected componentUi, then

d(f(x), f(y)) = di(f(x), f(y)) 1 12

dG(x, y) =1

2d(x, y).

Third, and last, the metric d generates the same topology on Uas the metric dP,

because the Hilbert metric dKand the metricdP are bi-Lipshitz equivalent on anycompact subset of the interior of the convex body K; see Remark 5 in Section13.

10. Proof that (4)(1) in Theorem 1 and the Proof of theUniqueness of Attractors

This section contains a proof that statement (4) implies statement (1) in Theorem1and a proof of Theorem2 on the uniqueness of the attractor.

A pointpf Pn is said to be an attractive fixed pointof the projective transfor-mation f iff(pf) = pf, and f is a contraction on some open ball centered at pf.Iffhas an attractive fixed point, then the real Jordan canonical form [24]can beused to show that the linear transformation Lf : Rn+1

Rn+1 has a dominant

eigenvalue. In the case that fhas an attractive fixed point, let Ef denote then-dimensional Lf-invariant subspace ofRn+1 that is the span of the eigenspacescorresponding to all the other eigenvalues. LetHf :=(Ef) be the correspondinghyperplane inPn. Note that Hf is invariant under f andpf /Hf. Moreover, thebasin of attraction ofxf forf isPn \ Hf.Lemma 7. [Theorem1 (4) (1)]: If there is a nonempty open setU Pn suchthatF is contractive onU, thenFhas an attractorA that avoids a hyperplane.


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PROJECTIVE IFS 17

Proof. We are assuming statement (4) in Theorem 1the IFSFis contractive onUwith respect to some metric d.. SinceU is compact and (Pn, dP) is a complete

metric space, (U , d) is a complete metric space. It is well known in this case[23]thatFhas an attractorA U. It only remains to show that there is a hyperplaneHsuch that A Pn \ H.

Let f be any function inF. Since f is a contraction on U, we know by theBanach contraction mapping theorem that fhas an attractive fixed point xf. Weclaim that xf A. If x Pn \Hf lies in the basin of attraction of A, thenxf = limk f

k(x) A. It now suffices to show that AHf =. By wayof contradiction, assume that x AHf. SinceF is contractive on U, it iscontractive on A. Since xf A, we have d(fk(x), xf) = d(fk(x), fk(xf)) 0 ask , which is impossible since fk(x) Hf andxf / Hf.

So now we have that Statements (1), (2), (3) and (4) in Theorem 1are equivalent.The proof of Lemma7 also shows the following.

Corollary 1. IfF is a contractive IFS, then each f F has an attractive fixedpointxfand an invariant hyperplaneHf.

Proposition 6. LetFbe a projective IFS containing at least one map that has anattractive fixed point. IfF has an attractor A, then A is the unique attractor inPn.

Proof. Assume that there are two distinct attractors A, A , and let U, U be theirrespective basins of attraction. IfU U =, then A = A, because if there isx U U then A = limkFk(x) = A, where the limit is with respect to theHausdorff metric. ThereforeU U = and A A = .

Iff Fhas an attractive fixed pointpf andp U\ Hf,and p U \ Hf, thenboth

pf = limk fk(p) lim Fk(p) = A, andpf = lim

kfk(p) lim Fk(p) = A.

But this is impossible since A A = . So Proposition6 is proved. We can now prove Theorem 2 - that a projective IFS has at most one attractor.

Proof of Theorem 2. Assume, by way of contradiction, that A and A are bothattractors. As in the proof of Proposition6, it must be the case that A A =.This fact implies that A and A cannot both contain a hyperplane. Otherwise,since any two hyperplanes have nonempty intersection, A A =. So, withoutloss of generality, A H = for some hyperplane H. Since it has already beenproved that (1)

(4) in Theorem 1,

F is contractive on some subset ofPn

\H.

By Theorem 1, each f F has an attractive fixed point. By Proposition 6 theattractor is unique.

11. Duals and Adjoints

Recall that dP(, ) is the metric on Pn defined in Section3.2. The hyperplaneorthogonal to p P is defined and denoted by

p = {q Pn : qp}.


