Fast Basins and Branched Fractal Manifolds of Attractors of Iterated Function Systems

The fast basin of an attractor of an iterated function system (IFS) is the set of points in the domain of the IFS whose orbits under the associated semigroup intersect the attractor. Fast basins can have non-integer dimension and comprise a class of deterministic fractal sets. The relationship between the basin and the fast basin of a point-fibred attractor is analyzed. To better understand the topology and geometry of fast basins, and because of analogies with analytic continuation, branched fractal manifolds are introduced. A branched fractal manifold is a metric space constructed from the extended code space of a point-fibred attractor, by identifying some addresses. Typically, a branched fractal manifold is a union of a nondenumerable collection of nonhomeomorphic objects, isometric copies of generalized fractal blowups of the attractor.


Introduction
We define and exemplify the continuations and the fast basin of an attractor of an IFS. Then we extend the standard symbolic IFS theory, based on Hutchinson's work [23] on the behaviour of a contractive IFS on its attractor, to a symbolic description of the behaviour of a invertible IFS on a set that contains the fast basin of a point-fibred attractor. We use this description to help define the fractal manifold, a new topological invariant, associated with the attractor of the IFS. We establish relationships between the fractal manifold, the fast basin, and the set of continuations of an attractor of IFS. For example, the projection of the fractal manifold onto the underlying space is the fast basin.
A fact that is fundamental in the theory of IFS on a complete metric space X is that, under very general conditions, there is a natural addressing system for the points of an attractor A of an IFS. In a sense, this is a generalisation of the decimal representation of the real numbers in the interval [0, 1]. Such an addressing scheme is possible for any point-fibred attractor of an IFS (see Definition 2).
A goal of this paper is to extend this addressing scheme, from just the attractor A, to as large a subspace of X as possible. This extended subspace is the fast basin of A. This extension, and the various consequences of this extension, is a subject of this paper. The phrase symbolic IFS is used to emphasize not only parallels with symbolic dynamics but also that the addresses of points in X, like decimals, are infinite words in the alphabet {1, 2, ..., N } where N is the number of functions in the IFS. For an invertible IFS, the fast basin (see Definition 3) is the union of continuations (see Definition 4) of the attractor. In the case of and invertible IFS, the basic idea of how to construct these continuations is to consider the images of an attractor by compositions of inverses of the functions of the IFS. This idea concerning continuations has recently been used both to construct tilings [7] and to construct continuations of fractal functions [14].
Our symbolic description of the behaviour of an invertible IFS F, on a set that contains a point-fibred attractor A is used to define the fractal manifold L, a metric space associated with A and F. It is a manifold in the sense that it has homeomorphically embedded in it all the continuations of the attractor, each continuation being locally homeomorphic to parts of A. There is a natural projection map π : L → X, and the image of the projection is the fast basin.
In [11] we discuss the basic topological and geometrical properties of fast-basins. In [16] we discuss the fractal tiling theory, which is related to fast basins and fractal manifolds. The symbolic IFS theory introduced here provides a canonical addressing structure for fractal tilings and fractal manifolds.
The organization of this paper is as follows. In Section 2 we establish notation and define concepts related to an iterated function system (IFS), an attractor of an IFS, the basin of an attractor, and the fast basin of an attractor. The concept of the fast basin has only recently been introduced [11], but plays a key role in the results of this paper. We also define invertible IFSs, the dual IFS, and the dual repeller of an attractor. In Section 3 we give examples of basins and fast basins of attractors.
Fractal continuation, the subject of Section 4, was introduced in [14], in the context of fractal interpolation. See also [13]. Here the notion of fractal continuation is generalized and related to fast basins. In some cases the continuations can be Riemann surfaces, as illustrated in Example E. For each infinite word generated by the alphabet {1, 2, ..., N } , where N is the number of functions in the IFS, there is a continuation of a point-fibred attractor A. We establish sufficient conditions under which a fractal continuation contains the basin of an attractor. A sufficient condition is that the attractor admits a "full" or, equivalently, "reversible" point, concepts which are defined in this section. The main results are Theorems 1 and 2.
In Section 5 we describe the basic symbolic spaces to be associated with a pointfibred attractor of an invertible IFS, and provide the properties of the shift map and its inverses, restricted to these spaces. These spaces are all subspaces of a product space of the form A ∞ where A is a finite alphabet, consisting of 2N symbols. In some cases the shift map is invertible and in others it is not. One of these spaces, called I + , corresponds to the alphabet {1, 2, ..., N } and is associated with the standard symbolic IFS theory discussed in Section 6. The largest symbolic space that we exploit is I, which, for N > 1, is a shift invariant proper subspace of A ∞ . The symbolic spaces we use are mainly different from the usual single-and double-sided shift spaces used in symbolic dynamics.
In Section 6 we present the "standard" relationship between I + and point-fibred attractors, see for example [21,Theorem 3.2] and [24]. This relationship is implicit in Hutchinson's work [23], for the case of contractive IFSs, and is summarized in the following commutative diagram, which is explained in Theorem 3.
In Section 7 we extend this standard relationship to provide a symbolic representation that relates the behaviour of the functions f ∈ F on A to their behaviour at points that lie in X\A. The new results, most importantly Theorem 5 and its corollaries, together with Theorem 3, provide a more complete symbolic description than the one provided by Theorem 3 alone, of the structure and dynamics of point-fibred attractors of invertible IFSs. First, in Section 7.1 we consider an invertible symbolic IFS Z on I, with a point-fibred attractor, and dual repeller that is an attractor for the dual IFS, and their symbolic fast basins I + and I − . The space I is the largest symbolic space that we use; we can variously project from I to the attractor, dual repeller, and fast basins of an invertible IFS. The main results of Section 7.1 are stated in Theorem 4. Then, in Section 7.2, we define a coding map on a shift invariant subspace J + of I that semi-conjugates the symbolic IFS Z with the IFS in the vicinity of a point-fibred attractor. The main result of this section is Theorem 5. Finally, in Section 7.3, we discuss the symbolic fast basin I + , introduced in Theorem 4, and its relationship with the fast-basin of a point-fibred attractor of an invertible IFS. The main result is Corollary 2.
