Iterated function systems have been at the heart of fractal geometry almost from its origins. The purpose of this expository article is to discuss new research trends that are at the core of the theory of iterated function systems (IFSs). The focus is on geometrically simple systems with finitely many maps, such as affine, projective and Möbius IFSs. There is an emphasis on topological and dynamical systems aspects. Particular topics include the role of contractive functions on the existence of an attractor (of an IFS), chaos game orbits for approximating an attractor, a phase transition to an attractor depending on the joint spectral radius, the classification of attractors according to fibres and according to overlap, the kneading invariant of an attractor, the Mandelbrot set of a family of IFSs, fractal transformations between pairs of attractors, tilings by copies of an attractor, a generalization of analytic continuation to fractal functions, and attractor–repeller pairs and the Conley “landscape picture” for an IFS.
Metric spaces such as Euclidean space, the sphere, and projective space possess rich families of simple geometrical transformations . Examples are affine transformations of Euclidean space, Möbius and quadratic conformal mappings on the sphere, and projective transformations on projective space. The space and the mappings are simple to describe explicitly, and they are smooth.
In this review the basic object, denoted and called an iterated function system (IFS), is a complete metric space together with a finite set of simple transformations on . Let be the collection of nonempty compact subsets of . Define by for all . Under fairly general conditions, the map has one or more attractors, an attractor being an attractive fixed-point of . Although and the functions may be smooth, an attractor can be geometrically complicated and rough; that is, it may have features which are non-differentiable, or have a non-integer Hausdorff–Besicovitch dimension, or have a dense set of singularities. Attractors and transformations between them comprise the principal objects of study in deterministic fractal geometry. They can be arcs of graphs of wavelets, Julia sets, Sierpinski triangles, or geometrical models for intricate biological structures such as leaf veins. Many textbooks use pictures of such objects to illustrate the idea of a fractal; Fig. 1 illustrates a few familiar IFS fractals, and also some newer fractal objects associated with simple geometrical IFSs.
Following the publication in 1983 of Benoit Mandelbrot’s book The Fractal Geometry of Nature , there has been a steadily increasing interest in the use of non-differentiable structures to model diverse natural objects and processes. In  H. Furstenberg observes that fractals have fundamentally changed the way that geometers look at space. For some, there has been a shift in viewpoint, away from the study of smooth structures such as differentiable manifolds to the study of rough non-differentiable objects such as fractal attractors of smooth dynamical systems. The rough objects are described in terms of the smooth systems that generate them. A key example is an attractor described in terms of the IFS that generates it. Questions regarding the topology (connectedness for example), geometry (Hausdorff dimension for example), invariant measures and other properties of the fractal objects are investigated, to reveal their relationships to the smooth objects that generate them.
Using these relationships, fractal objects can be used to model or approximate rough real-world data, from stock-market traces to turbulent wakes and cloud boundaries. In physics, for example, fractal geometry has played a role in mapping the seabed and in modelling the effect of the rough surfaces of small particles on heat absorption by the atmosphere . In engineering, fractal modelling has been successfully applied to image compression . In multimedia, fractal modelling can be used for real-time image synthesis . There are many more such examples, see .
Some fractal objects are self-referential. A self-referential object is a set or measure that can be defined in terms of a finite collection of geometrical transformations applied to . For example, an attractor of is self-referential since . According to a definition by Mandelbrot [87, p. 15], a fractal is a set for which the Hausdorff–Besicovitch dimension strictly exceeds the topological dimension. Note, however, that the attractor of an IFS may not have this property and may, in fact, be a classical geometric object such as a line segment or polygon.
Although fractals receive considerable current attention, they are a newcomer to the history of geometry and to the task of describing physical objects. Fractal geometry is an area of mathematics that is under construction. In his much cited 1981 paper , John Hutchinson formulated the concept of a contractive iterated function system, although not with that name, to unify and make rigorous, using geometric measure theory, some of the key ideas in Mandelbrot’s book . Related precursors to the concept of an attractor of a contractive system can be found, for example, in the work of Nadler  and Williams , and ideas concerning associated invariant measures can be found in [52, 99].
This paper is a survey of some recent trends in the subject of iterated function systems. The focus is two-fold. First, to emphasize new structural properties of affine, projective, analytic, and even more general non-conformal IFSs, adding to older publications concentrating on similitude and conformal IFSs. Second, to concentrate on the core of the subject—the role of contractivity on the existence of an attractor, addressing the points of an attractor, transformations between attractors, dynamical properties of an IFS, and tilings by copies of an attractor. Although these notions are at the historical foundation of the subject, new and exciting concepts, such as phase transition and fractal continuation, have recently come to light.
Section 2 provides definitions and notation associated with an IFS and its attractors. Examples of similitude, affine, projective and Möbius IFSs are given. Some concepts, such as an attractor–repeller pair, are introduced informally in this section, but are defined formally in later sections. The definition of an attractor of an IFS that is used in this review is in keeping with recent work in the area, and is a generalization of the concept for contractive systems.
In Sect. 3 the notion of a contractive IFS is defined, followed by Hutchinson’s theorem stating that a contractive IFS has a unique attractor. This theorem can be viewed as a generalization of Banach’s contraction mapping theorem, generalizing from the existence of an attractive fixed point of a single function to the existence of an attractor of a finite set of functions. It is known, under quite general conditions, that if a transformation on a metrizable space possesses a unique fixed point, then there exists a metric with respect to which the transformation is contractive. When does an analogous situation hold for an IFS with a unique attractor? Recent work has shown that contractivity is a necessary condition for the existence of an attractor for some basic types of IFSs, such as affine, projective and Möbius IFSs.
Section 4 contains recent advances concerning the applicability of the chaos game, one of two main algorithms for drawing an accurate approximation to an attractor of an IFS. The two approaches are the deterministic algorithm (illustrated in Fig. 2) and the more efficient chaos game algorithm . It has recently been proved that the chaos game algorithm converges to the attractor under general conditions.
The joint spectral radius of the linear parts of an affine IFS plays a crucial role in determining whether or not possesses an attractor. This role is explained in Sect. 5. Considering as a parameter, for linear IFSs there is a dramatic phase transition at the precise value of at which the IFS changes from not having an attractor to having a trivial attractor. The fractal figure that emerges at this phase transition is the main subject of Sect. 5.
