Attractors for classes of iterated function systems

Abstract

We consider various classes of iterated function systems, such as those comprised of similarities and those satisfying the open set condition, and weakly contractive systems. Necessary properties are developed for a compact set to be an attractor for the various classes of iterated function systems. Since the Hausdorff metric space is complete, we are able to develop several results concerning typical compact sets and their invariance with respect to iterated function systems.

1 Introduction

Let $X = (X,d)$ be a complete metric space, with ${\mathscr {S}} = \{S_{1}, \ldots , S_{m}\}$ a finite set of contraction maps $S_{i}:X \rightarrow X$. We call a set E an attractor (non-empty invariant set) of the iterated function system (IFS) ${\mathscr {S}}$ if ; that is, E is invariant with respect to ${\mathscr {S}}$. For any given finite set of contraction maps ${\mathscr {S}}$, there is a unique non-empty compact set E in X such that $E= {\mathscr {S}}(E)$. In what follows, we frequently take $\vert {\mathscr {S}} \vert$ to represent the necessarily unique attractor of the IFS ${\mathscr {S}}$. This and other results in [10] are recorded below.

Theorem 1.1

Let X be a complete metric space with ${\mathscr {S}} = \{S_{1}, \ldots , S_{m}\}$ a finite set of contraction maps $S_{i}:X \rightarrow X$.

• (1) There exists a unique non-empty compact set $E \subseteq X$ such that $E = {\mathscr {S}}(E)$.

• (2) The set E is the closure of the set of fixed points of the finite compositions of elements of ${\mathscr {S}}$.

• (3) If A is any non-empty compact set in X, then $\lim _{p \rightarrow \infty } {\mathscr {S}}^{p}(A)= E$ in the Hausdorff metric.

Now, suppose that the contraction maps found in ${\mathscr {S}}$ are similarities. That is, $d(S_{i}(x), S_{i}(y)) = r_{i} d(x, y)$ for all x, y in X, and $0< r_{i} < 1$. Each $S_{i}$ takes subsets of X and sends them into geometrically similar sets. This produces attractors that are self-similar. Should the images of the sets $S_{i}(F)$ not overlap “too much”, the self-similar attractor has Hausdorff dimension equal to the value of s satisfying .

The notion of insignificant overlap comes from the open set condition (OSC). Suppose ${\mathscr {S}} = \{S_{1}, \ldots , S_{m}\}$ is a finite set of contraction maps $S_{i}:X \rightarrow X$. The contractive system ${\mathscr {S}}$ satisfies the OSC if there exists a non-empty open set V for which and whenever $i \not = j$ [10, 15].

If each of the $S_{i}$ is a similarity, and $d(S_{i}(x), S_{i}(y)) = r_{i} d(x, y)$ for all x, y in X, where $0< r_{i} < 1$, then the Hausdorff dimension, $\dim _{\mathscr {H}} (E)$, is equal to s, and $0< {\mathscr {H}}^{s}(E) < \infty$, where $E = {\mathscr {S}}(E)$ is the attractor for ${\mathscr {S}}$, and .

As proved as an exercise in [8], if the unique non-empty invariant set E satisfies whenever $i \not = j$, then the system ${\mathscr {S}} = \{S_{1}, \ldots , S_{m}\}$ of contraction maps $S_{i}:X \rightarrow X$ satisfies the OSC.

The purpose of this paper is to study the attractors of the IFS that are comprised of similarities, that satisfy the OSC, or both. Let X be a complete metric space, take ${\mathscr {K}}(X)$ to be the collection of non-empty compact subsets of X, with ${\mathscr {S}} = \{S_{1}, \ldots , S_{m}\}$ a finite set of contraction maps $S_{i}:X \rightarrow X$. We set

Finally, let be the set of all attractors in ${\mathscr {K}}(X)$ generated by some ${\mathscr {S}}= \{S_{1}, \ldots , S_{m}\}$, a finite collection of contraction maps $S_{i} :X \rightarrow X$. We call a finite collection of contraction maps a contractive system and a finite collection of weak contractions a weakly contractive system.

We proceed through several sections. Definitions, notation and some previously known results are presented in Sect. 2. There, we also record some observations concerning the sets ${\mathscr {I}}_{1}$ and ${\mathscr {I}}_{2}$. Section 3 is dedicated to the study of countable attractors. If is homeomorphic to an element of ${\mathscr {I}}_{1}$ or ${\mathscr {I}}_{2}$, then E must be a singleton, or a Cantor set, or the closure of the union of countably many disjoint non-trivial closed intervals. Any nowhere dense element of ${\mathscr {I}}$ has a homeomorphic copy $E'$ in ${\mathscr {I}}_{3}$. If E is a countable attractor, then it has a homeomorphic copy F such that $F = {\mathscr {S}}(F)$ for some ${\mathscr {S}} = \{S_{1}, S_{2}\}$.

In Sect. 4, one studies weakly contractive systems. From [4], we know that the set of attractors in generated by contractive systems is a first category $F_{\sigma }$. This is also true for IFS composed of weak contractions. One also sees that the set of attractors generated by IFS of the form is nowhere dense in . Since the set of attractors is dense in , one concludes that most attractors can only be generated by IFS ${\mathscr {S}} = \{S_{1}, \ldots , S_{L}\}$, where $L > M$.

Not only is the typical element of not an attractor for any IFS, but one also can take that dense $G_{\delta }$ subset of so that ${\mathscr {H}}^{\phi }(E)= 0$ for each of its elements E, where $\phi$ is some fixed gauge function. In Sect. 5, one sees that attractors with “large” measure—that is, ${\mathscr {H}}_{m}^{\phi }(E) \geqslant c$—are also exceptional.

Section 6 is dedicated to attractors with non-empty interior. There, we develop a necessary condition for a compact set with non-empty interior to be an attractor. This is similar to that developed by Nowak for countable sets [1, 12].

2 Preliminaries

We take (X, d) to be a compact metric space. For A subset of X, we denote by $\vert A \vert$ its diameter, by $\overline{A}$ its closure, by its interior, by its convex closure (meaning the closure of its convex hull), and by its cardinality when A is finite.

Much of the analysis takes place in the Hausdorff metric space ${\mathscr {K}}(X) = ({\mathscr {K}}(X), {\mathscr {H}})$. Take ${\mathscr {K}}(X)$ to be the collection of non-empty compact subsets of X. Endow ${\mathscr {K}}(X)$ with the Hausdorff metric ${\mathscr {H}}$ given by

where the $\delta$-neighbourhood or $\delta$-parallel body, $B_{\delta }(A)$, of a set A is the set of points within distance $\delta$ of A. Thus, , where is the open ball in X centered at x of radius $\delta$. The Hausdorff metric space $({\mathscr {K}}(X), {\mathscr {H}})$ is also compact.