In Section 8, which lies at the core of this paper, we define and establish properties of the fractal manifold (f-manifold) associated with a point-fibred attractor of an invertible IFS. It is shown that the f-manifold is a topological invariant of (F, A); the sense in which this is meant is explained. Some other descriptive sentences concerning an f-manifold follow. An f-manifold is locally homeomorphic to the attractor that generates it, when appropriate care is taken regarding "branch sets". An f-manifold inherits, via its IFS, many of the properties of the attractor that generates it. It is a disjoint union of leaves, there being at most N +1 topologically distinct leaves. The f-manifold plays the role of a universal object with respect to the continuations of the attractor; a homeomorphic copy of each continuation is embedded in the f-manifold. Implicit in this Section 8 and subsequently, is the revelation that these are natural and beautiful entities, associated with the group generated by the IFS. The main results are Theorems 6 and 7, but clearly a great deal more could be said.
In Section 9 we introduce notation, needed later, concerning sections of the various projection maps, acting between code space, the fractal manifold, and the fast basin.
In Section 10 we summarise the theory of masks, sections, and fractal transformations (f-transformations) between point-fibred IFS attractors. This is needed to define, and establish properties of, transformations between continuations and between fractal manifolds.
In Section 11 we show how, given a section for an attractor of an invertible point fibred attractor of an IFS, sections of all of the projections, between the code space, the f-manifold, and the fast basin, can be induced. This has far reaching consequences including showing how a standard masked IFS yields a unique address for every point on the associated fractal manifold, a unique tiling of the manifold, the generalization of the practical theory of fractal transformations to manifolds and fast basins, and the feasibility of giving conditions under which two continuations are homeomorphic, by inspecting the local code space structure over the associated attractors.
Specifically, in Section 12, we use sections of projections to construct fractal transformations (f-transformations) between f-manifolds, between continuations, and between basins. Theorems 10 and 11, which concerns the topological properties of f-transformations, include conditions under which f-transformations (and their projections) are homeomorphisms. These are "inheritance" theorems, because properties concerning f-transformations between pairs of attractors are shown to also hold for the corresponding transformations between f-manifolds, and (as a consequence) between continuations. Potentially infinitely many continuous homeomorphisms between a fixed pair of continuations can, under fairly general conditions, be obtained.
1.1. Additional introductory remarks. In some cases a fractal manifold can be constructed by "gluing together" certain "fractal blow-ups". In such cases, the f-manifold comprises a canonical branched fractafold, in the sense of Stricharz [33]. An analytic fractal is an attractor of an analytic IFS. An analytic IFS is an IFS on a smooth manifold whose maps are analytic (i.e. possessing locally convergent multivariable Taylor series expansions). An example of an analytic IFS is a set of contractive affine transformations acting on R n or C n . It can occur that the set of continuations of an analytic fractal is essentially independent of the analytic IFS used to construct the continuations, and that, if the attractor is the graph of an analytic function, then the continuations coincide with the analytic continuations of the graph. In this way the present work is related to a generalization of the notion of analytic continuation: the role of the convergent power series or "germ" is played by the analytic fractal, while the role of the set of continuations is played by the fractal manifold.
The f-manifold is locally homeomorphic to the attractor, in a sense to be made precise, similarly to how a topological manifold is related locally to a region in R 2 . Fractal continuations and fractal manifolds are related to fractafolds, as defined by Strichartz [33]. Fractal manifolds occur naturally in the study of the sets of continuations of fractal functions, [14], where they appear as tree-like structures with infinitely many branches at each level. When an attractor of a similitude IFS (equal scalings) is post-critically finite, it is possible to define differential operators that act on functions defined on the attractor [34], and the theory extends to associated fractafolds [33]. This provides a motivation for introducing fractal manifolds. Another motivation is that they relate to a natural generalization of analytic continuation, discussed in [14]. where N ∈ N is fixed and X is a complete metric space. We use the same symbol F for the IFS and for the set of functions in the IFS. We write |F| = N to denote the number of functions in the IFS. Let H = H X be the collection of nonempty compact subsets of X and define for all C ∈ H. We also treat F as a map F : 2 X → 2 X , where 2 X is the collection of all subsets of X. For S ⊂ X, define F 0 (S) = S and let F k (S) denote the k-fold composition of F applied to S, namely, the union of f i1 • f i2 • · · · • f i k (S) over all finite words i 1 i 2 · · · i k of length k. Let d = d X be the metric on X, and let d H = d H X be the corresponding Hausdorff metric. Throughout, the topology on H X is the one induced by d H X . Key facts, proved in [22], for example, are that (H, d H ) is a complete metric space because (X, d) is complete, and that if (X, d) is compact then (H, d H ) is compact.
The union of all open sets U , such that Statement 2 of Definition 1 is true, is called the basin of the attractor A (w.r.t. F). If B = B F (A) denotes the basin of A, then Statement 2 of Definition 1 holds with U replaced by B. An IFS may not possess an attractor, or it may possess multiple attractors. Examples of attractors and their basins, and further discussion, can be found in [13], and in references therein.