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If (X, dX) denotes a compact metric space X with metric dX, then (H(X), hX)denotes the corresponding compact metric space that consists of the nonemptycompact subsets of

Xwith the Hausdorff metric hX derived from dX, defined by

hX(B, C) = max {supbB

infcC

dX(b, c), supcC

infbB

dX(b, c)}

for allB, C H.It is a standard result that ifF= (X; f1, f2,...,fM) is a contractiveIFS, thenF : H(X) H(X) is a contraction with respect to the Hausdorff metric.Definition 10. The dual spacePn of Pn is the set of all hyperplanes of Pn,equivalentlyPn ={p : p Pn}. The dual space is endowed with a metric dPdefined by

dP(p, q) = dP(p, q)

for allp, q

P. The mapD :Pn

Pn defined by

D (p) = p

is called theduality map. The duality map can be extended to a map D : H(Pn) H(Pn) between compact subsets ofPn andPn in the usual way.

Given a projective transformation f, its inverse f1 satisfies Lf1 =L1f . In a

similar fashion, define ft andft by

Lft =Ltf and Lft = (L

1f )

t = (Ltf)1,

wheret denotes the transpose matrix. For a projective IFS F, the following relatediterated function systems will be used in this section.

(1) The adjoint of the projective IFSFis denoted byFt and defined to be

Ft = Pn; ft1, ft2,...,ftM .

(2) The inverse of the projective IFSFis the projective IFSF1 = Pn; f11 , f12 ,...,f1M .

(3) IfF= (Pn; f1, f2,...,fM) is a projective IFS then the corresponding hyper-plane IFS is F= (Pn; f1, f2,...,fM),where fm :Pn Pn is defined by fm(H) ={fm(q) | q H}. Noticethat, whereasFis associated with the compact metric space (Pn, dP), thehyperplane IFSFis associated with the compact metric space (Pn, dP).

(4) The corresponding inverse hyperplane IFS isF1 = (Pn; f11 , f12 ,...,f1M ),wheref1m :Pn Pn is defined by f1m (H) = {f1m (q) | q H}.

Proposition 7. The duality mapDis a continuous, bijective, inclusion preservingisometry between compact metric spaces(Pn, dP) and

Pn, dP and also a continu-ous, bijective, inclusion preserving isometry between(H(Pn), hP) and

H(Pn), hP.


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20/31


Proof. The fact that the diagrams commute is easy to verify. Since the otherassertions are also easy to check, we prove only statement (3). Since D is inclusionreversing,

F(X)

Y implies that Y

[F

(X)] =F

(D(X)) =F

(X), theequality coming from the commuting diagram. The definition ofF then yieldsFt(Y) X. Proposition 9. IfF is a projective IFS, U Pn is open, andF(U) U, thenV =U

is open andFt(V) V.

Proof. From statement (3) of Proposition8it follows thatVis open. From F(U) U and from statement (2) of Proposition 8 it follows thatFt(U) U. Bystatement (4) we haveFt(V) = Ft(U) Ft(U) U =V. Lemma 8. [Theorem 1 (1)(5)]: A projective IFSFhas an attractorA thatavoids a hyperplane if and only ifFt has an attractorAt that avoids a hyperplane.Proof. Suppose statement (1) of Theorem 1 is true. By statement (2) of Theorem

1there is a nonempty open set Uand a hyperplane H such thatF(U) U andH U =. By Proposition 9 we haveFt(V) V where V = U is open.Moreover, there is a hyperplane Ht such thatHt V = : simply choose Ht =afor any a A U, where A is the attractor ofF. By the definition of the dualcomplement, a U = which, by statement (4) of Proposition8, implies thata V = a U =. So, as long asV=,Ft also satisfies statement (2) ofTheorem1. In this case it follows that statement (1) of Theorem1 is true forFt,and hence statement (5) is true.

We show that V=by way of contradiction. IfV =, then by the definitionof the dual complement, every y Pn is orthogonal to some point in U , i.e.

U

:= {y : y x for some x U} = Pn.

On the other hand, since Uavoids some hyperplane y

, we arrive at the contradic-tiony / U.

The converse in Lemma8 is immediate because (Ft)t = F. Definition 12. A setA Pn is called ahyperplane attractorof the projectiveIFSF if it is an attractor of the IFSF. A setR Pn is said to be a repellerofthe projective IFSF if R is an attractor of the inverse IFSF1. A setR Pnis said to be ahyperplane repellerof the projective IFSF if it is a hyperplaneattractor of the inverse hyperplane IFSF1.Proposition 10. The compact set A Pn is an attractor of the projective IFSFt that avoids a hyperplane if and only ifD(A) is a hyperplane repeller ofF thatavoids a point.

Proof. Concerning the first of the two conditions in the definition of an attractor,we have from the commuting diagram in Proposition7 thatFt(A) = A if and onlyifF1(D(A)) = D(Ft(A)) = D(A).

