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Real Projective Iterated Function Systems

This paper contains four main results associated with an attractor of a projective iterated function system (IFS). The first theorem characterizes when a projective IFS has an attractor which avoids a hyperplane. The second theorem establishes that a projective IFS has at most one attractor. In the third theorem the classical duality between points and hyperplanes in projective space leads to connections between attractors that avoid hyperplanes and repellers that avoid points as well as hyperplane attractors that avoid points and repellers that avoid hyperplanes. Finally, an index is defined for attractors which avoid a hyperplane. This index is shown to be a nontrivial projective invariant.


Introduction
This paper provides the foundations of a surprisingly rich mathematical theory associated with the attractor of a real projective iterated function system (IFS). .) In addition to proving conditions which guarantee the existence and uniqueness of an attractor for a projective IFS, we also present several related concepts. The first connects an attractor which avoids a hyperplane with a hyperplane repeller. The second uses information about the hyperplane repeller to define a new index for an attractor. This index is both invariant under projective transformations and nontrivial, which implies that it joins the cross ratio and Hausdorff dimension as nontrivial invariants under the projective group. Thus, these attractors belong in a natural way to the collection of geometrical objects of classical projective geometry.
The definitions that support expressions such as "iterated function system", "attractor", "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 analyzing self-referential sets such as deterministic fractals [2,3,23] and some types of random fractals [8]. Attractors of affine IFSs have many applications, including image compression [4,5,21] and geometric modeling [16]. They relate to the theory of the joint spectral radius [14] and to wavelets [15]. Projective IFSs have more degrees of freedom than comparable affine IFSs [7] while the constituent functions share geometrical properties such as preservation of straight lines and cross ratios. Projective IFSs have been used in digital imaging and computer graphics, see for example [6], and they may have applications to image compression, as proposed in [9, p. 10]. Projective IFSs can be designed so that their attractors are smooth objects such as arcs of circles and parabolas, and rough objects such as fractal interpolation functions.
The behavior of attractors of projective IFSs appears to be complicated. In computer experiments conducted by the authors, attractors seem to come and go in a mysterious manner as parameters of the IFS are changed continuously. See Example 4 in Section 4 for an example that illustrates such phenomena. The intuition developed for affine IFSs regarding the control of attractors seems to be wrong in the projective setting. Our theorems provide insight into such behavior.
One key issue is the relationship between the existence of an attractor and the contractive properties of the functions of the IFS. In a previous paper [1] we investigated the relationship between the existence of attractors and the existence of contractive metrics for IFSs consisting of affine maps on R n . We established that an affine IFS F has an attractor if and only if F is contractive on all of R n . In the present paper we focus on the setting where X = P n is real n-dimensional projective space and each function in F is a projective transformations. In this case F is called a projective IFS.
Our first main result, Theorem 1, provides a set of equivalent characterizations of a projective IFS that possesses an attractor that avoids a hyperplane. The adjoint F t of a projective IFS F is defined in Section 11, and convex body is defined in Definition 5. An IFS F is contractive on S ⊂ X when F(S) ⊂ S and there is a metric on S with respect to which all the functions of the IFS are contractive, see Definition 3. For a set X in a topological space, X denotes its closure, and int(X) denotes its interior.
Theorem 1. If F is a projective IFS on P n , then the following statements are equivalent.
(1) F has an attractor A that avoids a hyperplane.
(2) There is a nonempty open set U that avoids a hyperplane such that F(U ) ⊂ U . (3) There is a nonempty finite collection of disjoint convex bodies {C i } such that F(∪ i C i ) ⊂ int(∪ i C i ). (4) There is a nonempty open set U ⊂ P n such that F is contractive on U . (5) The adjoint projective IFS F t has an attractor A t that avoids a hyperplane. When these statements are true we say that F is contractive. Statement (4) is of particular importance because if an IFS is contractive, then it 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 various canonical measures, supported on its attractor, can be computed by means of the "chaos game" algorithm [2], and diverse applications, such as those mentioned above, become feasible. Note that statement (4) of Theorem 1 immediately implies uniqueness of an attractor in the set U , but not uniqueness in P n . See also Remark 2 in Section 13.
Our second main result establishes uniqueness of attractors, independently of whether or not Theorem 1 applies.
Theorem 2. A projective IFS has at most one attractor.
The classical projective duality between points and hyperplanes manifests itself in interesting ways in the theory of projective IFSs. Theorem 3 below, which depends on statement (5) in Theorem 1, is an example. It is a geometrical description The image on the right shows the attractor and repeller of the adjoint system. of the dynamics of F as a set operator on P n . The terminology used is provided in Section 11. Theorem 3. (1) A projective IFS has an attractor that avoids a hyperplane if and only if it has a hyperplane repeller that avoids a point. The basin of attraction of the 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 if it has a repeller that avoids a hyperplane. The basin of attraction of the hyperplane attractor is the set of hyperplanes that do not intersect the repeller. Figure 1 illustates Theorem 3. Here and in the other figures we use the disk model of the projective plane. Diametrically opposite points on the boundary of the disk are identified in P 2 . In the left-hand panel of Figure 1 the "leaf" is the attractor A of a certain projective IFS F consisting of four projective transformations on P 2 . The surrounding grainy region approximates the set R of points in the corresponding hyperplane repeller. The complement of R is the basin of attraction of A. The central green, red, and yellow objects in the right panel comprise the attractor of the adjoint IFS F t , while the grainy orange scimitar-shaped region illustrates the corresponding hyperplane repeller.
Theorem 3 enables us to associate a geometrical index with an attractor that avoids a hyperplane. More specifically, if an attractor A avoids a hyperplane then A lies in the complement of (the union of the hyperplanes in) the repeller. Since the connected components of this complement form an open cover of A and since A is compact, 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 13. 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 can take positive integer values other than one. Moreover, it is invariant under under P GL(n + 1, R), the group of real, dimension n, projective transformations. That is, index(A) = index(g(A)) for all g ∈ P GL(n + 1, R).
See Remark 3 of Section 13 concerning attractors and repellers in the case of affine IFSs. See Remark 4 in Section 13 concerning the fact that the Hausdorff dimension of the attractor is also an invariant under the projective group.