Since $({\mathscr {K}}(X), {\mathscr {H}})$ is complete, one can make good use of the Baire category theorem. A set is of the first category in the complete space (X, d) whenever it can be written as a countable union of nowhere dense sets; otherwise, the set is of the second category. A set is residual if it is the complement of a first category set, and an element of a residual set is called either typical or generic.

Theorem 2.1

(Baire category theorem) Let (X, d) be a complete metric space with B a first category subset of X. Then is dense in X.

Let $\Phi$ denote the set of functions $\phi$ that are continuous and increasing on , with $\phi (0) = 0$. For $\phi \in \Phi$ and $s \in {\mathbb N}$, set

Then defines a measure on the Borel sets in . In what follows, our concern will be primarily with closed sets. In the case that $\phi (x) = x^{s}$, one gets the usual s-dimensional Hausdorff measure [2, 4, 8, 9]. Define the Hausdorff dimension of a set E such that

In Sect. 3, we use ${\lambda }^{n}$ to refer to the Lebesgue n-dimensional measure.

Since our interest lies with contractions and weak contractions, the functions considered are all continuous. Let C(X, X) to be the set of continuous functions $f :X \rightarrow X$. Within C(X, X) we use the uniform metric: . Since X is compact, .

We recall that a topological space is a Cantor space if it is non-empty, perfect, compact, totally disconnected, and metrizable. In particular, a topological space is a Cantor space if it is homeomorphic to the Cantor ternary set.

A topological space X is said to be scattered if every non-empty subspace M has an isolated point in M. Every compact scattered Hausdorff space has a base consisting of clopen sets (is zero-dimensional), and a compact metric space is scattered if and only if it is countable. Let X be a compact scattered space. We let

be the Cantor–Bendixon derived set of X, and define inductively

$$\begin{aligned} X^{(\alpha + 1)} = {\bigl (X^{(\alpha )}\bigr )}{'};\qquad X^{\alpha } = \bigcap _{\beta < \alpha } X^{(\beta )} \quad \text { for a limit ordinal }\alpha . \end{aligned}$$

The height of X is

By the Mazurkiewicz–Sierpiński Theorem [11], every countable compact scattered space X is homeomorphic to the space , where and n is the number of elements of $X^{(\beta )}$.

Next we present some simple conclusions concerning elements in ${\mathscr {I}}_{1}$ and ${\mathscr {I}}_{2}$.

Proposition 2.2

Let ${\mathscr {S}}= \{S_{1}, \ldots , S_{N}\}$ be a finite set of similarities from ${\mathbb R}^{n}$ to ${\mathbb R}^{n}$ satisfying the OSC. Then the following are equivalent:

• (1) $F = {\mathscr {S}}(F)$ is nowhere dense and perfect,

• (2) $N \geqslant 2$ and .

Proof

(1)$\Rightarrow$(2): Suppose $F = {\mathscr {S}}(F)$ is nowhere dense and perfect. Then, $N \geqslant 2$. Suppose ${\mathscr {S}}$ satisfies the OSC with V open. Since F is nowhere dense, V must contain some G, an open ball contained in the complement of F. Since whenever $i \not = j$, it follows that $S_{{i_{1}} \ldots {i_{k}}}(V) \cap S_{{j_{1}} \ldots {j_{k}}}(V) = \varnothing$ whenever . Let $x \in F$. For each $k \in {\mathbb N}$, there exists a string of length k, say such that $x \in S_{i_{1} i_{2} \ldots i_{k}}(F)$. As $G \subseteq V$, $S_{i_{1} i_{2} \ldots i_{k}}(G) \subseteq S_{i_{1} i_{2} \ldots i_{k}}(V)$, $S_{i_{1} i_{2} \ldots i_{k}}(G)$ is complementary to F [10, p. 736], and the porosity

Thus, $\dim _{\mathscr {H}} (F) <n$, and ([9, Theorem 9.3], [17, Section 4F]).

(2)$\Rightarrow$(1): Suppose $F = {\mathscr {S}}(F)$ with $N \geqslant 2$ and . We show that it has no isolated point. Let $x \in F$ with $\epsilon >0$. Take ${i'_{1}}, \ldots ,{i'_{k}}$ so that $S_{{i'_{1}} \ldots {i'_{k}}}(F) \subseteq B_{\epsilon }(x)$. Since F is not a singleton, there exists $y \in S_{{i'_{1}} \ldots {i'_{k}}}(F)$ distinct from x. In particular, x is a limit point of F. So, F is perfect. Since $\dim _{\mathscr {H}}(\vert {\mathscr {S}} \vert ) <n$, F contains no ball. $\square$

Proposition 2.3

Let ${\mathscr {S}}= \{S_{1}, \ldots , S_{N}\}$ be a finite set of similarities from ${\mathbb R}^{n}$ to ${\mathbb R}^{n}$ satisfying the OSC. If $F = {\mathscr {S}}(F)$, then the following are equivalent:

• (1) F contains a ball in ${\mathbb R}^{n}$.

• (2) F is the closure of a countable union of non-degenerate closed balls in ${\mathbb R}^{n}$.

• (3) and $N \geqslant 2$.

Proof

(1)$\Rightarrow$(2): Let $x \in F$. It suffices to show that there is a sequence of balls converging to x. Let $\epsilon >0$, and take $i_{1} i_{2} \ldots i_{k}$ so that $S_{i_{1} \ldots i_{k}}(F) \subseteq B_{\epsilon }(x)$. Since F contains a ball and $S_{i_{1} \ldots i_{k}}$ is a composition of similarities, $S_{i_{1} \ldots i_{k}}(F)$ also contains a ball.

(2)$\Rightarrow$(3): Since F has non-empty interior, $N \geqslant 2$. Moreover, F is not nowhere dense, so by Proposition 2.2.

(3)$\Rightarrow$(1): Since , F is not nowhere dense by Proposition 2.2, so it must necessarily contain a ball in ${\mathbb R}^{n}$. $\square$

Proposition 2.4

Let ${\mathscr {S}} = \{S_{1}, S_{2}\}$ where each $S_{i}:{\mathbb R} \rightarrow {\mathbb R}$ is a similarity. If $r_{1} + r_{2} \leqslant 1$, and , then ${\mathscr {S}}$ satisfies the OSC with .

This allows us to characterize, up to homeomorphism, those elements of ${\mathscr {I}}$ generated by a pair of similarities.

Proposition 2.5

Let ${\mathscr {S}} = \{S_{1}, S_{2}\}$ where each $S_{i}:{\mathbb R} \rightarrow {\mathbb R}$ is a similarity. Necessarily, ${\mathscr {S}}$ satisfies the OSC so long as and $r_{1} + r_{2} \leqslant 1$. Set

Then is homeomorphic to an element of ${\mathscr {T}}$ if and only if one of the following occurs:

• (1) E is a singleton,

• (2) E is a Cantor set, or

• (3) E is an interval.