For an IFS consisting of functions, strings in the alphabet can be associated with points in an attractor of the IFS. This leads to the topic of addresses and coordinate and coding maps, which are the subject of Sect. 6. Mentioned in Sect. 6.1 is Kieninger’s classification of attractors according to their fibres and Kameyama’s fundamental question: When does the existence of a coding map ensure the existence of a contractive metric? In Sect. 6.2 we define the recently introduced notion of a section of a coordinate map, whereby each point on an attractor is assigned a unique address. Sections of coordinate maps play a key role in the construction of transformations between attractors, which is the subject of Sect. 9.
In Sect. 7, attractors of IFSs are classified according to how the images of the attractor, under the functions of the IFS, overlap. This leads to the notion of the kneading invariant and what it implies regarding the topology of an attractor.
The subject of Sect. 8 is the Mandelbrot set of a certain affine family of IFSs, an analog to the classical Mandelbrot set for . The Mandelbrot set is the set of those for which possesses a connected attractor.
While properties of classical geometrical transformations (for example affine, conformal, projective) are well known, much less is known about their rough counterparts (for example affine, conformal, and projective fractal transformations). Transformation between fractals is discussed in Sect. 9.
Tilings of the plane, and more generally Euclidean space, have been constructed from attractors of an IFS since the 1980s. Section 10 provides a new and unified approach to such tiling—generating both the classical self-similar IFS tilings and new tilings.
Fractal functions, introduced in the form considered here in 1986, have applications ranging from image compression to modeling brain waves. Fractal continuation of such functions, an analog to classical analytic continuation, is the subject of Sect. 11.
Based on the work of Conley on the dynamics of a single function and on the work of McGehee and Wiandt on iterated closed relations, a generalization of the notion of an attractor of an IFS is given in Sect. 12. This leads to the existence of attractor–repeller pairs, a characterization of the set of chain recurrent points of an IFS in terms of such pairs, and a “landscape picture” for an IFS.
This paper is not intended to be exhaustive. Many fascinating topics are omitted, and there is likely a bias towards the interests of the authors. In particular, “random fractals”, “statistically self-similar measures”, and related probabilistic entities are not discussed; our interest is in deterministic fractal geometry, where the information needed to describe the fractal object is low, and the geometrical complexity may appear to be high. Another topic not discussed is -variable fractals and superfractals [16, 26–28]. These entities serve several purposes; they provide a bridge between deterministic fractal geometry and random fractal geometry, and they enable the construction of rich simple geometrical families of fractal objects. Fractal dimensions are only touched on in this survey because there is already an ample literature of the subject.
2 IFSs and their attractors, basins, and dual repellers
In this section we define and provide examples of some fundamental notions. In so doing we introduce the concept of an attractor–repeller pair that is explained in more detail in Sect. 12, which deals with Conley’s “landscape picture” for an IFS.
An iterated function system (IFS) is a topological space together with a finite set of continuous functions .
We use the notation
to denote an IFS. Other more general definitions of an IFS are used in the literature; for example, the collection of functions in the IFS may be infinite, see for example , or the functions may themselves be set-valued [76, 79]. However, throughout this review, except where otherwise stated, is a finite positive integer and is a complete metric space. This fits with our philosophy that the system is geometrically simple. By slight abuse of terminology, we use the same symbol for the IFS and for the set of functions in the IFS.
Let denote the collection of nonempty compact subsets of and define the Hutchinson operator by
for all . By further slight abuse of notation we also treat as a map , where denotes the collection of all subsets of . For , define and let denote the -fold composition of applied to , namely, the union of over all finite words of length .
Let be the metric on , and let be the corresponding Hausdorff metric. The Hausdorff metric is defined on the space by
for all , where
Throughout we assume that the topology on is the one induced by . Key facts, discussed in , are that is a complete metric space because is complete, and that if is compact then is compact.
An attractor of the IFS is a set such that
there is an open set such that and for all with , where the limit is with respect to the Hausdorff metric on .
We remark that, since is a complete metric space, we have is continuous (w.r.t. the Hausdorff metric) see . So (1) in Definition 2.2 is superfluous. But we include (1) because it is key in more general settings, such as the one considered in Sect. 12.
The basin of an attractor of the IFS is the largest open set such that (2) in Definition 2.2 is true.
The dual repeller of an attractor of the IFS is the set where is the basin of .
Examples of attractors and their dual repellers appear below. The notion of a dual repeller arises as follows.
The IFS is said to be invertible when is a homeomorphism for all . If is invertible, then the IFS
is called the inverse IFS of .
Under reasonable conditions described in Sect. 12, an attractor of is a dual repeller of an attractor of .
If and the functions in are affine functions of the form , where is an matrix and , then is called an affine IFS. If the functions of are similitudes (i.e. affine with where is orthogonal and ,) then is called a similitude IFS.
An example of a similitude IFS is
The IFS (2.1) has a unique attractor because it is contractive; see Theorem 3.2. This attractor , approximated by the bottom picture of the third column of Fig. 2, is called the twindragon. The basin of is , and the dual repeller is the empty set, . The first three columns of Fig. 2 also demonstrates condition (2) in Definition 2.2 because they show the successive iterates for where is the rectangle at the upper left. If, in this example, we change the space to be , the one point compactification of where is “the point at infinity”, and we define then the basin is no longer the whole space and the dual repeller is . Remarks analogous to those mentioned in this paragraph hold, in fact, for all similitude IFSs. Note that can be represented as a sphere (with infinity at say the north pole) and is embedded in by stereographic projection. Then and are both metric spaces, with the Euclidean metric and with the spherical metric, and the two metrics give the same topology when restricted to . Both the case and case, of the IFS (2.1), are invertible. In the case the affine IFS does not possess an attractor. In the case the IFS has a unique attractor , the dual repeller of the attractor of .
If , real -dimensional projective space, and the functions in the IFS are projective functions of the form , where is an matrix and is given by homogeneous coordinates, then is called a projective IFS. An example of a projective IFS is
This IFS (2.2) possesses a unique attractor , illustrated in Fig. 3. This attractor, which is the union of a non-denumerable set of lines, is illustrated using a disk, with antipodal points identified, to represent . This image was computed using the chaos game algorithm, discussed in Sect. 4. The corresponding dual repeller is also shown: it is a spiral Cantor set in green, disjoint from the attractor. The basin of is The IFS (2.2) is invertible and its inverse possesses a unique attractor, equal to the dual repeller of , the spiral Cantor set to which we have just referred. The dual repeller of is the attractor of ; and the basin of the attractor of is
If , the extended complex plane, and the functions are Möbius functions of the form , where and are complex numbers, then is called a Möbius IFS. Möbius functions may equivalently be considered to act on the Riemann sphere or the complex projective line. The unique attractor (red) and the dual repeller (black) for a certain Möbius IFS () on the Riemann sphere are illustrated in Fig. 4. The basin of the attractor is the complement of the dual repeller. In this case the dual repeller is the attractor of the inverse IFS, and the attractor is the dual repeller of the inverse IFS. For further details on such Möbius examples see .