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 Theorem 1.
Section 3 contains definitions and notation related to iterated function systems, and background information on projective space, convex sets in projective space, and the Hilbert metric.
Section 4 provides examples that illustrate the intricacy of projective IFSs and the value of our results. These examples also illustrate the role of the avoided hyperplane in statements (1), (2) and (5) of Theorem 1.
The proof of Theorem 1 is achieved by showing that Section 5 contains the proof that (1) ⇒ (2), by means of a topological argument. Statement (2) states that the IFS F is 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 action on the connected set. This is a key result that is used subsequently.
Section 7 contains the proof that (2) ⇒ (3) by means of a geometrical argument, in Lemmas 2 and 3. 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 many convex bodies. What makes the proof somewhat subtle is that, in general, there is no single convex body that is mapped into its interior. Sections 8 and 9 contain the proof that (3) ⇒ (4). Statement (4) states that, with respect to an appropriate metric, each function in F is a contraction. The requisite 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 Section 3. How to combine these metrics into a single metric on the union of the convex bodies is what requires the two sections.
Section 10 contains both the proof that (4) ⇒ (1) and the proof of Theorem 2. Section 11 contains the proof that (1) ⇔ (5), namely that F has an attractor if and only if F t has an attractor. The adjoint IFS F t consists of those projective transformations which, when expressed as matrices, are the transposes of the matrices that represent the functions of F. The proof relies on properties of an operation, called the complementary dual, that takes subsets of P n to subsets of P n . Section 11 also contains the proof of Theorem 3, which concerns the relationship between attractors and repellers. The proof relies on classical duality between P n and its dual P n , as well as equivalence of statement (4) in Theorem 1. Note that, if F has an attractor A then the orbit under F of any compact set in the basin of attraction of A will converge to A in the Hausdorff metric. Theorem 3 tells us that if A avoids a hyperplane, then there is also a set R of hyperplanes that repel, under the action of F, hyperplanes "close" to R. The hyperplane repeller R is such that the IFS F −1 , consisting of all inverses of functions in F, when applied to the dual space of P n , has R as an attractor. The relationship between the hyperplane repeller of an IFS F and the attractor of the adjoint IFS F t is described in Proposition 10.
Section 12 considers properties of attractors that are invariant under the projective group P GL(n + 1, R) . In particular, we define index(A) of an attractor A that avoids a hyperplane, and establish Theorem 4 which shows that this index is a nontrivial group invariant.
Section 13 contains various remarks that add germane information that could interrupt the flow on a first reading. In particular, the topic of non-contractive projective IFSs that, nevertheless, have attractors is mentioned. Other areas open to future research are also mentioned. To define the attractor of an IFS, first define

Iterated Function
for any B ⊂ X. By slight abuse of terminology we use the same symbol F for the IFS, the set of functions in the IFS, and for the above mapping. For B ⊂ X, let F k (B) denote the k-fold composition of F, the union of f i1 • f i2 • · · · • f i k (B) over all finite words i 1 i 2 · · · i k of length k. Define giving the same topology as on U , such that, for each f ∈ F the restriction f | U of f to U is a contraction on U with respect to d.

3.2.
Projective Space. Let R n+1 denote (n+1)-dimensional Euclidean space and let P n denote real projective space. Specifically, P n is the quotient of R n+1 \ {0} by the equivalence relation which identifies (x 0 , . . . , x n ) with (λx 0 , . . . , λx n ) for all nonzero λ ∈ R. Let φ : R n+1 \ {0} → P n denote the canonical quotient map. The set (x 0 , . . . , x n ) of coordinates of some x ∈ R n+1 such that φ(x) = p is referred to as homogeneous coordinates of p. If p, q ∈ P n have homogeneous coordinates (p 0 , . . . , p n ) and (q 0 , . . . , q n ), respectively, and n i=0 p i q i = 0, then we say that p and q are orthogonal, and write p⊥q. A hyperplane in P n is a set the form for some p ∈ P n .

Definition 4.
A set X ⊂ P n is said to avoid a hyperplane if there exists a hyperplane H ⊂ P n such that H ∩ X = ∅.
We define the "round" metric d P on P n as follows. Each point p of P n is represented by a line in R n+1 through the origin, or by the two points a p and b p where this line intersects the unit sphere centered at the origin. Then, in the obvious notation, d P (p, q) = min { a p − a q , a p − b q } where x − y denotes the Euclidean distance between x and y in R n+1 . In terms of homogeneous coordinates, the metric is given by where ·, · is the usual Euclidean inner product. The metric space (P n , d P ) is compact.
A projective transformation f is an element of PGL(n + 1, R), the quotient of GL(n + 1, R) by the multiples of the identity matrix. A mapping f : P n → P n is well defined by f (φx) = φ(L f x), where L f : R n+1 → R n+1 is any matrix representing projective transformation f . In other words, the following diagram commutes: When no confusion arises we may designate an n-dimensional projective transformation f by a matrix L f ∈ GL(n + 1, R) that represents it. An IFS F = (P n ; f 1 , f 2 , ..., f M ) is called a projective IFS if each f ∈ F is a projective transformation on P n .