The IFS F X is said to be contractive if the maps that it comprises are Lipshitz with Lipshitz constant λ ∈ [0, 1); that is d X (f (x), f (y)) ≤ λd X (x, y), for all x, y ∈ X. A contractive IFS possesses a unique attractor A, with B(A) = X.
if lim k→∞ f σ|k (C) exists and is a singleton subset of A, independently of C ⊂ B(A) with C ∈ H X , where convergence is with respect to the Hausdorff metric.
The attractor of a contractive IFS is point-fibred [23]. Each attractor in this paper is assumed to be point-fibred with respect to its associated IFS.
A principal object of study in this paper is the fast basin of an attractor.
This paper is mainly concerned with "invertible IFSs". The IFS F X is said to be invertible if, for all f ∈ F X , f : X → X is a homeomorphism. If F X is an invertible IFS, then we define the dual IFS to be F * = F * X := {X; f −1 n , n = 1, 2, ..., N }. Some topological and geometrical properties of fast basins of invertible IFSs are discussed in [11].
If A is an attractor of the invertible IFS F X , then the set A * := X\B(A) is called the dual repeller of A (w.r.t. F). The dual repeller A * may or may not be an attractor of the dual IFS F * . It can occur, for example in cases where F C is a Möbius IFS on the Riemann sphere, C, that F C possesses a unique point-fibred attractor A with basin B(A) = C, and that A * is an attractor of F * C with basin B(A * ) = X\A [39].
In general, the relationship between the basin B and the fast basin B, is complicated because neither B ⊂ B nor B ⊃ B. If A has non-empty interior (in the topology of X), then B ⊃ B, see Proposition 1 and Theorem 1. If F is contractive and its attractor has non-empty interior (in the topology of X) then B = B. If F is contractive then B = X and A * = ∅.
Example B (Sierpinski triangle) The fast basin of a Sierpinski triangle ⊂R 2 , with vertices at a, b, c∈ R 2 , w.r.t. the IFS R 2 ; (x + a)/2, (x + b)/2, (x + c)/2 , is B = ∪ t∈G ( +t) where G is the group generated by the set of translations {a − b, b − c, c − a}. In Figure 1 we illustrate B when a = (0, 0), b = (1, 0) and c = (0, 1).  Example C (Other affine IFSs) Figure 2 illustrates part of the fast-basin of a contractive affine IFS (N = 2) of the form The attractor is the partial Koch curve near the center of the figure. The fast basin is the union of the boundaries of the tiles of a tiling of the plane by Koch snowflakes and other related tiles. Figure 3 illustrates part of the fast basin for the contractive affine IFS R 2 ; ( x 2 ± 1 2 , ± x 2 + 2y 5 + 1 4 ) whose attractor, at the center of the image, is the graph of a fractal interpolation function. At each branch point, there is a countable infinity of branches.
Example D (Topological dimension of a fast basin) We use this definition: an for some σ ∈ {1, 2, ..., N } ∞ . Let F R M be an affine IFS whose attractor A is contained in a linear subspace L of R M , where the topological dimension of L is less than M . Then it is easy to show that the fast basin of A is also contained in L because any orbit of any point not in L has empty intersection with A. It follows that the topological dimension of the fast basin of A is strictly less than that of the underlying space. For instance, if M = 2, N = 2, and Any orbit of any point in R 2 \R does not meet the attractor (although the closure of the orbit does meet the attractor) while, for any point in R, there exists an orbit that reaches A in finitely many steps, (i.e. the orbit intersects the attractor).
Example E ( Fast basin and analytic continuation) An example, illustrating the relationship between a fast basin and analytic continuation, is provided by the Fractal continuations were introduced in [14], in the context of fractal interpolation. See also [13,11]. Here the notion of fractal continuation is generalized and related to fast basins. We also establish a condition under which a continuation contains the basin. The condition is that the attractor admits a "full" or "reversible" point, notions which are defined in this section. The main results are Theorems 1 and 2.

Definition 4. Let A be an attractor of an invertible IFS
Note that the latter union is nested. The set B θ is referred to as a continuation of A or a fractal continuation of A. The family { B θ : θ ∈ I + } is referred to as the set of continuations of A (w.r.t. F). We can write The following result is readily verified.

Proposition 1. Let A be a point-fibred attractor of an invertible IFS
By the definition of attractor, for any > 0 there is an M such that if m ≥ M , then The concept of a full word was introduced in [7], as well as the equivalent notion of a reversible word, which is defined below. This equivalence is part of Theorem 2 below, which is proved in [7]. Definition 6. Call θ ∈ I + reversible w.r.t. a point-fibred attractor A and IFS F if there exists an ω ∈ I + such that ω is the address of some point in A • and, for every (In fact, if θ is disjunctive, then every finite word (built using the alphabet I + ) appears as a subword in σ infinitely many times.) With respect to A and F: (1) There are infinitely many disjunctive words in I + .
(3) A word is reversible if and only if it is full.
The paper [14] concerns the special case of fractal continuations of fractal functions. In that case it is proved, under special conditions the most important of which is that the IFS consists of analytic functions, that the set of continuations is uniquely determined by the attractor, independent of the analytic IFS used to generate the attractor. This suggests that the fast basin may be, under quite general conditions, uniquely determined by the attractor and the space in which it sits.

The code space of a point-fibred attractor of an invertible IFS
The standard relationships [23] between the space I + , shifts, and the attractor of a contractive IFS of N ∈ N maps, are detailed in Section 6. But it is efficient, to avoid duplication of notation, first to introduce both a larger symbolic space I, that lies between I + and I 0 , and generalized shifts.