3 Contractive IFSs
When does an IFS possess an attractor? We begin with the observation that if an IFS is contractive (defined below), then it possesses a unique attractor , with the basin , and dual repeller . Then we make the following points:
Contractivity is not integral to the existence of an attractor.
Contractivity is integral to the existence of an attractor for affine, Möbius, and many projective IFSs.
A function is a contraction with respect to a metric if there exists such that
for all . An IFS on a metric space is contractive if there is a metric , inducing the same topology on as the metric , with respect to which the functions in are contractions.
The following theorem is a generalization of the Banach contraction mapping theorem, from one function to finitely many functions.
(Hutchinson) If is a contractive IFS on a nonempty complete metric space , then has a unique attractor and the basin of is .
There are various versions of a converse to Banach’s fixed point theorem in the literature, for example [44, 55, 67, 80]. However, the converse to Hutchinson’s theorem is false. There exist non-contractive IFSs that have a unique attractor; for example, the IFS in (2.2) has a unique attractor , but there is no metric on any complete neighborhood of , generating the same topology as that of restricted to , such that restricted to is contractive.
Theorem 3.4 below addresses the role of contractivity, not only in the affine case, but also in the projective and Möbius cases. For a proof of the theorem, see  for the affine case,  for the projective case, and  for the Möbius case. The proof of the uniqueness result, Theorem 3.3, can also be found there. In the projective case, recall that a hyperplane in -dimensional real projective space is an -dimensional subspace. A set is said to avoid a hyperplane if . In the Möbius case, recall that the extended complex plane is essentially equivalent to the Riemann sphere via stereographic projection. For the Möbius case, the metric is the round metric on the sphere. Throughout, if then and denote, respectively, the closure and the interior of .
An affine, Möbius, or projective IFS can have at most one attractor.
An affine IFS has an attractor if and only is contractive.
A Möbius IFS has an attractor if and only there is a nonempty open set such that and restricted to is contractive.
A projective IFS has an attractor that avoids some hyperplane if and only if there is a nonempty open set such that is contractive on .
The attractor of the IFS (2.2) is a union of hyperplanes (lines). Since the intersection of two lines in the projective plane is nonempty, the attractor avoids no hyperplane, and, by Remark 1, this shows that the condition in part (3) of Theorem 3.4 cannot be removed.
An interesting aspect of Theorem 3.4, that also makes it somewhat difficult to prove, is that, in general, the IFS is not necessarily contractive with respect to the standard metric, even locally. In the affine case on , an IFS with an attractor is always contractive with respect to a Minkowski metric, i.e. a metric of the form
and where is a centrally symmetric convex body. In the projective case, an IFS with an attractor is always contractive with respect to a (modified) Hilbert metric. For a convex body , the Hilbert metric on the interior is defined in terms of the cross ratio as
where and are the points where the line joining and intersects the boundary of . In the Möbius case, an IFS with an attractor is always contractive on some open set with respect to the newly discovered metric on :
In each case, affine, projective, and Möbius, an IFS with an attractor is also contractive in a topological sense. This give rise to Theorem 3.6 below, a companion to Theorem 3.4. The proof appears in the same references cited for Theorem 3.4.
An IFS is said to be topologically contractive on a compact set if
An affine IFS has a attractor if and only is topologically contractive on some convex body.
A Möbius IFS on has an attractor if and only if is topologically contractive on some nonempty proper compact subset of .
A projective IFS has an attractor that avoids some hyperplane if and only if is topologically contractive on the union of a nonempty finite set of disjoint convex bodies.
4 Chaos game
Although condition (2) in Definition 2.2 can be used as the basis of an algorithm to draw arbitrarily accurate approximations to an attractor of an IFS, such a method is inefficient because the number of computations grows exponentially with . Also, it is necessary to have, a priori, an initial set that lies in the basin of the attractor, to initiate the iterative construction. More efficient is the chaos game algorithm, in which the attractor is approximated by a chaos game orbit. This has the benefits of low memory usage and more freedom in the choice of an initial set of the form .
Throughout this paper, the following notation is used. Let , and let denote the set of infinite strings on the alphabet .
Given an IFS and an , the chaos game orbit of a point with respect to is the sequence , where
for The chaos game orbit is called a random orbit of if there is such that, for each is selected randomly from with the probability that being greater than or equal to , regardless of the preceding outcomes, for all . (In terms of conditional probability, .) We say that a chaos game orbit yields if
where the limit is with respect to the Hausdorff metric.
An instructive example is provided by the IFS where is a circle, is an irrational rotation and is the identity. This IFS has unique attractor with basin . Let , and compute a chaos game orbit by tossing a coin at each step. As gets large the orbit approaches with probability one.
Theorems 4.2 and 4.3 below tell us how chaos game orbits reveal attractors under very general conditions. It is at first sight surprising that it was successfully used, for example, to calculate images of attractors of non-contractive IFSs such as the one in Fig. 7. Indeed, there is no contractivity assumption in either Theorem 4.2 or 4.3. The proof of Theorem 4.2 appears in . A metric space is proper if every closed ball is compact. Note that neither Definition 4.1 nor the following theorems say anything about convergence rates.
Let be a proper complete metric space and an IFS with attractor and basin . If is a random orbit of under , then with probability one this random orbit yields .
A deterministic version of Theorem 4.2 appears in [29, 31] and is stated as Theorem 4.3. A string is disjunctive if every finite string is a substring of . By a substring of we mean a string of the form for integers . In fact, if is disjunctive, then every finite string appears as a substring of infinitely many times. For example, the binary Champernowne sequence, formed by concatenating all finite binary strings in lexicographic order, is disjunctive. There are infinitely many disjunctive sequences if . Moreover, the set of disjunctive sequences is large in the sense that its complement in the natural metric space defined on is a meager set and even a -porous set. The metric space is defined formally at the beginning of Section 6. A meager set is a set of the first Baire category; the definition of -porous can be found in [93, 127] and results using porosity in fractal geometry in [46, 83]. The definition of a strongly-fibred attractor is given in Sect. 6, following Theorem 6.2.