3.3.
Convex subsets of P n . We now define the notions of convex set, convex body, and convex hull of a set with respect to a hyperplane. In Proposition 4 we state an invariance property that plays a key role in the proof of Theorem 1.
If H ⊂ P n is a hyperplane, then there is a unique hyperplane H ∈ R n+1 such that φ(H) = H. If p ∈ P n \ H, there is a unique 1-dimensional subspace p ∈ R n+1 such that φ(p) = p. Let u be a unit vector orthogonal to H and W = {x : x, u = 1} be the corresponding affine subspace of R n+1 . Define a mapping θ : P n \ H → W by letting θ(p) be the intersection of p with W . Now θ is a surjective mapping from P n \ H onto the n-dimensional affine space W such that projective subspaces of P n \ H go to affine subspaces of W . In light of the above, it makes sense to consider P n \ H as an affine space. Definition 5. A set S ⊂ P n \H is said to be convex with respect to a hyperplane H if S is a convex subset of P n \H, considered as an affine space as described above. Equivalently, with notation as in the above paragraph, S is convex with respect to H if θ(S) is a convex subset of W . A closed set that is convex with respect to 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 ⊂ P n with the property that if l is a line in P n then S ∩ l is connected. (See [18,22] for a discussion of related matters.) Definition 6. Given a hyperplane H ⊂ P n and two points x, y ∈ P n \H, the unique line xy through x and y is divided into two closed line segments by x and y. The one that does does not intersect H will be called the line segment with respect to H and denoted xy H .
Note that C is convex with respect to a hyperplane H if and only if xy H ⊂ C for all x, y ∈ C.
Definition 7. Let S ⊂ P n and let H be a hyperplane such that S ∩ H = ∅. The convex hull of S with respect to H is where conv(S) is the usual convex hull of S, treated as a subset of the affine space P n \H. Equivalently, with notation as above, if S = conv(θ(S)), where conv denotes the ordinary convex hull in W , then conv H (S) = φ(S ).
We can also describe conv H (S) as the smallest convex subset of P n \H that contains S, i.e., the intersection of all convex sets of P n \H containing S. The key result concerning convexity and projective transformations is Proposition 4 in Section 6.
3.4. The Hilbert metric. In this section we define the Hilbert metric associated with a convex body.
Let p, q ∈ P n , with p = q and with homogeneous coordinates p = (p 0 , . . . , p n ) and q = (q 0 , . . . , q n ). Any point r on the line pq has homogeneous coordinates r i = α 1 p i +α 2 q i , i = 0, 1, . . . , n. The pair (α 1 , α 2 ) is referred to as the homogeneous parameters of r with respect to p and q. Since the homogeneous coordinates of p and q are determined only up to a scalar multiple, the same is true of the homogeneous parameters (α 1 , α 2 ). Let be any four points on such a line in terms of homogeneous parameters. Their cross ratio R(a, b, c, d), in terms of homogeneous parameters on the projective line, is defined to be The key property of the cross ratio is that it is invariant under any projective transformation and under any change of basis {p, q} for the line. If none of the four points is the first base point p, then the homogeneous parameters of the points are (α, 1), (β, 1), (γ, 1), (δ, 1) and the cross ratio can be expressed as the ratio of (signed) distances: Definition 8. Let K ⊂ P n be a convex body. Let H ⊂ P n be a hyperplane such that H ∩ K = ∅. Let x and y be distinct points in int(K). Let a and b be two distinct points in the boundary of K such that xy H ⊂ ab H , where the order of the points along the line segment ab H is a, x, y, b. The Hilbert metric d K on int(K) is defined by A basic property of the Hilbert metric is that it is a projective invariant. See [13, p.105] for a more complete discussion of the properties of this metric. See Remark 4 in Section 13 concerning the relationship between the metrics d P and d K and its relevance to the evaluation and projective invariance of the Hausdorff dimension.

Examples
EXAMPLE 1 [IFSs with one transformation]: Let F = (P n ; f ) be a projective IFS with a single transformation. By Theorem 1 such an IFS has an attractor if and only if any matrix L f repesenting f has a dominant eigenvalue. (The map L f has a real eigenvalue λ 0 with corresponding eigenspace of dimension 1, such that λ 0 > |λ| for every other eigenvalue λ.) For such an IFS the attractor is a single point whose homogeneous coordinates are the coordinates of the eigenvector corresponding to λ 0 . The hyperplane repeller of F is the single hyperplane φ(E), where E is the span of the eigenspaces corresponding to all eigenvalues of L f except λ 0 . The attractor of the adjoint IFS is also a single point, φ(E ⊥ ), where E ⊥ is the unique line through the origin in R n+1 perpendicular to the hyperplane E. EXAMPLE 2 [Convex hull caveat ]: In Theorem 1 the implication (2) ⇒ (3) contains a subtle issue. It may seem, at first sight, to be trivial because surely one could choose C simply to be the convex hull of U ? The following example shows that this is not true.
In P 1 a hyperplane is just a point. Let H 0 = 0 1 and H ∞ = 1 0 be two hyperplanes and consider the four points p = −9 1 , q = −2 1 , r = 2 1 , and s = 9 1 in P 1 . Let C 1 be the line segment pq H0 and let C 2 = rs H0 . There are two possible convex hulls of C 1 ∪ C 2 , one with respect to the hyperplane H 0 for example and the other with respect to H ∞ for example. It is routine to check that either H 0 or H ∞ . Thus the situation is fundamentally different from the affine case; see [1].