For σ ∈ I k we define |σ| = k for all k ∈ N 0 . In particular |∅| = 0. For θ ∈ I k for all k ∈ N, we write θ = θ 1 θ 2 ...θ |θ| with the convention that if |θ| = 0 then θ = ∅. Let d I be the metric on I 0 such that for all σ, ω ∈ I 0 . The metric d I induces the product topology on I 0 . The space We use conventions such that (−j) ∈ I + and −(−j) = j for all j ∈ I − , and other similar expressions hold.
We define the shift map, S : The metric spaces (I, d I ),(I + , d I ), (I − , d I ) are compact subspaces of the metric space (I 0 , d I ). We have where U denotes the closure of the set U .
The subspaces I, J + , J − , I + , and I − , are shift invariant; consequently, the shift map, (domain and range) restricted to each of these spaces is well defined. Where it does not cause confusion, we will denote these restricted shift maps by the same symbol S; that is, we will write S : I → I, S : J + → J + , S : be an invertible IFS on a complete metric space. The space (I + , d I ) is sometimes called the code space or address space for F, because of its role in a fundamental result, Theorem 3, concerning the addresses of the points of an attractor.
We are going to need the maps introduced in Lemma 1.
Lemma 1. Let n ∈ I. Define s n : I → I by The map s n : I → I is a homeomorphism, with inverse s −n : I →I. Also, s n : for all σ, ω ∈ I.
Proof. It is readily verified that Equation ( Where the meaning is unambiguous, let s n : J + → J + and s n : J − → J − , be the maps s n | J+ : J + → J + and s n | J− : J − → J − , respectively, for n ∈ I. Notice that s n (I + ) ⊂ I + , for all n ∈ I + , and s m (I − ) ⊂ I − for all m ∈ I − . This allows us to introduce the (domain and range) restricted maps s n | I+ : I + → I + (n ∈ I + ) and s m | I− : Where the meaning is unambiguous, let s n : I + → I + (n ∈ I + ) and s m : I − → I − (m ∈ I − ), respectively, be the latter maps. These restricted maps are continuous, but they are not invertible, except under special conditions. Whereas s n : I →I is not a contraction mapping w.r.t. any metric d a on I (Proof: s +1 (+1) = +1,s +1 (−1) = −1, so d a (s +1 (+1), s +1 (−1)) = d a (+1, −1)), s n : I + → I + is a contraction mapping, with Lipshitz constant equal to one half, for all n ∈ I + .
6. Symbolic IFS theory (I): the standard relationship, between I + and point-fibred attractors The material in this Section 6 is of a standard, fairly well-known, nature. We include it to set the context for the extensions that follow.
Let A be a point-fibred attractor of F. A continuous map π F = π : I + → A ⊂ X is defined by It is readily proved that π is continuous and that π(I + ) = A. We refer to π : I + → A as defined here as the (standard) coding map. Here and in other definitions, we include parenthetical terms that may be dropped where the meaning is unambiguous. Theorem 3. Let A be a point-fibred attractor of the IFS F = {X; f n , n ∈ I + }. A uniformly continuous map π : I + → A F ⊂ X is provided by Equation (6.1). The IFS {I + ; s n , n ∈ I + } is contractive, with attractor I + . The following diagram, The code space I + provides addresses for the points of A; the addresses are continuously mapped onto A by π. For example, if f i (A) ∩ f j (A) = ∅ for all i = j, then π is a homeomorphism and it provides a unique address, that is, a unique point in the code space, for each point in the attractor; in this case we may say that π provides a conjugacy, rather than a semi-conjugacy, between f k ∈ F, acting on A, and s k ∈ {I + ; s n , n ∈ I + } acting on I + , for all k ∈ I + . In general, the set-valued inverse π −1 (a) comprises the set of addresses of the point a ∈ A, (w.r.t. F). These sets of addresses, and the semi-conjugacies provided by π, play an imporant role in the study of dynamics, invariant measures, laplacians on fractals, and other structures associated with attractors of IFSs.

Symbolic IFS theory (II): extensions of symbolic IFS theory (I) into basins
The goal of this Section 7 is to add to the known relationship between I + and a point-fibred attractor A of F, as in Theorem 3 in Section 6. Here we provide a symbolic representation that relates the behaviour of the functions f ∈ F on A to their behaviour at points that lie in X\A. The new results, most importantly Theorem 5 and its corollaries, together with Theorem 3, provide a more complete symbolic description than the one provided by Theorem 3 alone, of the structure and dynamics of point-fibred attractors of invertible IFSs. 7.1. A symbolic IFS with an attractor-repeller pair. We begin by introducing a symbolic invertible IFS Z that possesses a point-fibred attractor and dual repeller that is an attractor of the dual system. This IFS provides a symbolic basin as well as a canonical attractor-repeller pair. It also yields a symbolic description of the fast basin and its dual, the fast basin of the dual IFS. Proof. We prove that "The IFS Z := {I; s n , n ∈ I + } has point-fibred attractor I + with basin I\I − , and dual repeller I − ." The rest of the proof follows immediately, and is omitted. The compact nonempty set I + is contained in I, and we have I + = Z(I + ) := ∪ n∈I+ s n (I + ). An open neighborhood of I + is I\I − . If σ ∈ I\I − , then there is K = K(σ) ∈ N such that σ K ∈ I + , from which it follows that α 1 ∈ I + for all α ∈ Z K−1 ({σ}). In turn, this implies for all j ∈ N. This proves that Z k ({σ}) k∈N converges to I + for all σ ∈ I\I − . Now suppose that C ∈ H(I) and C ⊂ I\I − . Let σ ∈ C, and let K = K(σ) be as above. Then for all ω ∈ B(σ, 2 −K−1 ) we have ω K ∈ I + and hence which shows that Z k (B(σ, 2 −K(σ)−1 )) k∈N converges to I + . Using the compactness of C, we can cover C by a finite set of such balls, and we conclude that Z k (C) k∈N converges to I + . On the other hand, I − ⊂ Z(I − ) so I − ⊂ Z k (I − ) for all k ∈ N, so Z k (I − ) k∈N does not converge to I + , so we conclude that the basin of the attractor I + (w.r.t. Z) is I\I − . Hence, by definition, the dual repeller is I − .