Let be a complete metric space and an IFS with strongly-fibred attractor and basin . If and is a chaos game orbit of with respect to a disjunctive sequence, then this chaos game orbit yields .
Under certain additional conditions (see ), the initial point of the chaos game orbit is not restricted to lie in the basin of . There is a set whose complement is -porous and if , then the conclusion of Theorem 4.3 remains true.
5 Phase transition
Affine IFSs are basic to deterministic fractal geometry, perhaps because the first approximation to a non-linear map is affine, and because affine transformations, after similitudes, form a relatively simple group. Here we consider a family of affine IFSs depending on a parameter. The existence or non-existence of an attractor depends on the value of the parameter. As pointed out in Sect. 3, affine IFSs behave nicely in some ways; in particular, an affine IFS has a unique attractor if and only if is contractive. But there is an equivalent necessary and sufficient condition for to possess an attractor, a condition involving the positive real parameter called the joint spectral radius of . A dramatic geometric transition occurs at the precise value of where the IFS changes from having an attractor to not having an attractor.
5.1 Joint spectral radius
The joint spectral radius of a set of linear maps on was introduced by Rota and Strang , and the generalized spectral radius was introduced by Daubechies and Lagarias . Berger and Wang  proved that the two concepts coincide for bounded sets of linear maps. A set of linear maps is bounded if there is an upper bound on their norms. (Note that all norms are equivalent, in the sense that there are positive constants b such that for all .) The index set may be infinite, but it is assumed throughout this section that is compact (as a subset of . In particular, compact implies that is bounded. What follows is the definition of the joint spectral radius and the generalized spectral radius of . Let be the set of all words , of length , where . For , define
The generalized spectral radius is a generalization of the spectral radius of a single linear map , i.e. the maximum of the moduli of the eigenvalues of .
For any set of linear maps and any norm, the joint spectral radius of is
The generalized spectral radius of is
The following are well known properties of the joint and generalized spectral radii.
The joint spectral radius is independent of the particular norm.
For an IFS consisting of a single linear map , the generalized spectral radius is the ordinary spectral radius of , the maximum of the moduli of the eigenvalues of .
For any real we have and .
If is bounded, then the joint spectral radius equals the generalized spectral radius.
If is an affine IFS, infinite or not, the joint spectral radius is defined as the joint spectral radius of the set of linear parts of the functions in .
A set of linear maps is called reducible if these linear maps have a common nontrivial invariant subspace. The set is irreducible if it is not reducible. An affine IFS is reducible (irreducible) if the set of linear parts is reducible (irreducible).
In terms of matrices, a set of linear maps is reducible if and only if there exists an invertible matrix such that all can be simultaneously put in a block upper-triangular form:
for all , with and square, and is any matrix with suitable dimensions. The joint spectral radius is equal to .
An IFS, consisting of finitely or infinitely many affine functions, is compact if the set of matrices that represent the functions is compact as a subset .
The following theorem is proved in .
A compact affine IFS on has an attractor if and only if . Moreover, if , then the attractor is unique and the basin is ; if and is irreducible, then there does not exist a nonempty bounded set such that .
Theorem 5.4 states that is the transition value between having and not having an attractor. This transition is especially dramatic in the case of a linear IFS. Let be a linear IFS, an IFS all of whose functions are linear. In this case, if , then has a unique attractor , the origin. We say, in this case, that the attractor is trivial. If, on the other hand, , then by Theorem 5.4, the IFS has no attractor, in fact no invariant set, i.e., no set such that . What is surprising is that, at the transition value between the trivial invariant set and no invariant set, there is a geometric “blowup”, a non-trivial invariant set. This is the content of the following theorem .
A compact, irreducible, linear IFS with has a compact invariant set that is centrally symmetric, star-shaped, and full dimensional.
A set is centrally symmetric if whenever . A set is star-shaped if for all and all . And a set is full-dimensional if the affine hull of is . Figures 5 and 6 show two examples of the transition phenomenon at , transitioning between trivial invariant set and no invariant set. These two examples are further detailed below.
The theory outlined above can be reformulated as an analog to the eigenvalue problem in linear algebra.
Let be an IFS and consider the eigen-equation
where is the usual Hutchinson operator of , , and is a compact set in Euclidean space. The value in Eq. 5.1 above will be called an eigenvalue of , and a corresponding eigenset.
When consists of a single linear function , an eigenvalue-eigenvector pair of satisfies the eigen-equation. Even for a single linear function, however, there are more interesting eigenvalue-eigenset pairs. For example, let be a linear map with no nontrivial invariant subspace, equivalently no real eigenvalue. Although has no real eigenvalue, does have an eigen-ellipse, an ellipse , centered at the origin, such that , for some real . Although easy to prove, the existence of an eigen-ellipse is not universally known.
If Eq. 5.1 is rewritten as
then an eigenset for is an invariant set of , and Theorem 5.5 can be formulated as follows.
A compact, irreducible, linear IFS has exactly one eigenvalue which is equal to the joint spectral radius of . There is a corresponding eigenset that is centrally symmetric, star-shaped, and full dimensional.
Fig. 5 shows an eigenset for the IFS , where
The eigenvalue appears to be , the value of the largest eigenvalue of . The part of the set shown in red is, to viewing accuracy, the image of the whole set under . The part of the set shown in blue is, similarly, the image of the whole set under . The coordinate axes are indicated in black.
Figure 6 shows an eigenset for the IFS , where
The eigenvalue is . The coordinate axes are indicated in red.
Theorem 5.7 cannot be extended to affine IFSs. It is true that, for a compact, irreducible, affine IFS, a real number is an eigenvalue if and is not an eigenvalue if . However, if is not linear, there are examples where is an eigenvalue and examples where it is not.
6 Fibres and addresses
6.1 Coordinate maps and coding maps
This section concerns the assignment of addresses to the points in an attractor of an IFS . In the notation of Sect. 4, the set is called the code space for an IFS consisting of functions. For a string , denote the element in the string by , and denote by the string consisting of the first symbols in , i.e., . The code space is a compact metric space with distance function defined by
The space is homeomorphic to the classical Cantor set treated as a subset of the unit interval with the usual topology.