3 Countable attractors

In [12], Nowak completely characterizes, up to homeomorphism, countable attractors for iterated function systems. Nowak shows that every countable compact set of successor Cantor–Bendixon height with a single point of maximal rank has a homeomorphic copy in that is an attractor of an IFS consisting of two contractions. This IFS satisfies the OSC, and the Lipschitz constants of its two components can be chosen to be as small as one wishes. Moreover, Nowak shows that if a countable compact metric space is an IFS-attractor, then its Cantor–Bendixson height cannot be a limit ordinal.

Theorem 3.1

([12, Theorem 3]) A compact scattered metric space of limit Cantor–Bendixson height is not homeomorphic to any IFS-attractor consisting of weak contractions.

Theorem 3.2

([12, Theorem 4]) For every $\epsilon >0$ and every countable ordinal $\delta$ the scattered space is homeomorphic to the attractor of an iterated function system consisting of two contractions in the unit interval , such that

The main result of this section is the following theorem.

Theorem 3.3

Let be countable and compact. Then E is homeomorphic to an attractor $E{'}$ for some ${\mathscr {S}} = \{S_{1}, S_{2}\}$ which satisfies the OSC if and only if the height of E is a successor ordinal.

Theorem 3.3 is an immediate consequence of Theorem 3.2, and the following proposition. Frequently, we make use of the convex closure of a set E, and write . We use this notation even in the case that E is closed, in an effort to somewhat limit the various notations used.

Proposition 3.4

Suppose $E \in {\mathscr {I}}$, and $E = {\mathscr {S}}(E)$ where ${\mathscr {S}} = \{S_{1}, \ldots , S_{m}\}$. Then, for any $n \geqslant m$, there exists and which satisfies the OSC such that

• (1) $F = {\mathscr {S}}'(F)$,

• (2) , where each $E_{i}$ is similar to E ($\dot{\bigcup }$ means disjoint union), and

• (3) whenever $i \not =j$.

Proof

We give a proof in the case that $m=2$. The general case then follows easily. Suppose $E \in {\mathscr {I}}$, and for convenience and without loss of generality, , and $E = {\mathscr {S}}(E)$ for some ${\mathscr {S}} = \{S_{1}, S_{2}\}$. Take $E_{1},E_{2},\ldots ,E_{n}$ such that

• (i) each is similar to E,

• (ii) lies to the right of $E_{l}$ whenever $j < l$,

• (iii) , where , for any $1 < j \leqslant n$, and

• (iv) , where $G_{i}$ is the open interval lying between $E_{i}$ and $E_{i+1}$, that is $G_{i} = (\max E_{i+1}, \min E_{i})$.

Moreover, take . Define $T_{1}$ so that it is linear on each of the intervals and $G_{i}$. In particular, take $T_{1}$ so that

• (v) is mapped onto with $T_{1}(E_{i}) = E_{i+1}$ for $i=1,2,\ldots , {n-1}$,

• (vi) ,

• (vii) $G_{i}$ is mapped onto $G_{i+1}$ for $i=1,2, \ldots , n-2$, and

• (viii) $T_{1}(G_{n-1}) = \min E_{n}$, too.

Then is a contraction given the diameters of the sets $E_{i}$ and $G_{i}$, and .

Let us now define $T_{2}$. Recall that $E_{1}$ and $E_{2}$ are both similar to E and . Set and take ${\phi }_{i}:E \rightarrow E_{i}$ to be the line such that ${\phi }_{i}(0) = \min E_{i}$ and ${\phi }_{i}(1) = \max E_{i}$. Define $T_{2}$ on so that , and on , define $T_{2}$ so that . Set $T_{2}(x) = T_{2}(a_{2})$ for all $x \leqslant a_{2}$, and extend $T_{2}$ linearly on $G_{1}$. Then

and

It follows, then, that is a contraction map, and

$\square$

Remark 3.5

From [6, Theorem 5.4] and the construction found in its proof, if is nowhere dense and uncountable, then there exist $F'$, a homeomorphic copy of F, and an IFS ${\mathscr {S}}$ satisfying the OSC such that . This observation, coupled with Theorem 3.3, allows one to conclude that any nowhere dense attractor is homeomorphic to an element of ${\mathscr {I}}_{3}$.

4 Systems of weak contractions

In this section we investigate IFS consisting of weak contractions. First, we observe that the typical element of is not an attractor for any system of weak contractions. Then, we consider the function given by ${\mathscr {S}} \mapsto \vert {\mathscr {S}} \vert$, and investigate the images of , , and of . There, we focus on the case when .

In [4], we construct a residual subset ${\mathscr {K}}_{2}$ of with the property that if $E \in {\mathscr {K}}_{2}$, then $E \not = {\mathscr {S}}(E)$ for any finite set of contraction maps ${\mathscr {S}}$ defined on . The analysis developed for contractions is valid for weak contractions. This is recorded in the next result.

Theorem 4.1

There is a residual subset ${\mathscr {K}}_{2}$ of such that $E \not = {\mathscr {S}}(E)$ for any finite set of weak contraction maps ${\mathscr {S}}$ defined on , for any $E \in {\mathscr {K}}_{2}$.

For $s >0$, set

and, for $m \in {\mathbb N}$, set

As the following result in [6] shows, should we limit the number of contractions in our IFS to no more than N, and uniformly bound the Lipschitz constants of the contractions, then the set of attractors generated by such systems is closed.

Lemma 4.2

([6, Lemma 4.3]) Let (X, d) be a compact metric space. Let $N \in {\mathbb N}$ and $0< m < 1$. The set

is closed.

The following straightforward lemma shows not only that ${\mathscr {I}}_{1}$ is dense in the Hausdorff metric space , but also that the attractors for iterated function systems generated by contractions with arbitrarily small Lipschitz constants are dense in .

Lemma 4.3

The set of attractors ${\mathscr {I}}_{1}$ is dense in .