7.2. The (extended) coding map. Here we define a coding map on J + that semiconjugates the symbolic IFS Z with the action F in the vicinity of a point-fibred attractor.
The symbolic IFS Z = {I; s n , n ∈ I + } in Theorem 4 does not appear to be directly related, via a coding map, to F X when X is a finite-dimensional geometrical space. However, by restricting Z to an appropriate subspace of I, namely J + , we can obtain a coding map relationship. Recall from Section 5 that symbolic space J + = {σ ∈ I : ∃K ∈ N 0 S K (σ) ∈ I + } ⊂ I is dense in I and invariant under each of the homeomorphisms s n : J + → J + , n ∈ I. Definition 8. Let A be a point-fibred attractor of an invertible IFS F. The (extended) coding map π = π F : J + → X is for all σ ∈ J + , where b ∈ B(A).
The limit in (7.1) exists and is independent of b ∈ B(A) because, by the definition of J + , there is a K ∈ N 0 such that S K (σ) ∈ I + , which means that we can write π(σ) = f σ|K (π(S K (σ))) where π(S K (σ)) is well-defined by (6.1). Clearly, the extended coding map agrees with the standard coding map (6.1) on I + . We use the same notation for both maps.
When the dual repeller A * is a point-fibred attractor of the dual IFS F * , we denote the associated extended coding map by π * : J − → X.
Theorem 5. Let A be a point-fibred attractor of an invertible IFS F. The extended coding map π = π F : J + → X is continuous and agrees with the standard coding map on I + . The diagram If A * , the dual repeller of A, is a point-fibred attractor of the dual IFS F * , then the diagram is commutative for all n ∈ I.
Theorem 5 motivates the following definition.
is commutative, for all n ∈ I − . If A * = X\B(A) is a point-fibred attractor of the dual IFS F * then the following diagram, Proof. This is a corollary of Theorem 5. We simply note that s −n ( I) ⊂ I and s n ( I * ) ⊂ I * for all n ∈ I + .
The following result provides a symbolic model for attractor-repeller pairs, when there is one attractor and one dual repeller, for example, the loxodromic Möbius case as discussed in [39].

Corollary 2.
Let F X be an invertible IFS with a point-fibred attractor A with A • = ∅. If F is contractive then π F ( I) = X and the diagram (7.6) commutes for all n ∈ I − . If F is not contractive, let X be compact, let the dual repeller A * be a point-fibred attractor of F * X , and let (A * ) • = ∅. Then π( I) = π * ( I * ) = X and we have both ↓ π X → X f n for all n ∈ I − , and (7.7) s n I * → I * π * ↓ ↓ π * X → X f n for all n ∈ I + .
Proof. The main assertions follow from Corollary 1 once it is shown that the fast basin of A is X and that the fast basin of A * is also X. (In the case of a contractive IFS, the fact that the basin and the fast basin coincide is immediate, see the end of Section 2. The dual repeller in this case is empty.) In the case where X is compact, we show that the fast basin of A * is X; a similar argument shows that the fast basin of A is X. It is known [15, Theorem 5.2] that B(A * ) = X\A. So by Proposition 1 and Theorems 1 and 2, if x / ∈ A, then x is in the fast basin of A * . Lastly, if x ∈ A, let y = x be any other point of A. If σ is any address of y, then π(σ) = lim k→∞ f σ|k (A). Therefor there is a k such that x / ∈ f σ|k (A), which implies that Since z is in the basin of A * and (A * ) • = ∅, there is ω 1 ω 2 ...ω J for some J such that Hence x lies in the fast basin of A * .
We will use Theorem 5 in many contexts. Consequences of Theorem 5 are (i) the feasibility of continuous assignment of addresses to points in B F (A); (ii) symbolic dynamics on I − are semiconjugate to corresponding dynamics on B F (A); (iii) if A has non-empty interior, as a subset of X, then we obtain addresses and dynamics on B(A); (iv) extension of fractal homeomorphisms to basins of attractors, in some cases; (v) address structures for continuations, fractal manifolds (when A is a manifold), and tilings; (v) descriptions of the relationships between attractors and dual repellers.

Fractal Manifold
Throughout this Section 8, A is a point-fibred attractor of an invertible IFS F on a complete metric space X.
Definition 11. The metric space (L, d L ) is called the fractal manifold or fmanifold generated by (F, A).
The topology of L is the one induced by d L . Theorem 6 (v) tells how the topology of L is independent of the metric d X . Theorem 6 (vi) tells us that, with respect to this topology, the maps π : I → L and π : L → B are continuous. We refer to them as the projection maps (associated with the f-manifold, or equivalently, associated with (F, A)).
Using the notation of Equation These maps are a subject of Theorem 6 (vii).
Theorem 6. Let A be a point-fibred attractor of an invertible IFS F, with fast basin B ⊂ X. Let (L, d L ) be the f-manifold generated by (F, A). Let π : L → B, π : I→ L, and π = π • π : I→ B be the associated projection maps. The following statements are true.
(iv) The set of all leaves, {l(θ) : θ ∈ ∪ k∈N I k − }, is a partition of L. Each leaf is nonempty.