Let be an attractor of on a complete metric space . To each we associate a fibre according to
We have [71, Proposition 4.3.2]
Following Kieninger [71, p.89, Definition 4.2.7], define the coordinate map for (w.r.t. the IFS ) by
for all . The set of addresses of a point is defined to be
A point may have many addresses and many points may have the same address. To describe addresses, we use the notation to mean the periodic point . For example, . We also use the notation, when and ,
The notions of fibres and addresses are illustrated in Fig. 7, using the IFS
where is the filled triangle with vertices . The IFS (6.1) possesses a unique attractor , but there is no metric on generating the standard topology and such that the IFS (6.1) is contractive. Some of the fibres are line segments and others are points. For example, and is the point with coordinates . In order to illustrate some of the fibres of (w.r.t. ) we have rendered in two different ways in Fig. 7; the image on the left uses bright colours to emphasize fibres which are points while the image on the right emphasizes fibres which are line segments. In the figure, an address of a fibre that is point (on the left) and an address of a fibre that is a line segment (on the right) are given.
If is a singleton for all , then is said to be point-fibred (w.r.t. ).
When is point-fibred, by slight abuse of notation, we treat the coordinate map as a mapping
In this case
for where is the basin of . The following statement is an immediate consequence of [65, Section 3.1] and the definition of point-fibred.
If is contractive, then its attractor is point-fibred.
A point belonging to the attractor of a point-fibred IFS may have one or many addresses, but there is only one point with any given address. If, for every open set such that there is a fibre , then is said to be strongly-fibred. Figure 7 illustrates a strongly-fibred attractor that is not point-fibred. We have
An example of an IFS with a unique attractor that is not strongly-fibred is
where is a circle, is the point with angular coordinate , and is irrational: the attractor is and for all .
The coordinate map interacts with the shift map on code space. The shift map is defined by , and the inverse shift map is defined by for . Both and are continuous. A subset is called shift invariant if .
The following theorem is a simplified version of some results of Kieninger [71, Lemma 4.6.6 and Propositions 4.2.10 and 4.3.22].
Let be an attractor of an IFS on a complete metric space . The coordinate map is upper semicontinuous as a mapping from the metric space to the metric space . The following diagram commutes for all .
If is point-fibred, then is continuous.
This leads to the following definition.
Given an IFS on a complete metric space , a coding map for is a continuous surjection such that the following diagram commutes for each .
In this case is called a topological self-similar set.
If is a point-fibred attractor for then is a coding map for with . A point-fibred attractor is a topological self-similar set. In Definition 6.4 the nomenclature “topologically self-similar set” comes from the work of Kigami  and Kameyama , mainly in the context of analysis on fractals; it refers to the readily proved fact that is invariant for , i.e. .
Definition 6.4 leads to Kameyama’s Fundamental Question : Can a topological self-similar set be associated with a contractive IFS? This question can be reformulated: if is a coding map for an IFS on a complete metric space , then is , restricted to , contractive?
To prove that the answer to his question is “no”, Kameyama constructs an IFS where and the metric space is defined abstractly in terms of the code space . Kameyama’s IFS is non-geometrical, and his question remains open when and the transformations are geometrically simple, as described in the introduction. But for affine IFSs the answer is “yes”. Indeed, it is quite surprising that for an affine IFS, contractivity is assured solely by the existence of a coding map .
If is an affine IFS with a coding map , then is contractive when restricted to the affine hull of . In particular, if contains a non-empty open subset of , then is contractive on .
The affine hull of a subset of is the smallest affine subspace containing .
6.2 Sections of coordinate maps
If is point-fibred, then the coordinate map assigns a point in the attractor of to each infinite string. It would be convenient if, for an IFS with point fibred attractor , we had a map in the direction opposite to that of the coordinate map, assigning to each point a string in . Since the coordinate map need not be injective, the following definition is useful, for example in the construction of fractal transformations, discussed in Sect. 9.
A section of a coordinate map is a map such that is the identity. For , the string is referred to as the address of with respect to the section . The set is called the address space of the section . The section is called shift invariant when is shift invariant.
Properties of sections of coordinate maps are discussed in [21, Section 3].
Let where and . In this case the IFS possesses a unique attractor, the real interval . For , the “binary decimal expansion map” is a coding map. Note that . There are two shift invariant section maps. One is the map that sends each point of to its binary decimal expansion that contains no infinite string of consecutive zeros. The other is the map that sends each point of to its binary expansion that contains no infinite string of consecutive ones.
7 Classification of attractors
7.1 Set of overlap
In this section the attractor of an IFS is classified according to how the sets intersect. Throughout we assume that the functions are pairwise distinct when restricted to an attractor under discussion.
Let be an attractor of an IFS . The set of overlap of (w.r.t. ) is
is non-overlapping or disjoint(w.r.t. ) if ;
is overlapping (w.r.t. ) if ;
is slightly overlapping or just-touching (w.r.t. ) if it is not disjoint and the interior of (w.r.t. the subspace topology on ) is empty;
is strongly overlapping (w.r.t. ) if it is overlapping but not slightly overlapping, i.e. if the interior of (w.r.t the subspace topology on ) is nonempty;
obeys the open set condition (w.r.t. ) if there is a nonempty open set in the basin of such that and whenever , ;
is post-critically finite (w.r.t. ) if it is connected, is a contractive similitude IFS, and is a finite set;
is finitely ramified if comprises finitely many distinct points.
When the meaning is clear from the context we omit the caveat “w.r.t. ”. The terminology in Definition 7.1 mainly follows Kieninger  and Kigami [72, 73]. Note that Strichartz  uses a less general definition of post-critically finite.
Next, some properties of attractors with each of the properties in Definition 7.1, and some relationships between these properties, are briefly outlined. For ease of exposition, it is assumed that is contractive.
If is disjoint, then it is totally disconnected. If is a disjoint attractor of an injective IFS with , then it is perfect and it is homeomorphic to the classical Cantor set (where the relative topologies are understood). In this case the coordinate map is a homeomorphism. The green spiral Cantor set in Fig. 3 illustrates a totally disconnected attractor (of an inverse IFS).
If obeys the open set condition, then it is just-touching. The converse, however, is not true [71, Remark 5.3.2]. The open set condition, together with invariant measures associated with the IFS, are used in the analysis of the Hausdorff–Besicovitch dimension of an attractor . Assume that is an attractor that obeys the open set condition w.r.t. a similitude IFS on , with for all , where for all . Then (see for example [56, Theorem 9.3])
Bandt and Rao  have recently discussed the relationship between the open set condition and the size of the set of overlap. They provide an example of a similitude IFS in whose attractor is a Cantor set for which is arbitrarily small, but such that does not obey the open set condition. Various refinements and modifications of the open set condition are used to yield additional information about dimensions of attractors; see for example  and the references therein.