Concerning the second of the two conditions in the definition of an attractor,let B be an arbitrary subset contained in the basin of attraction U ofFt. Withrespect to the Hausdorff metric, limk(Ft)k(B) = A if and only if

limk

F1k(D(B)) = limk

D((Ft)k(B)) = D( limk

(Ft)k(B)) = D(A).


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PROJECTIVE IFS 21

Also, the attractorD(A) of

F1 avoids the point p if and only if the attractor A

ofFt avoids the hyperplanep.

Lemma 9. Let f : Pn Pn be a projective transformation with attractive fixedpoint pf and corresponding invariant hyperplane Hf. If f

1 :Pn Pn has anattractive fixed pointHf, thenHf=Hf.Proof. We use a change of basis. There is an invertible matrix M such that

Lf=M

L 00 1

M1,

whereL is a non-singular n n matrix whose eigenvalues satisfy|| 0, there is a positiveintegerK :=K() such thatFK()(B)A + , the setA dilated by an open ballof radius .

In the next paragraph we are going to show that, for sufficiently small > 0,

there is a metric onA+ such thatFis contractive onA+. For now, assumethatF is contractive onA+. This implies, by Theorems1and 2,thatFhas aunique attractor A and it is contained inA + . We now show that A =A. ThatF is contractive on

A + implies thatF, considered as a mapping on H

A +

,

is a contraction with respect to the Hausdorff metric. By the contraction mappingtheorem,Fhas a unique fixed point, so A =A. By choosing small enough thatA+ = A+ lies in the basin of attraction ofA, the fact thatFK(B)A+implies that limkFk(B) = A. Hence U lies in the basin of attraction of A,which concludes the proof of Proposition11.

To prove thatF is contractive onA+ for sufficiently small > 0, we followthe steps in the construction of the metric in statement (4) of Theorem 1, startingfrom the proof of Lemma 2. As in the proof of Lemma2, let U =U, where


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the U are the connected components ofU. Let{Oi} be the set ofU that havenonempty intersection with A. Since

A is compact and nonempty, we must have

( A + ) i

Oi

for allsufficiently small. We now follow the steps in the proof of Lemma 2, Lemma3, up to and including Lemma 6, to construct a metric on a finite set of convexbodies{Ci} such thatiOi iCi and such thatF is contractive oniCi. Notethat the metric is constructed on a set containing iOi, which in turn containsA + . This completes the proof.

We can now prove Theorem3.

Proof of Theorem 3. We prove the first statement of the theorem. The proof ofthe second statement is identical with

F replaced by

F1.

Assume that projective IFSF has an attractor that avoids a hyperplane. Bystatement (4) of Theorem1, the IFSFt has an attractor that avoids a hyperplane.Then, according to Proposition 10,F1 has an attractor that avoids a point. Bydefinition of hyperplane repeller,Fhas a hyperplane repeller that avoids a point.

Concerning the basin of attraction, let R denote the union of the hyperplanesinR and let Q = Pn R. We must show that Q = O, where O is the basin ofattraction of the attractor A ofF.

First we show that O Q, i.e. O R =. Consider any f : Pn Pn withf Fandf1 :Pn Pn. Since we have already shown thatF1 has an attractor,it satisfies all statements of Theorem 1. It then follows, exactly as in the proof

of Lemma 7, that f1 :

Pn

Pn has an attractive fixed point, a hyperplane

Hf R Pn. LetB=

k=1

fF

F1k (Hf) Pn and B= HB

H.

The fact thatHf = Hf (Lemma 9) and Hf O = for all f F implies thatO B =. We claim thatB=R and hence B = R, which would complete theproof that O R =. Concerning the claim, becauseR is the attractor ofF1,we have that

R = limk

F1kfF

Hf B.

SinceHf R for all f F, alsoB R, which completes the proof of the claim.Finally we show thatQ O. By statements (2) and (5) of Theorem1, Ft has an

attractorAt that avoids a hyperplane. Consequently there is an open neighborhoodV ofAt and a metric such thatFt is contractive onV ,and Vavoids a hyperplane.In particularFt is a contraction on H V with respect to the Hausdorff metric.Let denote a contractivity factor forFt|V. Let > 0 be small enough that theclosed set At + (the dilation of At by a closed ball of radius , namely the setof all points whose distance from At is less than or equal to ) is contained in V .


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PROJECTIVE IFS 23

Then

hP(At + , At) =

hP(Ft(At + ), At) = hP(Ft(At + ), Ft(At)) hP(At + , At)) = .