Proof

Let , and take $\epsilon >0$. We use the Euclidean metric within . Let $F \subseteq E$ be a finite set so that ${\mathscr {H}}(E,F) < {\epsilon }/{2}$; say . To each $x_{i}$ associate a non-degenerate closed ball $I_{i}$ and a similarity $S_{i}$ so that $x_{i} \in I_{i}$, whenever $i \not = j$, , and . It follows that $F \subseteq \vert \{S_{1}, S_{2}, \ldots , S_{n}\} \vert \subset B_{{\epsilon }/{2}}(F)$, and ${\mathscr {H}}(F, \vert \{S_{1}, S_{2}, \ldots , S_{n}\}\vert ) < {\epsilon }/{2}$. Now

$$\begin{aligned} {\mathscr {H}}(E, \vert \{S_{1}, S_{2}, \ldots , S_{n}\}\vert ) \leqslant {\mathscr {H}}(E, F) + {\mathscr {H}}(F, \vert \{S_{1}, S_{2}, \ldots , S_{n}\}\vert ) < \frac{\epsilon }{2} + \frac{\epsilon }{2} = \epsilon . \end{aligned}$$

The conclusion follows. $\square$

Set

and, for $m \in {\mathbb N}$, set

Call a weak contraction. We recall some simple observations.

Remark 4.4

Let .

• (1) The set is closed.

• (2) The set is a closed and nowhere dense subset of , for all $s \in (0,1)$.

• (3) Let

  

  Then is closed in $\mathrm{Lip}\,1$.

See also [7] for (2) in the case of maps on closed, convex and bounded subsets of a Hilbert space.

Should ${\mathscr {S}} = \{S_{1}, \ldots , S_{m}\}$ and ${\mathscr {T}} = \{T_{1}, \ldots , T_{m}\}$ be elements of , set , where . Take a and b in ${\mathbb R}$. By we denote the closed interval with endpoints a and b. That is, either or , should $a<b$ or $b <a$.

The next lemma shows that, while is a very small subset of for every $s \in (0,1)$, is considerably larger. This is similar to a result found in [13] for maps defined on closed, convex and bounded subsets of a Banach space.

Lemma 4.5

Let . The set of weak contractions , defined on X, is a dense $G_{\delta }$ subset of .

Proof

Let $\epsilon >0$. Suppose that . Then there exist points $x \not =y$ in such that $\vert f(x) - f(y) \vert = \vert x - y \vert$. Should $\vert f(x) - f(y) \vert = \vert x - y \vert$, then it follows that $\vert f(a) - f(b) \vert = \vert a -b \vert$ for all points a and b in , as . Let be an enumeration of and set . From Remark 4.4 it follows that $G_{n,m}$ is open in . Take such that $f(b) - f(a) = b-a$. Consider such that

and g is extended linearly on (a, b). Then $\Vert f - g \Vert < \epsilon$, and . The construction of g is similar, should $f(b) - f(a) = a - b$. We conclude that is dense. Since $G_{n,m}$ is both dense and open, one concludes that is a dense $G_{\delta }$ subset of . $\square$

Let us now turn our attention to attractors generated by sets of m-many weak contractions.

Lemma 4.6

Let X be a compact metric space. The function

given by ${\mathscr {S}} \mapsto \vert {\mathscr {S}} \vert$, is continuous.

Proof

Let be such that , or equivalently, ${\Vert {\mathscr {S}}_{k} - {\mathscr {S}} \Vert } \rightarrow 0$. Say ${\mathscr {S}}_{k}(E_{k}) = E_{k}$. Since $({\mathscr {K}}(X), {\mathscr {H}})$ is compact, we may assume that in the Hausdorff metric.

Say and ${\mathscr {S}} = \{S_{1}, \ldots , S_{m}\}$. Since , it follows that uniformly for each $1 \leqslant i \leqslant m$. Moreover, and , so that . Thus, if , then , and our conclusion follows. $\square$

Lemma 4.7

For any $s \in (0, 1)$, the set is closed and nowhere dense in .

Proof

Let $s \in (0,1)$. Since is compact, and ${\mathscr {T}}$ is continuous, it follows that is closed. Since , is nowhere dense. $\square$

It is easy to determine the size of since is compact in . The proof of the following is considerably more involved, as is a $G_{\delta }$ set.

The following observations are fundamental to the construction found in the proof of Theorem 4.8.

• (1) Let . If for any i, j and k in with $i \not =j$, one has , then for every weak contraction $f:E \rightarrow E$ and $1 \leqslant k \leqslant N$, it follows that $f(E_{k}) \subseteq E_{l}$, $1 \leqslant l \leqslant N$.

• (2) Let and . Suppose that for all $i \not = j$ in , and $\vert y_{l} - y_{k} \vert > \epsilon$ for all $l \not = k$ in $\{1, 2, \ldots , m\}$. If $f:E \rightarrow F$ is a weak contraction, then $f(E) = y_{i}$, for some $1 \leqslant i \leqslant m$.

Theorem 4.8

The set is nowhere dense in .

Proof

We can assume $m >1$ since the result is clear for $m=1$. Fix $\delta >0$. We begin by developing a set E.

• (1) , with for each $i \not = j$.

• (2) Each $E_{i}$ contains $2^{2^{i}}$ points that are distributed uniformly throughout the interval . Set , with , and .

• (3) lies to the right of $E_{l}$ whenever $j < l$ and .

• (4) for all i, where $d < {\delta }/{4}$.

The idea, now, is to associate to each point $x_{i,j}$ a non-empty compact neighborhood $E'_{i,j} \subseteq B_{r}(x_{i,j})$, where . More precisely, take such that , and set $E'_{i,j} = B_{r}(x_{i,j}) \cap E{'}$, for and , and .

Suppose is a weak contraction. Then

• (i) ; that is each can map into only one $E'_{l}$ as , and the smallest interval in complementary to the sets is of length at least .

• (ii) $S(E'_{k,j}) \subseteq E'_{l,p}$, so that each $E'_{k,j}$ can map into only one $E'_{l,p}$ as and the smallest interval complementary to the sets is at least of length .

• (iii) If $l >j$ and , then , for some s as for any i and p.

• (iv) If $S(E'_{l}) \subseteq E'_{l}$, then , whenever $j < l$ as $S(E'_{l}) \subseteq E'_{l}$ implies that , so that . Now, note that $S(E{'}) \subseteq E'$, and S is a weak contraction.

Let ${\mathscr {S}} = \{S_{1}, S_{2}, \ldots , S_{m}\}$, where each $S_{i}$ is a weak contraction, m is now a fixed number, and take $\epsilon >0$. Let . Since the collection of finite sets is dense in , there exists $F_{2}$ finite such that ${\mathscr {H}}(F_{1}, F_{2}) < {\epsilon }/{2}$. Say . Take $F_{3}$ finite in such that

• (v) , where ,

• (vi) , and

• (vii) there is a similar copy of E in $B_{\gamma }(x_{1})$. We take $k> m \geqslant 2$, so that

  • a. , and

  • b. whenever $k \geqslant j \geqslant k-m$.