EXAMPLE 3 [A non-contractive IFS with an attractor ]: Theorem 1 leaves open
the possible existence of a non-contractive IFS that, nevertheless, has an attractor. According to Theorem 1 such an attractor must have nonempty intersection with every hyperplane. The following example shows that such an IFS does exist. Let and θ/π is irrational. In terms of homogeneous coordinates (x, y, z), the attractor of F is the line x = 0.
Another example is illustrated in Figure 2, where Neither function f 1 nor f 2 has an attractor, but the IFS consisting of both of them does. The union A of the points in the red and green lines is the attractor. Since any two lines in P 2 have nonempty intersection, the attractor A has nonempty intersection with every hyperplane. Consequently by Theorem 1, there exist no metric 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 comprise the attractor. The color red is used to indicate the image of the attractor under f 1 , while green indicates its image under f 2 .

EXAMPLE 4 [Attractor discontinuity ]:
This example consists of a family F = {F(t) : t ∈ R} of projective IFSs that depend continuously on a real parameter t. The example demonstrates how behaviour of a projective family F may be more complicated than in the affine case. Let This family interpolates quadratically between three IFSs, F(0), F(1), and F(2), each of which has an attractor that avoids a hyperplane. But the IFSs F(0.5) and Proof. We prove (ii) first. The proof will make use of the function F −1 (X) = {x ∈ P n : f (x) ∈ X for all f ∈ F}. Note that F −1 takes open sets to open sets, Define V k , k = 0, 1, . . . , m, recursively, going backwards from V m to V 0 , as follows. Let V m = V and for k = m−1, . . . , 2, 1, 0, let To verify that a set O k with these properties exists, first note that property (4) holds for k = K. To verify the properties for all k = K − 1, . . . , 2, 1, 0 inductively, assume that O k , k ≥ 1, satisfies property (4). Using property (4) we have We claim that satisfies the properties in the statement of part (ii) of the lemma. (*) By property (5) we have U ∩ H = ∅. By properties (2) and (4) the first inclusion coming from property (6) and the second to last inclusion coming from property (4) applied to k = 0. This completes the proof that there is a nonempty open set U such that lies in the basin of attraction of A, which implies that U is contained in the basin of attraction of A. This completes the proof of (ii). The proof of (i) is the same as the above proof of (ii), except that P n \H is replaced by P n throughout, and the sentence (*) is omitted.

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.
Proposition 1 states that the property of being a convex subset (with respect to a hyperplane) of a projective space is preserved under a projective transformation.
Proposition 1. Let f : P n → P n be a projective transformation. For any two hyperplanes H, H with S ∩ H = ∅ and f (S) ∩ H = ∅, the set S ⊂ P n is a convex set with respect to H if and only if f (S) is convex with respect to H .
Proof. Assume that S is convex with respect to H. To show that f (S) is convex with respect to H it is sufficient to show, given any two points x , y ∈ f (S), that and y = f −1 (y ), then by the convexity of S and the fact that The converse follows since f −1 is a projective transformation.
Proposition 2 states that conv H (S) behaves well under projective transformation.
Proposition 2. Let S ⊂ P n and let H be a hyperplane such that S ∩ H = ∅. If f : P n → P n is a projective transformation, then However, if such a set exists, then by applying the inverse f −1 to the above inclusion, we have is convex by Proposition 1, we arrive at a contradiction to the fact that conv H (S) is the smallest convex set containing S.
In general, conv H (S) depends on the avoided hyperplane H. But, as Proposition 3 shows, it is independent of the avoided hyperplane when S is connected.
Proof. The fact that S is connected and S ∩H = ∅, implies that conv H (S)∩H = ∅. Therefore conv H (S) is the ordinary convex hull of S in (P n \ H)\H , which is an The key result, that will be needed, for example in Section 7, is the following.
Proposition 4. Let S ⊂ P n be a connected set and let H be a hyperplane.
Proof. This follows at once from Propositions 2 and 3.
(2.5) There is a hyperplane H and nonempty finite collection of nonempty disjoint connected open sets Proof. Let U = ∪ α U α , where the U α are the connected components of U . Let A = ∩ k F k (U ) and let {O i } be the set of U α that have nonempty intersection with A. This set is finite because the sets in {O i } are pairwise disjoint andÃ is compact.