It follows thatFt(At + ) int(At + ) and from Proposition8 (2,3) thatF((At + )) F(int((At + ))) (At + ).

LetQ := (At + ). It follows fromF(Q) Q and Proposition11 thatQ O.

LetR =D (At + ) and let R Pn be the union of the hyperplanes inR.By Proposition 10 and the definition of the dual complement, Q = Pn\R andQ = Pn\R. Since Q O it follows that R Pn\O. SinceD is continuous(Proposition7) and At +At, it follows thatR =D (At + ) D(At) =R.ConsequentlyR

Pn

\O, and therefore Q = Pn

\R

O.

12. Geometrical Properties of Attractors

The Hausdorff dimension of the attractor of a projective IFS is invariant underthe projective group P SL(n+ 1,R). This is so because any projective transfor-mation is bi-Lipshitz with respect to dP, that is, if f : Pn Pn is a projectivetransformation, then there exist two constants 0 < 1 < 2 < such that

1dP(x, y) dP(f(x), f(y)) 2dP(x, y).We omit the proof as it is a straightforward geometrical estimate.

The main focus of this section is another type of invariant.

Definition 13. LetFbe a projective IFS with attractorA that avoids a hyperplaneand letR denote the union of the hyperplanes in the hyperplane repeller ofF. Theindex ofF is

index(F) = # connected componentsO ofPn\R such thatA O= .Namely, the index of a contractive projective IFS is the number of components

of the open set Pn\R which have non-empty intersection with its attractor. Bystatement (1) of Theorem 3, we know that index(F) will always equal a positiveinteger.

Definition 14. Let A denote a nonempty compact subset of Pn, that avoids ahyperplane. IfFA denotes the collection of all projective IFSs for which A is anattractor, then the index ofA is defined by the rule

index(A) = minFFA{

index(F

)}

.

If the collectionFA is empty, then define index(A) = 0.

Note that an attractor A not only has a multitude of projective IFSs associatedwith it, but it may also have a multitude of repellers associated with it. Clearlyindex(A) is invariant under under P SL(n + 1,R), the group of real projectivetransformations. The following lemma shows that, for any positive integer, thereexists a projective IFSFthat has that integer as index.


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Figure 3. A projective IFS with index equal to four. The attrac-tor is sketched in white, while the union of the hyperplanes in thehyperplane repeller is indicated in red, blue, green and gray.

Proposition 12. LetF= (P1; f1, f2, f3,...,fM) be a projective IFS where

Lm:= Lfm =

2m 2m + 1 2m m 12 m (2m 1)

2 2 2m (2m 1)

.

For any integerM >1 and sufficiently large, the projective IFS has index(F) =M.

Proof. Topologically, the projective lineP1 is a circle. It is readily verified that

Lm= m m 12 1 m m 121 1 1 ,from which it can be easily checked that, for is sufficiently large,fmhas attractive

fixed point xm =

m1

and repulsive fixed point ym =

m 1

21

. In particular

Lm

m1

= 2

m1

and Lm

m 12

1

=

m 12

1

. Note that the pointsxi, i =

1, 2, . . . , M , and yi, i = 1, 2, . . . , M , interlace on the circle (projective line). Also,as increases, the attractive fixed points xm become increasingly attractive.

LetIk denote a very small interval that contains the attractive fixed point xk offk, for k = 1, 2,...,M. When is sufficiently large, fm(Ik) Im Ik. It followsthat the attractor ofFis a Cantor set contained inIk. Similarly, the hyperplanerepeller of

Fconsists of another Cantor set that lies very close to the set of points

{k 0.5 : k = 1, 2,...,M}. It follows that index(F) = M. Another example is illustrated in Figure3. In this case the underlying space has

dimension two and the IFSF has index(F) = 4.The previous result shows that the index of a contractive IFS can be any positive

integer. It does not state that the same is true for the index of an attractor. Thenext result shows that the index of an attractor is a nontrivial invariant in that itis not always the case that index(A) = 1.


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PROJECTIVE IFS 25

Theorem 4. IfF= (P1; f1, f2)is the projective IFS in Proposition12withM= 2,= 10, andA is the attractor ofF, then index(A) = 2.

Proof. LetF= (P1; f1, f2), whereL1 =

110 00 1

, L2 =

37 1854 26

.