Suppose that ${\mathscr {H}}(F_{4}, F_{3}) < \sigma$, where , , and , the distance between $E_{0}$ and $E_{1}$, the first two components of the similar copy of E found in $B_{\gamma }(x_{1})$. Thus and . We continue to use the notation of $F_{3}$, remembering that each point $x_{i}$ in $F_{3}$ now represents $F_{4} \cap B_{\sigma }(x_{i})$, a subset of $F_{4}$ both open and closed in the relative topology of $F_{4}$. Suppose ${\mathscr {S}}(F_{4}) = F_{4}$. By our choice of k (see (vii) b), there are so many components of $E'_{k}$ that it is not possible to cover $E'_{k}$ with m weak contractions defined on . Thus, there must exist some $S_{i}$—say $S_{1}$—such that $S_{1}(E'_{k}) \subseteq E'_{k}$. This implies $S_{1}(E') \subseteq E'_{k}$. Similarly, there exists some —say $S_{2}$—such that $S_{2}(E'_{k-1}) \subseteq E'_{k-1}$, which implies that . (It is also possible that $S_{2}(E'_{k}) \subseteq E'_{k-1}$.) Continuing, for each $0 \leqslant i \leqslant m-1$, we have, with possible renumbering of the elements of ${\mathscr {S}}$, that $S_{i+1}(E'_{k-i}) \subseteq E'_{k-i}$, so that . (Analogously to what we noted with $S_{2}$, it is also possible that there exists such that $S_{i+1}(E'_{l}) \subseteq E'_{k-i}$.) Note that can intersect at most i components of . This follows from (iii). The m contractions $\{S_{1}, \ldots , S_{m}\}$ acting on the $n-1$ components of can intersect at most $m(n-1) < mn$ components of $F_{4}$. We conclude that and can intersect at most components of . Since , we conclude that $F_{4} \nsubseteq {\mathscr {S}}(F_{4})$. $\square$

As a corollary, we have that the set of attractors generated by those IFS with m many contractions is a nowhere dense $F_{\sigma }$ set in .

Corollary 4.9

The set is a nowhere dense $F_{\sigma }$ set in .

Proof

We note that . $\square$

Remark 4.10

Let l and m be elements of ${\mathbb N}$. Then

where

• (1) is compact,

• (2) ${({\mathscr {K}}_{2})}^\mathrm{c}$ is a first category and dense $F_{\sigma }$ subset of ,

• (3) is a nowhere dense $F_{\sigma }$ subset of $({\mathscr {K}}([0,1]), {\mathscr {H}})$, and

• (4) ${\mathscr {T}}\bigl ({\bigl (\mathrm{Lip}\,\frac{l-1}{l}\bigr )}{}^{m}\bigr )$ is nowhere dense and closed.

In particular, the set of all attractors ${\mathscr {I}}$ is dense in , but, when we limit ourselves to only m maps, we are able to generate as attractors only a very small subset of .

5 The set of attractors with large measure

As in [4], we denote by ${\mathscr {I}R}$ the collection of all points in with all of the coordinates irrational. Set

We call a rational open interval in each set , where $a_{i}$ and $b_{i}$ are in for all i. We say that $E \cap J$ is a rational portion of if $E \cap J$ is non-empty and J a rational open interval. If $E \in {\mathscr {K}}_{1}$, then there exists some $\phi \in \Phi$ such that ${\mathscr {H}}^{\phi }(E \cap J) >0$ for every rational portion $E \cap J$ of E. Moreover, the set is a dense $G_{\delta }$ subset of [2, 4]. One concludes, then, that the typical element of has very small measure, and that the elements of which comprise the elements of ${\mathscr {K}}_{2}$ in Proposition 4.1 can be taken to have arbitrarily small measure. In what follows it will be convenient to use the following results. We let $\lambda$ be the Lebesgue measure in ${\mathbb R}$, and denote by ${\Phi }_\mathrm{c}$ the family of concave functions from $\Phi$.

If f is a finite function of a real variable defined in the neighbourhood of a point $x_{0}$, we denote the lower limit of $({f(x) - f(x_{0})})/({x - x_{0}})$ as x tends to $x_{0}$ by values $x > x_{0}$, that is the lower right Dini derivate of the function f at the point $x_{0}$, as $D_{+}f({x}_{0})$ [14].

Proposition 5.1

([16, Lemmas 3.3, 3.5, Proposition 3.2]) Let $\phi \in \Phi$.

• (1) If the lower right Dini derivate of $\phi$ at 0 is finite, for instance $D_{+} \phi (0) = s < \infty$, and for which $\lambda (E) =c$, then ${\mathscr {H}}^{\phi }(E) = sc$.

• (2) If $D_{+} \phi (0) = + \infty$, then there exists $\tilde{\phi } \in {\Phi }_\mathrm{c}$ such that ${\mathscr {H}}^{\phi }(E) = {\mathscr {H}}^{\tilde{\phi }}(E)$, for all .

• (3) If $\phi \in {\Phi }_\mathrm{c}$ and , then $L = \lim _{\delta \rightarrow 0} {\phi (M \delta )}/{\phi (\delta )} \not =0$ exists, and ${\mathscr {H}}^{\phi }(f(E)) \leqslant L {\mathscr {H}}^{{\phi }}(E)$. Moreover, should $\vert f(x) - f(y) \vert = M \vert x-y \vert$, then ${\mathscr {H}}^{\phi }(f(E)) = M {\mathscr {H}}^{{\phi }}(E)$.

For the remainder of this section, we presume that each $\phi$ taken is concave, so that we can apply Proposition 5.1 to our measures. The next result shows that the mass distribution within elements of ${\mathscr {I}}_{2}$ is relatively uniform.

Lemma 5.2

Let $X = {\mathbb R}$. Let $F \in {\mathscr {I}}_{2}$, and $\phi \in {\Phi }_\mathrm{c}$. Then one of the following is true:

• (1) For every portion P of F, ${\mathscr {H}}^{\phi }(P)=0$.

• (2) For every portion P of F, ${\mathscr {H}}^{\phi }(P)=\infty$.

• (3) For every portion P of F, $0< {\mathscr {H}}^{\phi }(P) < \infty$.

Proof

Let P, J be portions of F, where $F = {\mathscr {S}}(F)$ and ${\mathscr {S}} = \{S_{1}, \ldots , S_{N}\}$ and $\phi \in {\Phi }_\mathrm{c}$. There exists $i_{1} i_{2} \ldots i_{n}$ such that $S_{i_{1} i_{2} \ldots i_{n}}(P) \subset J$. Thus, there is some for which . There exists $j_{1} j_{2} \ldots j_{m}$ such that , which ensures the existence of for which . It follows that and . $\square$

In [3, 5] sets of arbitrary Hausdorff dimension are constructed which are not attractors of any finite family of weak contractions in and, more generally, in . Let $\phi \in {\Phi }_\mathrm{c}$. In [4] and Theorem 4.1, one sees that the typical element of has both ${\mathscr {H}}^{\phi }$-measure zero, and is not an attractor for any IFS. Working on the unit interval, we now show that the typical element of for which ${\mathscr {H}}_{m}^{\phi }$ is bounded away from zero also is not an attractor for any IFS. That is, the remainder of this section is dedicated to showing that while Lemma 5.2 provides a necessary condition for some compact set to be an element of ${\mathscr {I}}_{2}$, that condition is far from sufficient. As with Theorem 4.1, the typical “large” set that satisfies the necessary condition found in Lemma 5.2 is not an attractor for any IFS.