The topic of post-critically finite (p.c.f.) attractors is specialized, but central to analysis on fractals. Analysis on fractals involves the construction of Laplacians , harmonic analysis, Brownian motion , heat kernels  (see the review in ), and differential equations onfractals . See also Hambly  and Teplyaev . The attractor of the standard Sierpinski IFS
is an example of a p.c.f. fractal. We note that the structures related to analysis on fractals, such as random walks, invariant measures, Dirichlet kernels, and so on, can be described in terms of analysis on equivalence classes on the underlying code space, or in terms of analysis on limit sets of sequences of combinatorial graphs.
Recently, the relationship between the open set condition, the property of being p.c.f., the property of being finitely ramified, and other separation conditions, have been explored in the general setting of graph-directed IFSs ; see [51, 96].
Strongly overlapping point-fibred systems are related to -transformations , Bernoulli convolutions, and coding theory, areas which attract current attention; see for example [7, 64]. They are also related to fractal transformations as discussed in Sect. 9.
7.2 Kneading invariant
The topology of a point-fibred attractor is determined by the address structure of its critical set, a set related to the set of overlap. The definition of the critical set and the related address structure, called the kneading invariant, are defined below. The theory of kneading invariants developed from the study of the orbit of the critical point under an iterated unimodal map, by Collet and Eckman , in connection with dynamical systems and chaotic dynamics, and later substantially extended by Milnor and Thurston .
Bandt and Keller  initiated a systematic study of the topology of contractive IFS attractors based on the code space structure of the set of overlap. Here we adapt the more recent development by Kameyama  . If is a set with finitely many distinct elements then is the number of distinct elements in ; if has infinitely many distinct elements then .
Let be a point-fibred attractor of an IFS . We say that
is the critical set of (w.r.t. ), and
is the kneading invariant of (w.r.t. ).
From the continuity of the coordinate map in the point-fibred case, it follows that a point-fibred attractor is homeomorphic to the quotient space where if . Conversely, the equivalence relation derived from a set determines an “IFS” , but the space may not be a Hausdorff space and hence not metrizable. (If it is Hausdorff then it is metrizable with a metric constructed using the coding map.) Theorem 7.3 below tells us how the structure of is determined by the structure of the kneading invariant . It does so by characterizing the addresses of a point such that . We use the following notation: if for some and , then and , and .
Let be a point-fibred attractor of an IFS , and let be the kneading invariant of w.r.t. .
If and , then there uniquely exists and for some such that .
If , for some and , then . Moreover if , then and .
Theorem 7.4, a simplified version of a theorem of Kameyama , provides sufficient conditions under which the kneading invariant determines a canonical quotient space which is homeomorphic to the given attractor.
Conversely to Theorem 7.3, suppose that has the property (ii) in Theorem 7.3 and for each . Define an equivalence relation on by if either or for some , for some for some . Let be the quotient space and let be the projection. Then there exist continuous maps for such that the diagram (6.3), with , commutes. If the critical set is finite and is compact for each , then is an IFS with point-fibred attractor and kneading invariant .
Theorem 7.4 can be used to construct homeomorphisms between suitable pairs of point-fibred attractors whose kneading invariants are the same. For example, let attractors and of injective IFSs and obey the conditions of Theorems 7.3 and 7.4, i.e., assume that the attractors are point-fibred, that the critical sets are finite, and that is compact for every , where and are their respective coordinate maps. Let be a section of (w.r.t. ) and let be a section for (w.r.t. ), and assume that the critical sets are both finite, and that the kneading invariants are the same, say . Then, similarly to [17, Theorem 1], it follows from Theorem 7.4 that a homeomorphism is defined by
Transformations between attractors, including some of this kind, are discussed in Sect. 9.
An example of a homeomorphic pair of tree-like IFS attractors is illustrated in Fig. 8. The tree-like set in black in the top image represents the Julia set for the complex quadratic map where . This Julia set is the attractor of the IFS
where is any compact set such that for , and the two branches of the inverse of are continuous on . For an explanation of how such a Julia set can be viewed as the attractor of a suitable IFS, see . The bottom image in Fig. 8 is the attractor of the similitude IFS where and . The kneading invariant for each attractor (w.r.t. its IFS) is ; see .
8 Mandelbrot set for pairs of linear maps
Given a family of contractive IFSs, each consisting of a pair of similitudes that depend on a single parameter , the set of points in parameter space such that the attractor of is connected has emerged over the last twenty years as an area of interest. The set , called a Mandelbrot set, is the topic of this section.
If is a contractive IFS with attractor , then is connected if and only if, for any , there exists a sequence such that for any . Also, if is connected, then it is locally connected; see for example [73, Proposition 1.6.4].
For the case , there is a fundamental dichotomy for attractors: the attractor is connected if and only if
and is otherwise totally disconnected. This leads to the notion of Mandelbrot sets for parameterized families of pairs of transformations. Let denote the attractor of the contractive IFS
for The set
was introduced in  as an analog to the classical Mandelbrot set . Indeed, it is shown in  that attractors of corresponding to some values of that lie in the boundary of are homeomorphic to Julia sets for , where the corresponding values of belong to the boundary of . Such a pair of conjugate attractors is illustrated in Fig. 8. The set has connections to Bernoulli convolutions, Dirichlet forms on fractals, and zeros of polynomials with integer coefficients. See for example [6, 9, 39, 45, 110–112, 114].
To see the relationship between zeros of polynomials with integer coefficients and , note that finite compositions of the maps in give rise to polynomials of the form , where each coefficient is or . It follows that for all . It, in turn, follows from the remark containing equation (8.1), that the attractor of is connected, for given , if and only if there exists with such that
Since all such sequences of zeros and ones are permissible, the attractor is connected if and only if is a solution of equation (8.2) for some distinct pair of such sequences. The above implies
On the left in Fig. 9 we illustrate the density of zeros in shades of gray, where black means no zeros/pixel, in the complex plane, for all polynomials of degree , with each coefficient equal to or . Since we do not include coefficients (other than the leading one) equal to , the points which are not black, which lie inside the circle , lie in . (The centers of the black disks lie on the circle ). On the right in Fig. 9 we show a rough plot of , in white. This was computed by “brute force” – compute a digital approximation to and then see whether or not (digitized (digitized . The common features, and the differences, of the two images, the one on the right and the one on the left, are striking.