It is easy to check that f1 = f f1 f1 and f2 = f f2 f1 wheref1 andf2 arethe functions in Proposition12when = 10, andfis the projective transformation

represented by the matrix Lf =

1 11 12

. It is sufficient to show that ifA is the

attractor of

F, then index(

A) = 2. From here on the IFSF is not used, so we

drop the hat fromF, f1, f2,A. Also to simplify notation, the set of points of theprojective line are taken to be P = R{}, where x1 is denoted as the fraction

x1x0

, and,

10

is denoted as. In this notationf1(x) = 110 x and f2(x) = 37x1854x26

when restricted to R. The following are properties ofF.(1) The attractorCofFis a Cantor set.(2) index(F) = 2.(3) The origin a = 0 is the attractive fixed point of f1 while its repulsive

hyperplane is.(4) The attractive fixed point off2 is at c= 2/3 and its repulsive hyperplane

at 1/2.(5) C

[a, b]

[c, d], whereb= 1140

1120

609 (= 0.069351) andd= 114

112

609

(= 0.69351) are the attractive fixed points off1 f2 and f2 f1 respectively.(6) If h is any projective transformation taking C into itself, then h([a, b]

[c, d]) [a, b] [c, d].(7) The symmetric group ofCis trivial, i.e., the only projective transformation

h such that h(C) = Cis the identity.

Property (1) is in the proof of Proposition12, and property (2) is a consequenceof Proposition12. Properties (3) and (4) are easily verified by direct calculation.Property (5) can be verified by checking thatF([a, b] [c, d]) [a, b] [c, d].

To prove property (6), let I denote a closed interval (on the projective line,topologically a circle,) that contains C. Its image h1(I) is also a closed interval.Since h(C) C, it follows that C h1(C). Since C contains{a,b,c,d} andsome points between a and b, h1(I) must contain a, b and some points between

a and b. It follows that h1(I) [a, b]. Similarly h1(I) [c, d]. Thereforeh1(I) [a, b] [c, d], and hence h([a, b] [c, d]) I. Now choose I to be [a, d]to get (A) h([a, b] [c, d]) [a, d]. Choose I to be [c, b] (by which we mean theline segment that goes from c through d then = then through a to endat b,) to obtain (B) h([a, b][c, d]) [c, b]. It follows from (A) and (B) thath([a, b] [c, d]) [a, d] [c, b] = [a, b] [c, d].

To prove property (7), assume thath(C) = C. We will show that h must be theidentity. By property (6)h([a, b] [c, d]) = [a, b] [c, d]. Taking the complement, we


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have h ((b, c) (d, a)) = (b, c) (d, a), and so h([b, c] [d, a]) = [b, c] [d, a]. Henceh([a, b]

[c, d])

h([b, c]

[d, a])

= ([a, b] [c, d]) ([b, c] [d, a]) .It follows thath({a,b,c,d}) = {a,b,c,d}. Any projective transformation that maps{a,b,c,d} to itself must preserve the cross ratio of the four points, so the onlypossibilities are (i) h(a) = a, h(b) = b, h(c) = c, h(d) = d, in which case h is theidentity map; (ii) h(a) = b, h(b) = a, h(c) = d, h(d) = c; (iii) h(a) = c, h(b) =d, h(c) = a, h(d) = b, and (iv) h(a) = d, h(b) = c, h(c) = b, h(d) = a. In eachcase one can write down the specific projective transformation, for example, (iii) isachieved by

h(x) = (d c)(b c)(x a)

(b a + d c)(x c) (d c)(b c)+ c.

The other two specific transformations can be deduced by permuting the symbols

a,b,c,d. In each of the cases (ii), (iii) and (iv) it is straightforward to check nu-merically that h(x) does not map C into C. (One compares the union of closedintervals

[f1(a), f1(b)] [f1(c), f1(d)] [f2(a), f2(b)] [f2(c), f2(d)],whose endpoints belong toCand which contains C, with the union

[h(f1(a)), h(f1(b))] [h(f1(c)), h(f1(d))] [h(f2(a)), h(f2(b))] [h(f2(c)), h(f2(d))].)It follows that h must be the identity map, as claimed.

LetG= (P1; g1, g2,....,gL) be any projective IFS with attractor equal to C. Wemake two claims.

(i) For anyg G

we haveg = f1

f2

...

fk

, for some k , where each i iseither 1 or 2.

(ii) The IFSsF andG have the same hyperplane repeller.The proof proceeds by first proving claim (i), then showing that claim (i) implies

claim (ii). If claim (ii) is true, then index(G) = index(F) = 2, the last equality byproperty (2) above. This would complete the proof of Theorem 4 because it showsthat any IFS with attractorChas index 2, i.e. index(C) = 2.