Proof. Assume that there is a nonempty finite collection of nonempty disjoint connected open sets
it must be the case that, for each f ∈ F and each i, there is an index that we denote by However, it may occur, for some i = j, that C i ∩ C j = ∅. In this case C i ∪ C j is a connected set that avoids the hyperplane H, and is such that f (C i ∪ C j ) also avoids H. It follows again by Proposition 4 that Define C i and C j to be related if C i ∩ C j = ∅, and let ∼ denote the transitive closure of this relation. (That is, if C i is related to C j and C j is related to C k , then C i is related to C k .) From the set {C i } define a new set U whose elements are U = conv C∈Z C : Z is an equivalence class with respect to ∼ .
By abuse of language, let {C i } be the set of convex sets in U . It may again occur, for some i = j, that C i ∩ C j = ∅. 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 {C i } such that F(∪C i ) ⊂ int(∪C i ). The standing assumption in this section is that statement (3) of Theorem 1 is true. We begin to develop a metric with respect to which F is contractive. The final metric is defined in the next section.
Let U := {C 1 , C 2 , ..., C q } be the set of nonempty convex connected components in statement (3) of Theorem 1. Define a directed graph (digraph) G as follows. The nodes of G are the elements of U. For each f ∈ F, there is an edge colored f directed from node U to node V if f (U ) ⊂ int(V ). Note that, for each node U in G, there is exactly one edge of each color emanating from U . Note also that G may have multiple edges from one node to another and may have loops. (A loop is an edge from a node to itself.) A directed path in a digraph is a sequence of nodes U 0 , U 1 , . . . , U k such that there is an edge directed from U i−1 to U i for i = 1, 2 . . . , k. Note that a directed path is allowed to have repeated nodes and edges. Let p = U 0 , U 1 , . . . , U k be a directed path. If f 1 , f 2 , . . . , f k are the colors of the successive edges, then we will say that p has type f 1 f 2 · · · f k . Lemma 4. The graph G cannot have two directed cycles of the same type starting at different nodes.
Proof. By way of contradiction assume that U = U are the starting nodes of two paths p and p of the same type f 1 f 2 · · · f k . Recall that the colors are functions of the IFS F. If g = f k • f k−1 • · · · • f 1 • f 0 , then the composition g takes the convex set U into int(U ) and the convex set U into int(U ). By the Krein-Rutman theorem [19] this is impossible. More specifically, the Krein-Rutman theorem tells us that if K is a closed convex cone in R n+1 and L : R n+1 → R n+1 is a linear transformation such that L(K) ⊂ int(K), then the spectral radius r(L) > 0 is a simple eigenvalue of L with an eigenvector v ∈ int(K). Every finite partially ordered set has a linear extension (see [17] for example), i.e. there is an ordering < of the nodes of G 2 :  (2) and (3) guarantee that d G is a metric on N (G). The fact 2 ≤ d G (U i , V i ) ≤ 4 for all i guarantees the triangle inequality. If then 0 < s < 1 and, for any f ∈ F, we have

Part 2 of the proof that (3)⇒(4) in Theorem 1
In this section we construct a metric d i on each component C i of the collection {C i } = {C i : i = 1, 2, ..., q} in statement (3) of Theorem 1. We will then combine the metrics d i with the graph metric d G in Section 8 to build a metric on ∪ i C i such that statement (4) in Theorem 1 is true. Proofs that a projective transformation is contractive with respect to the Hilbert metric go back to G. Birkhoff [11]; also see P. J. Bushell [12]. The next lemma is used to compute the contraction factor for projective maps under the Hilbert Metric. Lemma 6. If r ≥ α ≥ 0, t ≥ α, and h, h , s, s ∈ (0, 1), where s = 1 − s, h = 1 − h, and s ≤ h, then log( (r+h)(t+s ) (r+s)(t+h ) ) ≤ log( (α+h)(α+s ) (α+s)(α+h ) ) ≤ 1 α+1 log( hs sh ). Proof. Since we are assuming that s ≤ h, s(1 − h) > 0, and α ≥ 0, it is an easy exercise to show that (α+h)(α+s ) (α+s)(α+h ) ≥ 1. A bit of algebra can be used to show that N : . If we let α = 0 in the above expression, we x j j , whenever |x| < 1, for a logarithm of any base we see that Note that the above inequality holds because the assumption s ≤ h implies s = 1 − s ≥ 1 − h = h and (1 − s) j + h j ≥ (1 − h) j + s j , for all positive integers j. Thus, the series in the numerator and denominator can be compared term by term. Finally, it is a straightforward argument to show the numerator N (α) has the property that if r ≥ α ≥ 0 and t ≥ α ≥ 0, then (r+h)(t+s ) (r+s)(t+h ) ≤ (α+h)(α+s ) (α+s)(α+h ) . Thus, log( (r+h)(t+s ) (r+s)(t+h ) ) ≤ log( (α+h)(α+s ) (α+s)(α+h ) ). Proposition 5. Let F be a projective IFS and let there be a nonempty finite collection of disjoint convex bodies {C i : i = 1, 2, ..., q} such that F(∪ i C i ) ⊂ int(∪ i C i ) as in statement (3)  2. there is a real 0 ≤ α < 1 such that for all x, y ∈ C i , for all i = 1, 2, ...q, for all f ∈ F; and 3. d i (x, y) ≤ 1 for all x, y ∈ C i and all i = 1, 2, ...q.

Proof. For each
Given arbitrary x, y ∈ int C i , let a, b be the points where the line xy intersects We claim that for all x, y ∈ int C i , for all f ∈ F, and all i = 1, 2, .... Here | · | denotes Euclidean distance as discussed in Section 3. The second to last equality is the invariance of the cross ratio under a projective transformation. Concerning the inequality, let, without loss of generality, |f for all x, y ∈ C i , for all i = 1, 2, ...q, for all f ∈ F. Since C i ⊂ int C i it follows that statement (2) in Proposition 5 is true. Statement (1) follows at once from the fact the topology generated by the Hilbert metricd i on C i as defined above is bi-Lipschitz equivalent to d P ; see Remark 4.
Sinced i : C i × C i → R is continuous and C i × C i is compact, it follows that there is a constant J i such thatd i (x, y) ≤ J i for all x, y ∈ C i . Let J = max i J i , and define a new metric d i by d i (x, y) =d i (x, y)/J for all x, y ∈ C i . We have that d i satisfies (1), (2) and (3) in the statement of Proposition 5. (3) of Theorem 1, then there is a nonempty open set U ⊂ P n and a metric d : U → [0, ∞), generating the same topology as d P on U , such that F is contractive on U .