The observation (8.3) was made by Bousch , who used it to show that is both connected and locally connected. (Douady and Hubbard  proved the connectivity of the classical Mandelbrot set . Their conjecture, made in 1982, that is locally connected, remains open.) The equivalence (8.3) was also used by Bandt  as the basis of a fast algorithm for the computation of high resolution pictures of . This revealed surprising features and led to interesting questions and conjectures concerning the geometry and topology of . For example, Bandt noted that the complement of splits into many components and that is not the closure of its interior; and he conjectured that is the closure of its interior. Bandt also noted a relationship between and Bernoulli convolutions, reviewed in . Bandt’s ideas were further developed by Solomyak and Xu , who made progress on Bandt’s conjecture and studied properties of complex analogues of Bernoulli convolutions. Loci of connectedness of attractors of IFSs continues to be an active area of research; see for example .
9 Fractal transformations
While properties of classical geometrical transformations (e.g. affine, conformal, projective) are well known, much less is known about their rough counterparts, fractal transformations. Fractal transformations were introduced in 2005 [15, 16], and they have applications to image synthesis , photography , and digital imaging . A fractal transformation takes the attractor of one IFS to the attractor of another IFS. Figures 10 and 11, explained in more detail later in this section, provide visualizations of fractal transformations.
A fractal transformation is defined in terms of a coordinate map, discussed in Sect. 6.1, and a section of a coordinate map, discussed in Sect. 6.2. Here it is assumed that all IFSs are point-fibred so that each possesses a coordinate map, as given by Eq. 6.2. Given two point-fibred IFSs and with respective attractors and , a fractal transformation is basically a function that sends a point in to the point in with the same address. More precisely, assume that and have the same number of functions, and let and be the respective coordinate maps.
A transformation is called a fractal transformation if
for some shift invariant section of . If is a homeomorphism, then is called a fractal homeomorphism.
If a fractal transformation is a homeomorphism, then there exists a shift invariant section of such that the following diagram commutes:
Conversely, if there exists sections and and a homeomorphism such that the above diagram commutes, then and .
The commuting diagram means that the homeomorphism takes each point with address to the point with the same address . Two basic questions are (1) given an IFS, how to construct a shift invariant section, and (2) when is a fractal transformation a homeomorphism.
An IFS is said to be injective if the map is injective, for .
We consider the first question in the case that is an injective IFS.
For an IFS with attractor , a mask is a partition of such that for all . Given an injective IFS and a mask , consider the function defined by
when . The itinerary of a point is the string , where is the unique integer such that
A particular mask for an IFS with functions and attractor is defined by and
for . This is called the tops mask. There is a tops mask for every permutation of the functions, i.e., every permutation of .
The following theorem provides an answer to the first question posed above; it states that any shift invariant section is constructed from a mask.
Let be a contractive and injective IFS.
If is a mask, then is a shift invariant section of .
If is a shift invariant section of , then for some mask .
The section from a tops mask is given by
where the maximum is with respect to the lexicographic order on .
Question (2) involves establishing conditions under which and have related kneading invariants; see the discussion following Theorem 7.4. In the following subsection we consider two geometrically simple types of IFS, for which interesting families of fractal transformations can be established.
9.1 Bi-affine IFSs on
A bi-affine function from to is more general than an affine function but less general than a quadratic function.
A function is called bi-affine if it has the form
where bold face letters represent vectors in . Call a bi-affine function non-degenerate if and neither nor is a scalar multiple of . In particular, neither nor is the zero vector. A description of the geometric degeneracies that occur in these situations is described in . An IFS is called bi-affine if each function in is a non-degenerate bi-affine function.
It is easy to verify that a function is bi-affine if and only if, for a fixed or a fixed , it is affine in the other variable:
for all . Interpreted geometrically, these equations mean that
horizontal and vertical lines are taken to lines, and
proportions along horizontal and vertical lines are preserved.
This elementary class of functions, with connections to classic geometric results of Brianchon and Lambert dating back to the 18th century, proves extremely versatile for applications involving fractal transformations. A photography application , for example, is based on it. See also .
Note that the unique bi-affine function taking to the points , respectively, is
Let denote the unit square with vertices . The following Theorem provides sufficient conditions for a bi-affine function to be a contraction.
Let be a non-degenerate bi-affine function such that is a convex quadrilateral . If there is an , such that (1) each side of has length less than or equal to , (2) each diagonal has length less than or equal to , and (3) the vector sum of any two incident sides has length less or equal to , then is a contraction on .
Attention is now restricted to the case of a non-degenerate bi-affine function taking into itself. For such a bi-affine function, the conditions in Theorem 9.8 are not too restrictive. The intention is to form an IFS consisting of such functions, which must, according to Theorem 3.4, have a unique attractor. Given two such bi-affine IFSs with the same number of functions, fractal transformations, such as those illustrated in Fig. 11, can be produced as described below.
Consider a bi-affine IFS , where the four functions are determined by the images of the four vertices of as shown in Fig. 12. Each of the four images , of the square contains exactly one vertex of . The attractor of is itself. For “most” choices of the center and side points, satisfies the conditions of Theorem 9.8 and hence is a contractive IFS.
Next let be the tops mask for . Explicitly, is the closed quadrilateral , is the open quadrilateral together with the segments , is the open quadrilateral together with the segments , and is the open quadrilateral together with the segments .
Now consider a second bi-affine IFS of the same type with points replacing , and with mask defined exactly as it was for . The masks and induce shift invariant sections and , respectively, as guaranteed by Theorem 9.6.
For the pair of IFS described above, the maps and its inverse are homeomorphisms.
In this case the attractors of both IFSs are identical, the square. The fractal homeomorphism can be visualized as follows. Define an image as a function , where denotes the color palette, for example . If is any homeomorphism from onto , define the transformed image by
When is a fractal homeomorphism as constructed via Theorem 9.9, we get a visualization of the fractal homeomorphism by looking at how an image is transformed. This is done, for example, in Fig. 11.
9.2 Three dimensional tri-affine fractal homeomorphisms
9.3 Special overlapping IFSs
The bi-affine and tri-affine IFSs described above are non-overlapping, in the sense that if is the attractor of the IFS , then for all distinct . To determine whether a strongly overlapping IFS (in the sense of Definition 7.1d) fractal transformation is a homeomorphism is, in general, difficult. In this section a criterion is provided for a simple family of overlapping IFSs.
A special overlapping IFS is an IFS
on the unit interval, where the functions are continuous, increasing, contractions that satisfy
the last condition guaranteeing that is strongly overlapping.
To construct a fractal transformation from the attractor of one special overlapping IFS to the attractor of another, a section of the coordinate map must be constructed, which in turn requires designating a mask. A single point determines a pair of related masks. The partitions are of the form
where . The point will be called the mask point. For a masked special overlapping IFS, the two sections and , one corresponding to and the other corresponding to , are as follows: and , where
For a special overlapping masked IFS , the itineraries
of the mask point will be called the critical itineraries.