To prove claim (i), consider the IFSH = ([a, b] [c, d]; f1, f2, g) where g is anyfunction in IFS G. By property (6) g ([a, b] [c, d])[a, b] [c, d]. SoH is indeeda well-defined IFS. It follows immediately from the fact that both F andG haveattractor equal to C thatH also has attractor C. It cannot be the case thatg([a, b])[a, b] and g ([c, d])[c, d]) since then g would have two attractive fixedpoints which is impossible. Similarly, it cannot occur that g([c, d])

[a, b] and

g([a, b]) [c, d] for then g2 would have two attractive fixed points, which is alsoimpossible. It cannot occur thatg(a) [a, b] andg(b) [c, d] for theng([a, b][c, d])would not be contained in [a, b][c, d], contrary to property (6). Similarly, werule out the possibilities that g(a) [c, d] and g(b) [a, b]; that g(c) [a, b]and g(d) [c, d]; and that g(d) [a, b] and g(c) [c, d]. It follows that eitherg([a, b] [c, d]) [a, b] f1([a, d]), or g([a, b] [c, d]) [c, d] f2([c, b]) where [c, b]denotes the interval from c tothen fromto b. (Here, the containments [a, b] f2([c, b]) and [c, d]f2([c, b]) are readily verified by direct calculation.) It now


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PROJECTIVE IFS 27

follows that either g (C)C f1([a, d]) =f1(C) or g(C)C f2([c, b]) =f2(C).Hence

g(C) C1 :=f1(C)for 1 {1, 2}. Ifg(C) = C1 then h(C) = C where h is the projective transfor-mationf11 g. In this case property (7) implies that h must be the identity map.Therefore

g = f1 .

If, on the other hand, g (C) f1(C) then we consider the IFS

H1 = (f1([a, b] [c, d]); f1 f1 f11 , f1 f2 f11 , g f11 ),It is readily checked that the functions that comprise this IFS indeed map f1([a, b][c, d]) into itself. The attractor ofH1 is C1 =f1(C) because

H1(C1) = f1 f1 f11 (f1(C)) f1 f2 f11 (f1(C)) g f11 (f1(C))=f1(f1(C)

f2(C))

g(C) = f1(C)

g(C)

=C1 (because g(C) f1(C)).Let

a1 < b1 < c1 < d1

denote the endpoints of the two intervals f1([a, b]) and f1([c, d]), and write ournew IFS as

H1 = ([a1 , b1 ] [c1 , d1 ]; f(1)1, f(1)2, g1),where

f(1)2 =f1 f2 f11 , and g1 =g f11 .Repeat our earlier argument to obtain

g1([a1 , b1 ] [c1 , d1 ]) f2([a1 , b1 ] [c1 , d1 ]),and in particular that

g1(C1) C12 :=f(1)2(C1) = f1 f2 f11 f1(C) = f1 f2(C)for some 2 {1, 2}. Ifg1(C1) = C12 then g1(f1(C)) = f1 f2(C) whichimplies g f11 f1(C) = f1 f2(C) which implies, as above, that

g= f1 f2 .Ifg1(C1) C12 then we construct a new projective IFS H12 in the obviousway and continue the argument. If the process does not terminate with

g= f1 f2 fkfor somek, theng(C) is a singleton, which is impossible becauseg is invertible. Weconclude that

G = (P; f1 , f2 , . . . , f L)where

fl =fl1

fl2

flkl

in the obvious notation. This concludes the proof of claim (i).It only remains to prove that claim (i) implies claim (ii). Assuming (i), we

must show thatRG =RF, whereRF is the hyperplane repeller ofF andRG isthe the hyperplane repeller ofG . Let = 12 and l1l2 be strings of


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symbols in{1, 2} and{1, 2, . . . , L}, respectively. Using claim (1), define: {1, 2, . . . , L} {1, 2} by letting

(l) = l1 l2 ljl ,where

fl =fl1

fl2

fljl ,and letting

(l1l2 ) = (l1) (l2) .We claim that the mapping is surjective. In order to show this, we need somebasic facts about the relationship between infinite strings of symbols and points ofthe Cantor set C. SinceF is a contractive IFS consisting of two injective maps, f1andf2,with a totally disconnected attractor, each point x Ccan be representedby a unique string = 12 {1, 2} in the sense that

(12.1) x= F() = limk f1 f2 fk(x0),where x0 is any point in C; see for example [3, Chapter 4]. Indeed, the mappingF :{1, 2} C is a (continuous) bijection. Let = 12 {1, 2} andlet x = limk f1 f2 fk(x0). Since C is also the attractor of theIFS G = (P; f1 , f2 , . . . , f L), it is likewise true that there is at least one string= l1l2 {1, 2, , L} such that

x= limk

fl1 fl2 flk (x0)= lim

k(fl11

fl1jl1

) (fl21

fl2jl2

) (flk1

flkjlk

)(x0).