Lemma 7. [Theorem 1 (3)⇒(4)]: If there is a nonempty finite collection of disjoint convex bodies
where the metrics d i and d G are defined in Lemma 5 and Proposition 5.
First we show that d is a metric on U . We only need to check the triangle inequality. If x, y and z lie in the same connected component of C i , the triangle inequality follows from Proposition 5. If x, y and z lie in three distinct components, the triangle inequality follows from Lemma 5. If x, y ∈ C i and z ∈ C j for some i = j, then Second we show that F is contractive with respect to d. By Proposition 5 there is 0 ≤ α < 1 such that, if x and y lie in the same connected component of U and y). If x and y lie in different connected components of U , then there are two cases. If f (x) and f (y) lie in different connected components, then by Lemma 5, where α G is the constant guaranteed by Lemma 5. If f (x) and f (y) lie in the same connected component U i , then Third, and last, the metric d generates the same topology on U as the metric d P , because, for any convex body K, the Hilbert metric d K and the metric d P are bi-Lipshitz equivalent on any compact subset of the interior of K; see Remark 4 in Section 13.

Proof that (4)⇒(1) in Theorem 1 and the Proof of the Uniqueness of Attractors
This section contains a proof that statement (4) implies statement (1) in Theorem 1 and a proof of Theorem 2 on the uniqueness of the attractor.
A point p f ∈ P n is said to be an attractive fixed point of the projective transformation f if f (p f ) = p f , and f is a contraction with respect to the round metric on some open ball centered at p f . If f has an attractive fixed point, then the real Jordan canonical form [24] can be used to show that any matrix L f : R n+1 → R n+1 representing f has a dominant eigenvalue. In the case that f has an attractive fixed point, let E f denote the n-dimensional L f -invariant subspace of R n+1 that is the span of the eigenspaces corresponding to all the other eigenvalues. Let H f := φ(E f ) be the corresponding hyperplane in P n . Note that H f is invariant under f and p f / ∈ H f . Moreover, the basin of attraction of p f for f is P n \ H f .

Lemma 8. [Theorem 1 (4) ⇒ (1)]:
If there is a nonempty open set U ⊂ P n such that F is contractive on U , then F has an attractor A that avoids a hyperplane.
Proof. We are assuming statement (4) in Theorem 1 that the IFS F is contractive on U with respect to some metric d.. Since U is compact and (P n , d P ) is a complete metric space, (U , d) is a complete metric space. It is well known in this case [23] that F has an attractor A ⊂ U . It only remains to show that there is a hyperplane H such that A ⊂ P n \ H. Let f be any function in F. Since f is a contraction on U , we know by the Banach contraction mapping theorem that f has an attractive fixed point x f . We claim that x f ∈ A. If x ∈ P n \ H f lies in the basin of attraction of A, then So now we have that Statements (1), (2), (3) and (4) in Theorem 1 are equivalent. The proof of Lemma 8 also shows the following. Corollary 1. If F is a contractive IFS, then each f ∈ F has an attractive fixed point x f and an invariant hyperplane H f . Proposition 6. Let F be a projective IFS containing at least one map that has an attractive fixed point. If F has an attractor A, then A is the unique attractor in P n . Proof. Assume that there are two distinct attractors A, A , and let U, U be their respective basins of attraction.
where the limit is with respect to the Hausdorff metric. Therefore U ∩ U = ∅ and A ∩ A = ∅.
If f ∈ F has an attractive fixed point p f and p ∈ U \ H f , and p ∈ U \ H f , then both But this is impossible since A ∩ A = ∅. So Proposition 6 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 distinct attractors of F in P n . As in the proof of Proposition 6, it must be the case that A ∩ A = ∅ and hence that their respective basins of attraction are disjoint.
By Let L f be a matrix for f ∈ F in real Jordan canonical form and such that the largest modulus of an eigenvalue is 1. Let W denote the L f -invariant subspace of R n+1 corresponding to the eigenvalues of modulus 1, and let L denote the restriction of L f to W . If E is the subspace of P n corresponding to the subspace W of R n+1 , then, by use of the Jordan canonical form, A(x) ⊂ E and A(x ) ⊂ E. Together with the inclusions above, this implies that A ∩ E = ∅ and A ∩ E = ∅. Hence Each Jordan block of L can have one of the following forms where R is a rotation matrix of the form cos θ − sin θ sin θ cos θ , 0 denotes the 2 × 2 zero matrix, and I denotes the 2 × 2 identity matrix.
Case 1. L : W → W is an isometry. This is equivalent to saying that each Jordan block of L is of type (a) or (b). The fact that | det L| = 1, and regarding L as acting on the unit sphere in W , implies that But W is invariant under L and, if L is the restriction of L to W , then L is an isometry. We now arrive at a contradiction exactly as was done in Case 1.