We begin by recalling the following result from [2], which describes a typical element of .

Proposition 5.3

([2, Proposition 2.2 (1)]) Let

Then ${ \Sigma }_{0}$ is a dense $G_{\delta }$ subset of , hence topologically complete.

Lemma 5.4

Let $\gamma >0$. Then

is dense and open in the complete metric space .

Proof

The density of ${ \Sigma }_{1}$ follows from the fact that each compact set can be approximated by finite sets in the Hausdorff metric. Let $E \in { \Sigma }_{1}$. To show that ${ \Sigma }_{1}$ is open, take ${\{U_{i} \}}_{i=1}^{\infty }$ such that

• (1) ,

• (2) $\vert U_{i} \vert \leqslant {1}/{m}$ for all i, and

• (3) .

If $F \subseteq \bigcup _{i=1}^{\infty } U_{i}$, then ${\mathscr {H}}_{m}^{\phi }(F) < \gamma$, too. $\square$

Proposition 5.5

Let $\gamma >0$. The collection is topologically complete with respect to the Hausdorff metric ${\mathscr {H}}$.

Proof

Let $\gamma >0$. Then , so that ${ \Sigma }_{2}$ is closed in the topologically complete metric space ${ \Sigma }_{0}$. $\square$

Proposition 5.6

Fix $F \in { \Sigma }_{2}$. The set is closed in ${\Sigma }_{2}$.

Proof

Let $\{E_{n}\} \subseteq { \Sigma }_{3}$ with such that $f_{n}(E_{n}) \supseteq F$. Since is compact and $\{f_{n}\}$ is uniformly bounded and equicontinuous, there exist in ${\mathbb N}$ and such that and $f_{n_{k}}$ converges uniformly to some f. Then

Since $F \subseteq f_{n_{k}}(E_{n_{k}})$, it follows that $F \subseteq f(E)$. $\square$

Theorem 5.7

Let . The typical $E \in { \Sigma }_{2}$ is not an attractor of any system of weak contractions.

Proof

Suppose E and F are nowhere dense subsets of , and is such that $f(E) =F$. If is complementary to F, then there exists complementary to E such that $f((a,b)) \supseteq (c,d)$ and $({d-c})/({b-a}) \leqslant \rho$. Let $\{J_{k}\}$ be an enumeration of the intervals complementary to F such that $\vert J_{k} \vert \geqslant \vert J_{k+1} \vert$ for all k. Similarly, let $\{I_{k}\}$ be an enumeration of the intervals complementary to E. There exists a subsequence $\{I_{k_{s}}\}$ such that ${\vert J_{s} \vert }/{\vert I_{k_{s}} \vert } \leqslant \rho$ for all i [16, Lemma 3.6]. This observation provides a necessary condition on the sets E and F in order for there to exist some with the property that $f(E) \supseteq F$. We use this observation below. Let I and J be disjoint rational intervals. Our goal is to show that the typical element of ${ \Sigma }_{2}$ is not an attractor of any system of weak contractions. A critical step in this direction is establishing that given some fixed $\rho >0$, for the typical $E \in { \Sigma }_{2}$ and disjoint clopen portions $E \cap I$ and $E \cap J$, one has that whenever f is in . We restrict our attention to disjoint portions as the identity function $i_{d}$ is in , and for all E in ${ \Sigma }_{2}$ and rational intervals I. Set . Then ${\Sigma }_{4}^{I,J}$ is closed. This can be proved as for ${\Sigma }_{3}$ in Proposition 5.6. We wish to show that ${\Sigma }_{4}^{I,J}$ is nowhere dense. Since ${\Sigma }_{4}^{I,J}$ is closed, it is sufficient to show that its complement, necessarily open in ${ \Sigma }_{2}$, is also dense. Let $E \in {\Sigma }_{4}^{I,J}$ and $\epsilon >0$. We modify E to get $E'$ in ${ \Sigma }_{0}$ such that

• (1) , so that E and $E'$ are the same outside of I;

• (2) Take $E'$ on I so that

  • a. and . Thus, ${\mathscr {H}}(E, E') < \epsilon$. Since $E \subseteq E'$, it follows that , too.

  • b. Consider $\{J_{k}\}$ for $J \cap E$ and $\{I_{k}\}$ for $I \cap E$. Suppose is a complementary interval of $E \cap I$. Then c and d are elements of E and if $d-c < \epsilon$, then $(c,d) \subseteq B_{\epsilon }(c) \subseteq B_{\epsilon }(E)$. It follows that there exists some N such that $\vert I_{k} \vert < \epsilon$ whenever $k >N$, and hence . Take such that if $\{I_{k}'\}$ are the complementary intervals of , then for $1 \leqslant i \leqslant N$, and ${\vert J_{i} \vert }/{\vert I_{i}' \vert } > \rho$ for all $i >N$. Thus , so ${\Sigma }_{4}^{I,J}$ is nowhere dense.

Set . Then ${\Sigma }_{5}$ is a first category $F_{\sigma }$ subset of ${ \Sigma }_{2}$. If , and , then whenever I and J are any disjoint rational intervals. In particular, must be nowhere dense in $E \cap J$ as is a continuous image of the compact set $E \cap I$, hence is itself compact. We show that E is not the attractor of any weakly contractive system. Since we are interested in weak contractions, in what follows, we may fix $\rho =1$ so as to work within . Let $\{T_{1}, T_{2}, \ldots , T_{s}\}$ be a set of weak contractions, $T_{i}:E \rightarrow E$. It suffices to show that $T_{i}(E)$ is nowhere dense in E, for any i. In fact, if it is so, then also is nowhere dense in E. Thus is a proper subset of E, and, hence, E is not an attractor of $\{T_{1}, T_{2}, \ldots , T_{s}\}$. To this end, fix , a weak contraction, with the unique fixed point of T in . Let $y \not =x$, $y \in E$. Since T is a weak contraction,

$$\begin{aligned} \vert T(y) - T(x) \vert = \vert T(y) - x \vert < \vert y - x \vert . \end{aligned}$$