(v) If the metric spaces (X, d X ) and (X, d X ) have the same topology, then (L, d L ) and (L, d L ) have the same topology, where the metric d L is defined by Equation (8.2) wherein d X is replaced by d X .
(ix) If A is pathwise connected, then L is pathwise connected.
(v) In the metric space (X,d X ) let B (X,d X ) (x, r) denote a ball of radius r > 0 centered at x. Two metrics d X and d X on X generate the same topology if, given B (X,d X ) (x, r), there is a positive radius r such that B (X,d X ) (x, r ) ⊂ B (X,d X ) (x, r) and, given any B (X,d X ) (x, r ), there is a positive radius r such that B (X,d X ) (x, r) ⊂ B (X,d X ) (x, r ). In the present situation, let the metrics on L defined using d X and d X be denoted by d L and d L . Let r > 0 be given and let α ∈ L be given. We will show that there is r > 0 so that B (L,d L ) (α, r ) ⊂ B (L,d L ) (α, r). The rest of the argument is then obtained by switching the roles of the key players. Let l([α]) = l(α 1 α 2 ...α kα ) be the unique leaf, by (iv), such that α ∈ l([α]).
Theorem 7. Let A be a point-fibred attractor of an invertible IFS F. For the fmanifold generated by (F, A) there are at most |F| + 1 topologically distinct leaves. Specifically, the leaf l(∅) = K is homeomorphic to A and the leaf l(θ) for θ = ∅ is homeomorphic to A\f i θ (A) where −i θ is the last component of θ.
Example G (An f-manifold when the attractor is the standard Cantor set) The attractor of the IFS {R; f 1 , f 2 } where f 1 (x) = 1 3 x and f 2 (x) = 1 3 x + 2 3 , is the standard Cantor set C. The leaves of the f-manifold are magnified (by some power of 3) and translated copies of C. Moreover, L is a disjoint union of its leaves. In this example the leaves are compact.

Sections of projections
To avoid repetition, in Sections 11, 12, and 13, of closely related ideas, we make the following definitions, mainly concerning notation.
Definition 12. If f is a function, then the domain of f is D(f ) and the range of

the identity on D(t).
A section may or may not be continuous, unlike the situation in differential geometry, where sections are required to be continuous.
Definition 13. Let A be a point-fibred attractor of an invertible IFS F on a complete metric space X. Let (L, d L ) be the f-manifold generated by (F, A), with projection maps π : I→ L, π : L → B, and π : I→ B, where π = π • π. Let τ : B ⊂ X → I be a section of π, i.e. π • τ = id B . Let τ : L → I be a section of π,i.e. π • τ = id L . Let τ : B ⊂ X → L be a section of π, i.e. π • τ = id B . Let τ : A → I + be a section of π : I + → A, i.e. π • τ = id A . We have the following diagram, where horizontal arrows denote inclusion maps, Similarly, let π also be the restricted projection π : I + → K =l(∅). Let τ : K → I + be a section of π : I + → K, so that π • τ = id K . Similarly, let π be the restricted projection π : K → A. Let τ : A → K be a section of π : L → B, so that π • τ = id A . The notatation is summarized in the following diagram.
Horizontal arrows indicate inclusion maps. Primed functions are restrictions of unprimed functions. Tildas denote functions between code space and either K or L.
To emphasize that sections are associated with (F, A),

Masks, sections, and fractal transformations
In this Section 10 we summarise the theory of masks, sections, and fractal transformations (f-transformations) between point-fibred IFS attractors. This simplifies the subsequent development, in Sections 11 and 12, of transformations between fractal manifolds and between continuations. Of particular relevance is Theorem 9. Our approach follows [9] and differs from [12]. Let F = F X , G = G Y be invertible IFSs, on complete metric spaces (X, d X ) and (Y, d Y ), with point-fibred attractors A F , A G . We assume that |F| = |G| = N .
Let τ F : A F → I + be such that π F • τ F (x) = x for all x ∈ A F , so that τ F is a section for π F : I → B F . Using any such a section τ F , a fractal transformation (f-transformation), generated by (F,A F ) and (G,A G ), is a map T F G : A F → A G that can be written in the form An f-transformation may be into but not onto, surjective, bijective, continuous, or a homeomorphism.
The simplest examples of f-transformations relate to topologically conjugate pairs of IFSs. Two IFSs F = {X; f i , i ∈ I + } and G = {Y; g i , i ∈ I + } are said to be topologically conjugate when there is a homeomorphism h : X → Y, such that g i = hf i h −1 for all i ∈ I + . In this case we write G =h(F).
Theorem 8 states some properties shared by topologically conjugate pairs of IFSs. Statement (iv) expresses how an f -manifold is a topological invariant of the IFS that generates it. Statement (v) is interesting because, as we will see later, it can occur that more than one distinct onto f-transformation may be generated by A F ) and (G, A G ). In such cases, the two systems are necessary not topologically conjugate.
While it is easy to construct an f-transformation by applying a simple geometrical transformation (e.g. affine) to an IFS attractor (e.g. a filled square in R 2 ), the interesting case is when an f-transformation is generated by two given simple geometrical IFSs, [5] . The resulting f-transformation may be geometrically complicated; for example, it may change the Hausdorff dimensions of some objects upon which it acts. ( Proof. Statements (i), (ii), (iii), and (iv), express changes of coordinates and are readily verified.