Duals and Adjoints
Recall that d P (·, ·) is the metric on P n defined in Section 3.2. The hyperplane orthogonal to p ∈ P is defined and denoted by p ⊥ = {q ∈ P n : q⊥p}.
If (X, d X ) denotes a compact metric space X with metric d X , then (H(X), h X ) denotes the corresponding compact metric space that consists of the nonempty compact subsets of X with the Hausdorff metric h X derived from d X , defined by It is a standard result that if F = (X; f 1 , f 2 , ..., f M ) is a contractive IFS, then F : H(X) → H(X) is a contraction with respect to the Hausdorff metric.
Definition 9. The dual space P n of P n is the set of all hyperplanes of P n , equivalently P n = {p ⊥ : p ∈ P n }. The dual space is endowed with a metric d P defined for all p ⊥ , q ⊥ ∈ P. The map D : P n → P n defined by is called the duality map. The duality map can be extended to a map D : H(P n ) → H( P n ) between compact subsets of P n and P n in the usual way.
Given a projective transformation f and any matrix L f representing it, the matrix L f −1 := L −1 f represents the projective transformation f −1 that is the inverse of f . In a similar fashion, define the adjoint f t and the adjoint inverse transformation f −t as the projective transformations represented by the matrices , respectively, where t denotes the transpose matrix. It is easy to check that the adjoint and adjoint inverse are well defined. For a projective IFS F, the following related iterated function systems will be used in this section.
(1) The adjoint of the projective IFS F is denoted by F t and defined to be The inverse of the projective IFS F is the projective IFS .., f M ) is a projective IFS then the corresponding hyperplane IFS is where f m : P n → P n is defined by f m (H) = {f m (q) | q ∈ H}. Notice that, whereas F is associated with the compact metric space (P n , d P ), the hyperplane IFS F is associated with the compact metric space ( P n , d P ).
(4) The corresponding inverse hyperplane IFS is The duality map D is a continuous, bijective, inclusion preserving isometry between compact metric spaces (P n , d P ) and P n , d P and also a continuous, bijective, inclusion preserving isometry between (H(P n ), h P ) and H( P n ), h P . Moreover, the following diagrams commute for any projective transformation f and any projective IFS F: Proof. Clearly D maps P n bijectively onto P n and H(P n ) bijectively onto H( P n ). The continuity of D and the inclusion preserving property are also clear. The definition of d P in terms of d P implies that D is an isometry from P n onto P n . The definition of h P in terms of d P and the definition of h P n in terms of d P implies that D is an isometry from H(P n ) onto H( P n ). The compactness of (P n , d P ) implies that ( P n , d P ) is a compact metric space.
To verify that the diagrams commute it is sufficient to show that, for all x ∈P n and any projective transformation f , we have But, using the ordinary Euclidean inner product, Let S(P n ) denote the set of all subsets of P n (including the empty set).
Definition 10. The complementary dual of a set X ⊂ P n is For an IFS F define the operator F : S(P n ) → S(P n ) by for any X ∈ S(P n ).
Proposition 8. The map * : S(P n ) → S(P n ) is an inclusion reversing function with these properties: 1. The following diagram commutes * S(P n ) → S(P n ) F ↓ ↓ F S(P n ) → S(P n ). *

If
Proof. The fact that the diagrams commute is easy to verify. Since the other assertions are also easy to check, we prove only statement (3). Since * is inclusion reversing, F(X) ⊂ Y implies that Y * ⊂ [F(X)] * = F(X * ), the equality coming from the commuting diagram. The definition of F then yields F t (Y * ) ⊂ X * .
Moreover, there is a hyperplane H t such that H t ∩ V = ∅: simply choose H t = a ⊥ for any a ∈ A ⊂ U , where A is the attractor of F. By the definition of the dual complement, a ⊥ ∩ U * = ∅ which, by statement (4) of Proposition 8, implies that So, as long as V = ∅, F t also satisfies statement (2) of Theorem 1. In this case it follows that statement (1) of Theorem 1 is true for F t , and hence statement (5) is true. We show that V = ∅ by way of contradiction. If V = ∅, then by the definition of the dual complement, every y ∈ P n is orthogonal to some point in U , i.e.
On the other hand, since U avoids some hyperplane y ⊥ , we arrive at the contradiction y / ∈ U ⊥ .
The converse in Lemma 9 is immediate because (F t ) t = F. Definition 11. A set A ⊂ P n is called a hyperplane attractor of the projective IFS F if it is an attractor of the IFS F. A set R ⊂ P n is said to be a repeller of the projective IFS F if R is an attractor of the inverse IFS F −1 . A set R ⊂ P n is said to be a hyperplane repeller of the projective IFS F if it is a hyperplane attractor of the inverse hyperplane IFS F −1 .
where L is a non-singular n × n matrix whose eigenvalues λ satisfy |λ| < 1. Then If x = (0, 0, ..., 0, 1), then by Proposition 7 Proposition 11. If F is a projective IFS and U is an open set such that U avoids a hyperplane and F(U ) ⊂ U , then F has an attractor A and U is contained in the basin of attraction of A.
Proof. We begin by noting that F(U ) ⊂ U implies that F k (U ) ∞ k=1 is a nested sequence of nonempty compact sets. So is also a nonempty compact set. Using the continuity of F : In the next paragraph we are going to show that, for sufficiently small ε > 0, there is a metric on A ε such that F is contractive on A ε . For now, assume that F is contractive on A ε . This implies, by Theorems 1 and 2, that F has a unique attractor A and it is contained in A ε . We now show that A = A. That F is contractive on A ε implies that F, considered as a mapping on H A ε , is a contraction with respect to the Hausdorff metric. By the contraction mapping theorem, F has a unique fixed point, so A = A. By choosing ε small enough that A ε = A ε lies in the basin of attraction of A, the fact that F K (B) ⊂ A ε implies that lim k→∞ F k (B) = A. Hence U lies in the basin of attraction of A, which concludes the proof of Proposition 11.
To prove that F is contractive on A ε for sufficiently small ε > 0, we follow the steps in the construction of the metric in statement (4) of Theorem 1, starting from the proof of Lemma 2. As in the proof of Lemma 2, let U = ∪ α U α , where the U α are the connected components of U . Let {O i } be the set of U α that have nonempty intersection withÃ. Since A is compact and nonempty, we must have for all ε sufficiently small. We now follow the steps in the proof of Lemma 2, Lemma 3, up to and including Lemma 7, to construct a metric on a finite set of convex bodies {C i } such that ∪ i O i ⊂ ∪ i C i and such that F is contractive on ∪ i C i . Note that the metric is constructed on a set containing ∪ i O i , which in turn contains A ε . This completes the proof.
We can now prove Theorem 3.
Proof of Theorem 3. We prove the first statement of the theorem. The proof of the second statement is identical with F replaced by F −1 .
Assume that projective IFS F has an attractor that avoids a hyperplane. By statement (4) of Theorem 1, the IFS F t has an attractor that avoids a hyperplane. Then, according to Proposition 10, F −1 has an attractor that avoids a point. By definition of hyperplane repeller, F has a hyperplane repeller that avoids a point.
Concerning the basin of attraction, let R denote the union of the hyperplanes in R and let Q = P n R. We must show that Q = O, where O is the basin of attraction of the attractor A of F.
First we show that O ⊂ Q, i.e. O ∩ R = ∅. Consider any f : P n → P n with f ∈ F and f −1 : P n → P n . Since we have already shown that F −1 has an attractor, it satisfies all statements of Theorem 1. It then follows, exactly as in the proof of Lemma 8, that f −1 : P n → P n has an attractive fixed point, a hyperplane H f ∈ R ⊂ P n . Let The fact that H f = H f (Lemma 10) and We claim that B = R and hence B = R, which would complete the proof that O ∩ R = ∅. Concerning the claim, because R is the attractor of F −1 , we have that Since H f ∈ R for all f ∈ F, also B ⊂ R, which completes the proof of the claim. Finally we show that Q ⊂ O. By statements (2) and (5) of Theorem 1, F t has an attractor A t that avoids a hyperplane. Consequently there is an open neighborhood V of A t and a metric such that F t is contractive on V , and V avoids a hyperplane. In particular F t is a contraction on H V with respect to the Hausdorff metric. Let λ denote a contractivity factor for F t | V . Let ε > 0 be small enough that the closed set A t ε (the dilation of A t by a closed ball of radius ε, namely the set of all points whose distance from A t is less than or equal to ε) is contained in V . If and let R ε ⊂ P n be the union of the hyperplanes in R ε . By Proposition 10 and the definition of the dual complement, Q ε = P n \R ε and Q = P n \R. Since Q ε ⊂ O it follows that R ε ⊂ P n \O. Since D is continuous (Proposition 7) and A t ε → A t as ε → 0, it follows that R ε = D (A t ε ) → D(A t ) = R. Consequently R ⊂ P n \O, and therefore Q = P n \R ⊂ O.