Let $\epsilon = \vert T(y) -y \vert$. Since T is continuous, there exists $\delta >0$ such that $\vert T(z) - T(y) \vert < {\epsilon }/{2}$ whenever $\vert z - y \vert < \delta$. Set . Then $T(B_{{\delta }'}(y)) \subseteq B_{{\epsilon }/{2}}(T(y))$, and $B_{{\delta }'}(y) \cap B_{{\epsilon }/{2}}(T(y)) = \varnothing$. Now, take (a, b) a rational interval of such that $y \in (a,b) \subseteq B_{{\delta }'}(y)$. Since $E \in {\Sigma }_{2}$, it follows that $E \cap {\mathbb Q} = \varnothing$, so that a and b are not in E. Now, . Moreover, and $T((b,1] \cap E)$ are both nowhere dense in $(a,b) \cap E$ since and $T((a,b) \cap E) \cap ((a,b) \cap E)= \varnothing$. We conclude that there exists a neighborhood of y, namely (a, b), such that T(E) is nowhere dense in $(a,b) \cap E$. Since this is true for any $y \not = x$, T(E) is nowhere dense in E. $\square$

6 Attractors with non-empty interior

In Proposition 2.3, one sees that when working within ${\mathscr {I}}_{2}$ and $X = {\mathbb R}^{n}$, an attractor F has non-empty interior if and only if it is the closure of its components with non-empty interior. Simple examples, as our first example below, show that in ${\mathscr {I}}_{2}$ we can have attractors composed of an arbitrarily large number of non-trivial components.

Example 6.1

We show that is an attractor for , where each $S_{i}$ is a similarity and ${\mathscr {S}}$ satisfies the OSC.

Construction. Let $S_{1}(x)= x/{4}$, $S_{2}(x) = x/{4} + {1}/{12}$, $S_{3}(x)= x/{4} + {2}/{3}$, $S_{4}(x)= x/{4} + {3}/{4}$ and set . Then

This gives . Moreover, ${\mathscr {S}} = \{S_{1}, S_{2}, S_{3}, S_{4}\}$ satisfies the OSC with $G = (0, {1}/{3}) \cup ({2}/{3}, 1)$.

This construction can be generalized to ${\mathbb R}^{n}$ for any number of components k. That is, for any n and k in ${\mathbb N}$, there exists a disjoint union, where each $F_{i}$ is an n-cube, where each is a similarity, ${\mathscr {S}}$ satisfies the OSC, and $F= {\mathscr {S}}(F)$.

As we remarked in Proposition 2.5, the only type of attractor with non-empty interior formed by two similarities is an interval. The next example shows that, with three similarities, one can form an attractor that is the closure of the union of countably many disjoint and non-degenerate closed components. In what follows, we use $\dot{{\bigcup }}$ to indicate the union of pairwise disjoint subsets of . Each of the intervals or $(a_{i}, b_{i})$ taken is non-degenerate, so that $b_{i} > a_{i}$.

Example 6.2

We develop an attractor of type for ${\mathscr {S}} = \{S_{1}, S_{2}, S_{3}\}$ composed of similarities that satisfy the OSC.

Construction. Fix $0< r < 1$.

• 1. Set .

• 2. Set $S_{1}(x) = r x$. Then .

• 3. Set $S'_{2}(x) = \sqrt{r} x$. Then .

• 4. Let $h:{\mathbb R} \rightarrow {\mathbb R}$ be the linear homeomorphism taking to : $h(0)= \sqrt{r}$ and $h(1)= 1$ so that .

• 5. Set and $S_{3} = h$. Then as and $i_{d}(E)= E$ imply that , and .

• 6. We conclude that $E = {\mathscr {S}}(E)$, with ${\mathscr {S}} = \{S_{1}, S_{2}, S_{3}\}$, where $S_{1}(x) = rx$, , with $S'_{2}(x) = \sqrt{r} x$, and .

• 7. The collection ${\mathscr {S}}$ satisfies the OSC with .

• 8. If we let $m_{i}$ be the slope of $S_{i}$, then .

Let ${\mathscr {I}}$ be the set of attractors contained in and set . For A and B subsets of , we take . In what remains of this section, we study the elements of ${\mathscr {L}}$. Definition 6.3 and Lemma 6.4 provide a topological characterization of the elements of ${\mathscr {L}}$ analogous to that for countable and compact sets described in Theorems 3.1 and 3.2. Definition 6.3 gives us a tool that allows us to relate elements of ${\mathscr {L}}$ with nowhere dense attractors.

In what follows, let be a pairwise disjoint enumeration of the components of E in ${\mathscr {L}}$ with non-empty interior. Should the collection be finite, then $P \in {\mathbb N}$. Otherwise, $P = \infty$.

Definition 6.3

(collapse morphism) Let $E \in {\mathscr {L}}$, with a pairwise disjoint enumeration of the components of E with non-empty interior. We define ${\phi }$ as the uniform limit of a sequence of functions. Let ${\phi }_{0}(x) = x$ be the identity function. Define ${\phi }_{1}$ such that

In general, define ${\phi }_{n}$ such that

Should $P \in {\mathbb N}$, set $\phi = \phi _{P}$.

By our construction, $\Vert {\phi }_{n} - {\phi }_{n+1} \Vert = \vert b_{n+1} - a_{n+1} \vert$. Thus, ${\phi }_{n}$ converges uniformly to $\phi$ on . The following are consequences of this uniform convergence:

• (1) Since ${\phi }_{n}$ is continuous for all n, $\phi$ is continuous.

• (2) Since for all n, .

• (3) Since ${\phi }_{n}$ is non-decreasing for all n, $\phi$ is non-decreasing.

• (4) Since ${\phi }_{k}$ is constant on for all $k \geqslant n$, $\phi$ is constant on for all n.

• (5) If is complementary to E, then ${\phi }_{n}$ is linear with slope $m=1$ on (a, b), for all n. Thus, $\phi$ is linear with slope $m=1$ on (a, b).

• (6) If , then $\lambda (A)= \lambda ({\phi }_{n}(A))$, for all n. Thus, $\lambda (A)= \lambda (\phi (A))$.

Lemma 6.4

Suppose that the pairwise disjoint sequence is contained in $E \in {\mathscr {L}}$ and . Then as $i \rightarrow \infty$ if and only if as $i \rightarrow \infty$.

Proof

Suppose is contained in E and that . By definition, for any $\epsilon >0$, there is N a natural number so that whenever $i >N$. Since for any , it follows that whenever $i >N$. We conclude that as $i \rightarrow \infty$.