To prove statement (v), let T F ,h(F ) : A → h(A) be an onto f-transformation generated by (F,A) and (h(F),h(A)). It is readily verified that The section τ F : A mask defines a dynamical system (we call it a masked dynamical system) T : A F → A F according to n (x) for x ∈ M n , for all n ∈ I + . By [9], a mask for A F determines a section for A F , as follows: A section τ F : A F → I + of π F is said to be a masked section when it is constructed in this manner. It is straightfoward to prove that a section τ F is shift invariant if and only if it is a masked section, and in this case the following diagram commutes.
If τ F is a masked section for A F , then π G • τ F may be referred to as a masked f-transformation.
An f-transformation π G • τ F : A F → A F is onto, i.e. π G • τ F (A F ) = A G , if and only if there exists a masked section τ G : A G → I + such that τ G (A G ) = τ F (A F ), and in this case there exists the inverse f-transformation and the diagram (10.2) commutes. (10.2) In [12] we restrict attention to onto masked f-transformations.
Conditions under which an f-transformation is continuous are of special interest.
(ii) If A F and A G are just-touching (w.r.t. F and G resp.), Proof. (i), (iii), and (iv) are essentially [9,Theorems 3.4 and 4.5]. Here we have added the "only if" assertions to (iii) and (iv); we omit the proofs. Part (ii) is a consequence of (i) together with the observation that, in the envisaged situation, S k−1 τ F (x) is disjunctive for all k and hence π F (S k−1 τ F (x)) lies in the interior of A F for all k. Now use the commutative diagram (10.1) to conclude that (T ) If an f-transformation is a homeomorphism, we refer to it as an f-homeomorphism (fractal homeomorphism). A masked f-homeomorphism is a masked f-transformation that is a homeomorphism. Consistently with [12] we have the following observation. In Section 12 we use these sections to construct fractal transformations between f-manifolds, between continuations, and between basins. In Sections 13 we relate these sections to tilings of f-manifolds, tilings of continuations and tilings of basins. 11.1. How τ : A → I + induces a section τ : L → I. The projection π : K →A is a homeomorphism; σ ∈ A is related to σ ∈ l(∅) = K according to σ = π( σ). Here the notation is σ ∈ I, π(σ) = σ ∈ L, π( σ) = π(σ) = σ ∈ A. The given section τ : A → I + , which obeys π • τ = id A , uniquely defines a section τ : K → I + according to τ = τ • π. We have, for all σ ∈ K, We use the section τ : K → I + to define a section τ : L → I. Let α ∈ L. Then α ∈ l([ α]) and S k α ( α) ∈ K. A map τ : L → I is well defined, for all α ∈ L, by Proposition 2 asserts that τ : L → I, thus defined, is a section τ : L → I which agrees with τ • π on K. Proposition 2. Let τ : A → I + be a section for π : I + → A. Then τ := τ • π : K → I + is a section for π : I + → A. Let τ : L → I be defined by for all α ∈ L. Then τ is a section for π : I → L. Moreover, τ | K = τ and S k α τ ( α) ∈ I + .
In this case we say that τ is a masked section for π : I → L. We justify this terminology next. We can construct a mask on L starting from a masked section τ : L → I by defining a partition {M k : k ∈ I − } of L according to We define a symbolic masked dynamical system T : L → L by This makes sense, in view equation (11.3). Clearly, the itinerary of a point σ ∈ L (obtained, as in symbolic dynamics, by treating {M k } as a Markov partition of L) yields the value of τ ( σ) ∈ I. 11.2. How τ : A → I + induces a section τ : B −θ → L, for each θ ∈ I − . For given ω ∈ I, define −ω ∈ I by (−ω) n = −ω n for all n ∈ N.

f-transformations between f-manifolds
Let L F and L G be f-manifolds of point-fibred attractors A F and A G of invertible IFSs F = F X and G = G Y , respectively, where X and Y are complete metric spaces, and |F| = |G| = N .
Definition 14. An f-transformation (between f-manifolds) generated by (F,A F ) and (G,A G ), is a map T F G : L F → L G that can be written in the form where τ F : L F → I is a section of π F : I → L defined, as in Equation (11.2) in Proposition 2, uniquely, in terms of a given masked (shift invariant) section τ F : Theorem 10 notes examples of properties of T F G that are inherited from T F G = π G • τ F . Clearly, there are many other such properties.
Theorem 10. Let (F, A F ) and (G, A G ) be invertible IFSs with point-fibred attractors, and let T F G = π G • τ F : A F → A G be a masked f-transformation between attractors, and let T F G : L F → L G be the corresponding f-transformation between f-manifolds. The following statements are true.
(i) T F G : L F → L G is into but not onto if and only if T F G : Proof. Omitted because it is a book-keeping exercise.
Theorem 9 tells us when T F G is bijective; it also tells us when T F G it is into but not onto.
Let θ ∈ I − and let L θ = ∪ k∈N0 l(θ|k) be a sheet of L. The restricted map π F | L θ : L θ ⊂ L →B −θ ∈ X is a homeomorphism. Given an f-transformation T F G : L F → L G , we can construct a transformation T F Gθ : B F −θ → B G −θ between the continuation B F −θ of A F and the continuation B G −θ of A G according to Using the observations of Section 11, given any masked section τ : A → I + and any θ ∈ I − , we can construct the f-transformation T F Gθ . We refer to T F Gθ : as an f-transformation (between continuations). Theorem 11 notes important examples of properties of T F Gθ that are inherited from T F G .
Theorem 11. Let (F, A F ) and (G, A G ) be invertible IFSs with point-fibred attractors, let θ ∈ I − , let T F G : A F → A G be a masked f-transformation between A F and A G , and let T F Gθ = π G • T F G • τ F θ be the corresponding f-transformation between the continuations B F −θ of A F and B G −θ of A G . The following statements are true.