The following theorem  states that, for two special overlapping IFSs, whether or not a fractal transformation is a homeomorphism depends only on these two critical itineraries.
Given two special overlapping masked IFSs and with respective mask points and , coordinate maps and , sections and , and critical itineraries and , the fractal transformations and are homeomorphisms if and only if and .
10 IFS tiling
Computer generated drawings of tilings of the plane by self-similar figures appear in papers beginning in the 1980s and 1990s, for example the lattice tiling of the plane by copies of the twindragon. Tilings constructed from an IFS often possess global symmetry and self-replicating properties. Research on such tilings include the work of Bandt, Gelbrich, Gröchenig, Hass, Kenyon, Lagarias, Madych, Radin, Solomyak, Strichartz, Thurston, Vince, and Wang and the more recent work of Akiyama and Lau; see for example [3, 5, 58–60, 70, 75, 78, 104, 113, 116, 119, 122] and the list of references in . Even aperiodic tilings can be put into the IFS context; see for example the fractal version of the Penrose tilings .
The use of the inverses of the IFS functions in the study of tilings is well-established, in particular in many of the references cited above. However, in this section we introduce a simple and yet unifying technique for constructing both the classical and new IFS tilings. Theorems 10.4 and 10.5 show that the resulting tiling very often covers the entire basin of the attractor. In the case of tilings of Euclidean space, the basin is the entire Euclidean space.
Throughout this section, is a contractive IFS. By tile we simply mean a compact set. If is the attractor of an IFS , it is sometimes possible to tile the basin of with non-overlapping copies of or perhaps with non-overlapping tiles of several shapes related to . The attractor of any IFS in this section will be just-touching and such that the interior is nonempty. Let be an invertible IFS with just-touching attractor . For any string , a tiling will be constructed as follows. Extending the notation used in Section 6, let
Note that, in general, . Given a positive integer and any , let
Since, for any , we have
holds for all . Therefore
is a tiling of
where the union is a nested union.
Under certain, not too restrictive conditions on , conditions given in Theorem 10.4, tiles the entire basin of the attractor .
The IFS where and has attractor . The tiling is the tiling of by unit intervals. The tiling is the tiling of by unit intervals. If , then tiles by unit intervals.
For an invertible IFS with attractor , call full if there exists a compact set such that, for any positive integer , there exist such that
Call reversible for IFS if, for every positive integer , there exists an such that
where is the address of some point in . The following notation will be used: , so condition (10.1) can be expressed as .
A string is periodic of period if for all . Recall from Sect. 4 that a string is disjunctive if every finite word is a substring of . The set of disjunctive sequences is a large subset of in a topological, in a measure theoretic, and in an information theoretic sense .
Let be an invertible IFS with attractor with non-empty interior.
There are infinitely many disjunctive strings in if .
Every disjunctive string is reversible.
Every reversible string is full.
Let be a non-overlapping invertible IFS with attractor with non-empty interior and with basin . If is full, then covers .
By the above proposition and theorem (which are proved in ), full strings are plentiful. In fact, according to the next result, also proved in , any string is full with probability . Define a string to be a random string if it is chosen as follows: there is a such that, for each , is selected randomly from , where the probability that is greater than or equal to , independently of , for all . (See also Definition 4.1.)
Let , where is compact, be an invertible IFS with just-touching attractor with non-empty interior and with basin . If is a random string, then, with probability , the tiling covers .
(Digit tiling of ) The terminology “digit tiling”comes from the data used to construct the tiling, which is analogous to the usual base and digits used to represent the integers. An expanding matrix, i.e., linear function on , is an matrix such that the modulus of each eigenvalue is greater than 1. Let be an expanding integer matrix. A set of coset representatives of the quotient is called a digit set. It is assumed that . By standard algebra results, for to be a digit set it is necessary that
Consider an affine IFS , where
Since is expanding, it is known that, with respect to a metric equivalent to the Euclidean metric, is a contraction. Since is contractive, there is a unique attractor called a digit tile. The basin of is all of . Note that a digit tile is completely determined by the pair and will be denoted . It is known  that a digit tile is the closure of its interior and its boundary has Lebesque measure . The iterated function system is non-overlapping. If is full, then is a tiling of called a digit tiling. It is straightforward to show that, up to a rigid motion of , a digit tiling does not depend on as long as it is full. Under fairly mild assumptions [123, Theorem 4.3], a digit tiling is a tiling by translation by the integer lattice with the following self-replicating property: for any tile , it’s image is the union of tiles in . For this reason, such a tiling is often referred to as a reptiling of .
(Crystallographic tilings of ) Gelbrich  generalized digit tiling from the lattice group to any crystallographic group . Let be an expanding linear map such that . If is a set of right coset representatives of , then
is a contractive IFS with attractor with non-empty interior, called a crystallographic tile. The Levy curve is an example of such a crystallographic tile (for the 2-dimensional crystallographic group ). A tiling of is called a crystallographic reptiling.
(Chair tilings of ) The IFS where
is an IFS whose attractor is a filled square with a missing quadrant, i.e., a chair shaped polygon. There are numerous other such polygonal tiles that are the attractors of just touching IFSs. For the chair tile, there are uncountably many tilings , i.e. uncountably many full strings .
(Triangular affine tilings of ) Although we do not do so here, the significance of this tiling is that it can be used to extend a fractal homeomorphism between two triangular attractors, to a fractal homeomorphism of the Euclidean plane. Consider the IFS where each is an affine transformation defined as follows. Let , , and denote three noncollinear points in . Let denote a point on the line segment , a point on the line segment , and a point on the line segment , such that ; see panel (i) of Fig. 15. Let denote the unique affine transformation such that
by which we mean that maps to , to , and to . Using the same notation, let affine transformations , , and be the ones uniquely defined by
Panel (ii) of Fig. 15 shows the images of the points under the four functions of the IFS, illustrating the special way that the four functions fit together. The attractor of is the filled triangle with vertices at , , and , illustrated in (iii) in Fig. 15.
If is full, then according to Theorem 10.4, is a tiling of by triangles. One such affine tiling is illustrated in Fig. 16. A related projective tiling is also shown in that figure. Note that, if is a common edge of two triangles in , then is the image of the same edge of the original triangle from and , and if is a common vertex in of two triangles in , then is the image of the same vertex of the original triangle from and .