Suppose that as $i \rightarrow \infty$. Then, for any $\epsilon >0$, there exists $N(\epsilon )$ such that whenever $i >N(\epsilon )$. Since is a pairwise disjoint sequence contained in , the series converges. Take M such that . Let , and set . If $i > N(\delta )$ and , then $l >M$. Let $k > \max \{N(\epsilon ), N(\delta )\}$. Then

This follows from (4), (5) and (6) in the observation following Definition 6.3. $\square$

Example 6.5

Let F be a countable space homeomorphic to the space . Suppose $x \in F$ is isolated; say $\epsilon >0$ such that $B_{\epsilon }(x) \cap F = \{x\}$. To each such isolated point x, associate the interval $[x - {\epsilon }/{2},x + {\epsilon }/{2}]$. Let

Then $\phi (E)$ is homeomorphic to the space .

Example 6.6

Let Q be the middle thirds Cantor set with C the set of midpoints of the intervals complementary to Q in . Let $x \in (c,d)$ be in C, where (c, d) is an interval complementary to Q. Take so that . Then , and $\phi (E)$ is uncountable.

Let $E \in {\mathscr {L}}$ with an enumeration of the components of E with non-empty interior and $\phi$ the collapse morphism defined on E. Say . In the following proposition, we make use of ${\phi }^{-1}$ defined on $E'$, which is intended to reverse the collapse affected by $\phi$ on . If , then there exists a unique element of such that . This follows from the observations (3)–(6). If $x \in E'$ and , then there exists a unique element $y \in E$ such that $\phi (y) =x$. This follows from (3), (5) and (6). Let be the power set of . We define such that , the unique element of such that , whenever , and ${\phi }^{-1}(x) = y$, the unique point in E such that $\phi (y) = x$, whenever .

Proposition 6.7

If $E \in {\mathscr {L}}$ is an attractor for some IFS, then $\phi (E)$ is an attractor, too.

Proof

Suppose $E \in {\mathscr {L}}$, with an enumeration of the components of E with non-empty interior, and $E = {\mathscr {S}}(E)$ where ${\mathscr {S}} = \{ S_{1}, S_{2}, \ldots , S_{N}\}$. Let $\phi$ be the collapse morphism as in Definition 6.3, and set $\phi (E) = E'$. Take . We first show that if , then , too. We make frequent reference to the observations following Definition 6.3. Let . Take . Then from (3),

with

and

is the length of . This follows from (4), (5) and (6). As ,

so that

which again follows from (4), (5) and (6). Or, alternatively, we can, again, write

so that

and hence

Therefore

From (5) and (6), one sees that $\phi$, on , preserves measure off the set . Since ${\mathscr {A}}$ is countable and is a singleton for each i, we have . This follows from (6). Thus .

We now show that . Let $x \in E'$, and take $y \in {\phi }^{-1}(x)$. Then $y \in E$, so there exists $1 \leqslant j \leqslant N$ such that . Say $z \in E$ for which . Then $\phi (z) \in E'$. Now, , where , so that , and . Finally, . We conclude that . Thus, .

We now show that $S'(E') \subseteq E'$. Let $x \in E'$. It suffices to show that , for each $1 \leqslant j \leqslant N$. From the construction of $\phi$, one sees that . Since for any j, it follows that , too. Again, by the construction of $\phi$, $\phi (E) = E'$, so . $\square$

The above proposition allows us to show that certain classes of elements of are not attractors for any IFS, using Theorem 3.1.

Corollary 6.8

A compact set $E \in {\mathscr {L}}$ such that $\phi (E)$ is of limit Cantor–Bendixon height is not homeomorphic to any IFS-attractor consisting of weak contractions.

As the next example shows, and as one would expect, Corollary 6.8 provides a necessary but not a sufficient condition for a set of the form to be an attractor for some IFS.

Example 6.9

A compact set $E \in {\mathscr {L}}$ with the property that $\phi (E)$ has height of a successor ordinal can be embedded into the real line so that it is not an attractor of any iterated function system consisting of weak contractions defined on E.

Construction. We begin with the example found in [12, Theorem 2]. Nowak proves that a compact scattered metric space with successor height can be embedded topologically in the real line so that it is not an attractor of any iterated function system consisting of weak contractions. Consider the scattered set , where $\{0\} \cap X_{n} = \varnothing$ for all n, each of the $X_{n}$ is homeomorphic to and whenever $n \not = m$, and $d(\{0\}, X_{n}) \rightarrow 0$ as $n \rightarrow \infty$. Since , it follows that X is homeomorphic to . Let be the collection of isolated points found in X. Let . Take $E_{1}$ so that , is a translation 1 / 2 units to the right of , and . That is, we start our construction with $E_{1}$ using $E_{0}$, replacing $x_{1}$ with an interval of length 1 / 2 so that $a=x_{1}$, and that part of $E_{0}$ to the right of $x_{1}$ is translated to the right a distance of 1 / 2. In general, we take $E_{n+1}$ so that , , and is a translation ${1}/{2^{n+1}}$ units to the right of .

Since $\{E_{n}\}_{n=0}^{\infty }$ is Cauchy in $({\mathscr {K}}({\mathbb R}), {\mathscr {H}})$, exists. Since , and , we see that . Moreover, $\phi (E)= X$, where $\phi$ is the collapse morphism of E. Now, suppose $E = {\mathscr {S}}(E)$ for some collection ${\mathscr {S}} = \{ S_{1}, S_{2}, \ldots , S_{N}\}$ of weak contractions. From Proposition 6.7, it follows that $\phi (E)=X$ is the attractor of the IFS . This contradicts the fact that X is not an attractor for any IFS defined on . We conclude that E is not an attractor for any IFS.

The next example constructs an attractor E of the form such that $\phi (E)$ is homeomorphic to $Q \cup C$, where Q is the middle thirds Cantor set, and C is the set of midpoints of the intervals complementary to Q in .

Example 6.10

There exists an IFS ${\mathscr {S}} = \{S_{1}, S_{2}, S_{3}, S_{4}\}$ satisfying the OSC for which $E = \vert {\mathscr {S}} \vert$, and $\phi (E)$ is homeomorphic to $Q \cup C$, where Q is the middle thirds Cantor set, and C is the set of midpoints of the intervals complementary to Q in .

Construction. Let ${\mathscr {S}} = \{S_{1}, S_{2}, S_{3}, S_{4}\}$, where $S_{1}(x) = {1}/{3} x$, $S_{2}(x) = x/{3} + {2}/{3}$, and $S_{3}$ and $S_{4}$ are piecewise linear functions such that . Let . Then , and is composed of three intervals, where K is centered at 1 / 2 of length 1 / 6, $S_{1}(K)$ is centered at 1 / 6 of length 1 / 18, and $S_{2}(K)$ is centered at 5 / 6, also of length 1 / 18. Moreover, . As one can easily verify, , each $E_{k}$ is composed of disjoint closed intervals centered in intervals complementary to Q, the middle thirds Cantor set, and in .