Proof. Straightfoward, using the fact that each panicle is homeomorphic to A.
Theorem 11 together with Theorem 9 provides a description of conditions under which f-transformations between continuations are, for example, homeomorphisms. Successful techniques, previously used, for the construction of f-transformations between attractors [5,9], can now be applied to f-manifolds and to continuations.

Tilings of f-manifolds
The tilings of a metric space X composed of transformed copies of an attractor A of an IFS F on X has been the topic of active research over the past couple of decades [1,3,17,19,18,20,26,28,30,31,32,36,38] and the list of references in [37]. The object of this Section 13 is to show that potentially an infinite number of such tilings are inherent in the f-manifold L generated by (F, A).
Let A have non-empty interior. Let τ : L → I be a section of π : I → L, constructed, starting from any masked section τ : A → I + , as described earlier. Let θ ∈ ∪ k∈N0 I k − . Let l(θ) ⊂ L be a leaf. We recall these points. (1) L is a disjoint union of the set of all leaves generated by (F, A).
For the purposes of this presentation, a tiling of a set is a partition of the set. For example, the leaf tiling of L is the one provided by its partition into leaves.
Clearly, from the above remarks, any tiling of L induces many tilings of B, one for each reversible point −σ ∈ I + . For example, a family of tilings of B is provided by the leaf tiling of L; we call these the leaf tilings of B.
The problem with leaf tilings is that the tiles, in familiar cases, become larger and larger, the further away the tile is from the attractor. We are interested in tilings with certain additional properties. We want tilings that can be produced systematically, starting from a small and natural set of information. The information that we choose comprises (F, A) together with a mask for A. Note that a mask for A (w.r.t. F) is a tiling of A. We want our tilings to reduce to familiar Figure 4. A tiled "sheet", one of the (non-denumerably) many tilings of R 2 generated by an affine IFS whose attractor is a triangle, see text. The associated fractal manifold is obtained by gluing together, appropriately, all of the tiled sheets. The attractor belongs to all sheets.
self-similar tilings when B = X = R m and F consists of similitudes, all with the same scaling factor s.
To understand the next step, imagine painting L by "numbers", the "number" of a point σ ∈ L being the address τ ( σ), where τ : L → I is the section of π : I → L induced from the mask (equivalently the corresponding section of π : I + → A), as described in Section 11.1. Each colour corresponds to a tile of L. We simply need to explain which sets of numbers in I have the same colour; i.e. we need a partition of I. (Actually, it would suffice to define which sets of addresses in {σ ∈ I : S kσ σ ∈ I + } have the same colour because, by the last remark in Proposition 2, τ (L) is contained in {σ ∈ I : S kσ σ ∈ I + }, which is usually a proper subset of I.) For σ ∈ I let l σ = min{l : S k σ ∈ I + } ∈ N 0 . We define an equivalence relation on I according to (13.1) θ ∼ ω ⇐⇒ θ|(2l θ ) = ω|(2l ω ).
We assign a different colour to each equivalence class. Let c(σ) denote the colour of the equivalence class to which σ ∈ I belongs. Finally, as stated above, we assign the colour c( τ ( σ)) to σ ∈ L. Points of the same colour comprise a tile (equivalence class) of L. T Theorem 12. Let A be a point-fibred attractor of an invertible IFS F, and let L be the associated f-manifold. Let M be a mask for A w.r.t. F and let τ : L → I be the corresponding section of π : I → L, defined in Proposition 2, so that each point in L has a unique address in τ (L). The equivalence relation (13.1), restricted to τ (L) yields a partition (i.e. a tiling) of L, one for each mask M. If A has nonempty interior and −σ ∈ I + is reversible (w.r.t. See also [16] where a different, but related, point-of-view, not using the fmanifold, is adopted.
Observe that if F is a an IFS of similitudes on R m , then each tile yielded by Theorem 12 is congruent to a subset of A. In this case, when the IFS is justtouching, standard fractal tilings are produced. Also in this case, when the IFS is overlapping and other geometical constraints on the maps and masks are imposed, Rauzy tilings are produced. Also, by choosing masks that look like elegant drawings on A, diverse tilings of B can be produced, with boundaries that are unions of parts of the boundary of A and parts of the drawings. The work of Esher, generalized, comes to mind.
In Figure 4 we illustrate a simple affine tiling based on a just-touching affine IFS attractor. The IFS is F = {R 2 ; f i , i = 1, 2, 3, 4} is affine, with the following property. There is a triangle ABC with points c ∈ AB, a ∈ BC,b ∈ CA, where XY is the line segment joining the points X and Y , and the triangles abc, ABC, Abc, aBc, abC are non-degenerate. Moreover, f 1 (ABC) = Abc, f 2 (ABC) = aBc, f 3 (ABC) = abC, and f 4 (ABC) = abc. IFSs of this kind are discussed in [5]. The IFS possesses a unique attractor, the filled triangle with vertices ABC. If θ ∈ I + is reversible, then B θ = R 2 . Consequently the f-manifold consists of non-denumerably many copies of R 2 glued together appropriately. Each region of glue is triangular.
In Figures 5, 6, 7, and 8 illustrate the continuations B ijkl for a fixed choice of ijk and l = 1, 2, 3, 4 after they have been tiled. To form part of the tiled f-manifold, the four objects must be thought of as being glued together on B ijk and otherwise thought of as distinct. (If you flick through the the pages of the PDF file, you may obtain the impression of the layers being put down one upon another, being glued at their common triangular internal region. The attractor itself is illustrated in multicolours, very small, in the middle of all pictures.)   ..the process is going to be repeated many times, with lots of glue being used.