By the uniqueness of in equation 12.1, we have () = , showing that issurjective.

We are now going to show that RF RG .Letr RF.Note that the hyperplanesofP are simply the points ofP. Moreover, the hyperplane repeller RFofFis simplythe attractor of the IFSF1 := (P; f11 , f12 ) and the hyperplane repellerRG ofG is the attractor ofG1 := (P; f11 , f12 , . . . , f 1L ). Let r0 be the attractive fixedpoint off11 . Note that r0 lies in bothRG and inRF. According to Theorem1and Theorem3, bothF1 andG1 are contractive. Therefore

r= limk

f11

f12 f1k (r0)

for some = 12 {1, 2}. Since is surjective, there is a stringl1l2 lm {1, 2, . . . , L} such that

r = limk

f11

f12 f1k (r0)

= limk

fk fk1 f11 (r0)= lim

m

flm

flm1 fl11

(r0)

= limk

f1l1

f1l2 f1lk

(r0) limk

(G1)k(r0) = RG .

A similar, but easier, argument shows thatRG RF. HenceF andG have thesame hyperplane repeller.


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PROJECTIVE IFS 29

13. Remarks

Various remarks are placed in this section so as to avoid interrupting the flow of

the main development.

Remark 1. Example 3 in Section4 illustrates that there exist non-contractive pro-jective IFSs that, nevertheless, have attractors. Such IFSs are not well understoodand invite further research.

Remark 2. It is well known [27] that if each function of an IFS is a contractionon a complete metric spaceX, thenFhas a unique attractor inX. So statement(4) of the Theorem1 immediately implies the existence of an attractorA, but notthat there is a hyperplaneH such thatA H= .Remark 3. IfF is a projective IFS onP0 then it possesses the unique attractorA= P0. This attractor consists of a single point, andFis contractive with respectto any metric onP0.

Remark 4. LetF be a contractive IFS. By Corollary 1, each f F has aninvariant hyperplaneHf. If all these invariant hyperplanes are identical, sayHf=H for allf F, then the projective IFSFis equivalent to an affine IFS acting onthe embedded affine spacePn H. More specifically, letG = (Rn; g1, g2,...,gM) bean affine IFS wheregi(x) = L

i(x) + ti and whereL

i is the linear part and ti the

translational part. A corresponding projective IFS isF= (Pn; f1, f2,...,fM) where

Lfi

x0x1.

xn

= 1 0ti Li

x0x1.

xn

,Here Rn corresponds to P\H with H the hyperplane x0 = 0. In this case thehyperplane repeller of

F isH.

Remark 5. Straightforward geometrical comparisons betweendK(x, y)anddP(x, y)show that (i) the two metrics are bi-Lipshitz equivalent on any convex body containedinint (K)and (ii) iffis any projective transformation onPn then the metricdf(P)defined by df(P)(x, y) = dP(f(x), f(y)) for all x, y Pn is bi-Lipschitz equivalentto dP. A consequence of assertions (i) and (ii) is that the value of the Hausdorffdimension of any compact subset ofint (K) is the same if it is computed using theround metricdP or the Hilbert metricdK; see[20, Corollary 2.4, p.30], and its valueis invariant under the group of projective transformations onPn. In particular, theHausdorff dimension of an attractor of a projective IFS is a projective invariant.

Remark 6. Theorem 1 provides conditions for the existence of a metric with respectto which a projective IFS is contractive. In so doing, it invites other directions ofdevelopment, including IFS with place-dependent probabilities [10], graph-directedIFS theory[25], projective fractal interpolation, and so on. In subsequent papers wehope to describe a natural generalization of the joint spectral radius and applicationsto digital imaging.

Remark 7. Definition2of the attractor of an IFS is a natural generalization of thedefinition[3, p.82] of the attractor of a contractive IFS. Another general definition,in the context of iterated closed relations on a compact Hausdorff space, has beengiven by McGehee [26]. He proves that his definition is equivalent to Definition2,


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for the case of contractive iterated function systems. However, readily constructedexamples show that McGehees definition of attractor is weaker than Definition2.