Geometrical Properties of Attractors
The Hausdorff dimension of the attractor of a projective IFS is invariant under the projective group P GL(n + 1, R). This is so because any projective transformation is bi-Lipshitz with respect to d P , that is, if f : P n → P n is a projective transformation, then there exist two constants 0 < λ 1 < λ 2 < ∞ such that We omit the proof as it is a straightforward geometrical estimate.
The main focus of this section is another type of invariant that depends both on the attractor and on a corresponding hyperplane repeller. It is a type of Conley index and is relevant to the study of parameter dependent families of projective IFSs and the question of when there exists a continuous family of IFS's whose attractors interpolate a given set of projective attractors, as discussed in Example 4 in Section 4. Ongoing studies suggest that this index has stability properties with respect to small perturbations and that there does not exist a family of projective IFSs whose attractors continuously interpolate between attractors with different indices.
Definition 12. Let F be a projective IFS with attractor A that avoids a hyperplane and let R denote the union of the hyperplanes in the hyperplane repeller of F. The index of F is Namely, the index of a contractive projective IFS is the number of components of the open set P n \R which have non-empty intersection with its attractor. By statement (1) of Theorem 3, we know that index(F) will always equal a positive integer.
Definition 13. Let A denote a nonempty compact subset of P n , that avoids a hyperplane. If F A denotes the collection of all projective IFSs for which A is an attractor, then the index of A is defined by the rule If the collection F A is empty, then define index(A) = 0.
Note that an attractor A not only has a multitude of projective IFSs associated with it, but it may also have a multitude of repellers associated with it. Clearly index(A) is invariant under under P GL(n + 1, R), the group of real projective transformations. The following lemma shows that, for any positive integer, there exists a projective IFS F that has that integer as index.
For any integer M > 1 and sufficiently large λ, the projective IFS has index(F) = M. Let I k denote a very small interval that contains the attractive fixed point x k of f k , for k = 1, 2, ..., M. When λ is sufficiently large, f m (∪I k ) ⊂ I m ⊂ ∪I k . It follows that the attractor of F is a Cantor set contained in ∪I k . Similarly, the hyperplane repeller of F consists 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 Figure 3. In this case the underlying space has dimension two and the IFS F 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. The following Theorem 4 shows that the index of an attractor is a nontrivial invariant in that it is not always the case that index(A) = 1. To prove it we need the following definition and result. Definition 14. A set C ⊂ P n is called a Cantor set if it is the attractor of a contractive IFS (P n ; f 1 , f 2 , ..., f M ), M ≥ 2, such that each point of C corresponds to an unique string σ = σ 1 σ 2 · · · ∈ {1, 2, ..., M } ∞ such that where x 0 is any point in C.
Lemma 11. Let F = (P n ; f 1 , f 2 , ..., f M ) be a projective IFS whose attractor is a Cantor set C. Let the projective IFS G = (P n ; f ω1 , f ω2 , . . . , f ω L ) have the same attractor C, where each f ω l is a finite composition of functions in F, i.e.
f ω l = f σ l 1 • f σ l 2 • · · · • f σ l j l in the obvious notation. Then F and G have the same hyperplane repeller and index(F) = index(G).
In this notation f 1 (x) = 1 10 x and f 2 (x) = 37x−18 54x−26 when restricted to R. The following are properties of F.
(1) The attractor C of F is a Cantor set.
(3) The origin a = 0 is the attractive fixed point of f 1 while its repulsive hyperplane is ∞.
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 numerically that h(x) does not map C into C. (One compares the union of closed intervals