References

[1] Ross Atkins, M. F. Barnsley, David C. Wilson, Andrew Vince, A characterization of point-

fibred affine iterated function systems, Topology Proceedings (to appear) (2009).[2] M. F. Barnsley and S. G. Demko, Iterated function systems and the global construction of

fractals, Proc. Roy. Soc. London Ser. A 399 (1985) 243275.[3] M. F. Barnsley, Fractals Everywhere, Academic Press, Boston, MA, 1988.

[4] M. F. Barnsley and L. P. Hurd, Fractal Image Compression.Illustrations by Louisa F. Anson.

A. K. Peters, Ltd., Wellesley, MA, 1993.[5] M. F. Barnsley, Fractal image compression, Notices Am. Math. Soc. 43 (1996) 657-662.[6] M. F. Barnsley, The life and survival of mathematical ideas, Notices Am. Math. Soc. 57

(2010) 12-24.

[7] M.F. Barnsley, Superfractals, Cambridge University Press, Cambridge, 2006.

[8] M. F. Barnsley, J. Hutchinson, O. Stenflo, V-variable fractals: fractals with partial self sim-ilarity, Advances in Mathematics, 218 (2008) 2051-2088.

[9] M. F. Barnsley,Introduction to IMA fractal proceedings, Fractals in Multimedia (Minneapolis,MN, 2001), 1-12, IMA Vol. Math. Appl., 132, Springer, New York, 2002.

[10] M. F. Barnsley, S. G. Demko, J. H. Elton, J. S. Elton, Invariant measures for Markov

processes arising from iterated function systems with place-dependent probabilities,Ann. Inst.

H. Poincare Probab. Statist. 24 (1988), 367-394.[11] G. Birkhoff,Extensions of Jentzschs theorem, Trans. Amer. Math. Soc. 85 (1957), 219-227.[12] P. J. Bushell, Hilberts metric and positive contraction mappings in Banach space, Arch.

Rat. Mech. Anal. 52 (1973), 330-338.G

[13] Herbert Buseman,The Geometry of Geodesics, Academic Press, New York, 1955.[14] M. A. Berger and Y. Wang, Bounded semigroups of matrices, Linear Algebra and Appl. 166

(1992), 21-27.[15] M. A. Berger,Random affine iterated function systems: curve generation and wavelets,SIAM

Rev. 34 (1992), 361-385.[16] J. Blanc-Talon,Self-controlled fractal splines for terrain reconstruction, IMACS World Con-

gress on Scientific Computation, Modelling, and Applied Mathematics114(1997), 185-204.[17] R. A. Brualdi,Introductory Combinatorics, 4th edition, Elsevier, New York, 1997.

[18] D. Dekker,Convex regions in projective N-space, Am. Math. Monthly 62 (1955), 430-431.[19] M. G. Krein and M. A. Rutman, Linear operators leaving invariant a cone in a Banach

space, Translations American Mathematical Society, Number 26, New York, New York, 1950.

(Translated from Uspehi Matem. Nauk (N.S.)3, no 1(23), (1948) 3-95.)[20] Kenneth Falconer, Fractal Geometry: Mathematical Foundations and Applications, John

Wiley and Sons, Ltd., Chichester, 1990.[21] Yuval Fisher, Fractal Image Compression: Theory and Application, Springer Verlag, New

York, 1995.

[22] J. de Groot and H. de Vries, Convex sets in projective space, Compositio Mathematica 13(1957) 113-118.

[23] J. E. Hutchinson,Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981) 713747.[24] G. E. Shilov, Linear Algebra, Dover Publications, Inc., Minneola, New York, 1997.

[25] R. Daniel Mauldin and S. C. Williams,Random recursive constructions: asymptotic geomet-

rical and topological properties, Trans. Amer. Math. Soc. 295(1986), 325-346.[26] Richard McGehee,Attractors for closed relations on compact Hausdorff spaces,Indiana Univ.

Math. J. 41(1992), 1165-1209.[27] R. F. Williams, Composition of contractions, Bol. da Soc. Brazil de Mat. 2 (1971) 55-59.


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PROJECTIVE IFS 31

Department of Mathematics, Australian National University, Canberra, ACT, Aus-tralia

Department of Mathematics, University of Florida, Gainesville, FL 32611-8105, USA

Department of Mathematics, University of Florida, Gainesville, FL 32611-8105, USA

E-mail address: [email protected]

URL: http://www.superfractals.com

Real Projective Iterated Function Systems

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