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Alternative characterization of hyperbolic affine infinite iterated function systems
J. Math. Anal. Appl. 407 (2013) 56–68
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Alternative characterization of hyperbolic affine infinite iterated function systems Radu Miculescu, Alexandru Mihail ∗ Bucharest University, Faculty of Mathematics and Computer Science, Str. Academiei 14, 010014 Bucharest, Romania
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Article history: Received 11 February 2013 Available online 9 May 2013 Submitted by R.M. Aron Keywords: Affine infinite iterated function system Comparison function Attractor of an affine infinite iterated function system ϕ -hyperbolic affine infinite iterated function system Uniformly point-fibre affine infinite iterated function system Convex body Strictly topologically contractive affine infinite iterated function system
abstract In this paper we present a characterization of hyperbolic affine infinite iterated function systems defined on an arbitrary normed space. Our result is a generalization of Theorem 1.1 from the paper ‘‘A characterization of hyperbolic affine iterated function systems’’, Topology Proceedings, 36 (2010), 189–211, by R. Atkins, M. Barnsley, A. Vince and D. Wilson. Some examples are presented. © 2013 Elsevier Inc. All rights reserved.
1. Introduction Motivated by the following question of A. Kameyama (see [10]): Given a topological self-similar set, does there exist an associated system of contraction mappings?, in a recent paper, R. Atkins, M. Barnsley, A. Vince and D.C. Wilson [1] presented a theorem that characterizes hyperbolic affine iterated function systems defined on Rm . More precisely, Rm being endowed with the Euclidean metric, if N > 0 is an integer and fn : Rm → Rm , n ∈ {1, 2, . . . , N }, are continuous mappings, then F = (Rm ; f1 , f2 , . . . , fN ) is called an iterated function system (IFS). If each fn is an affine map on Rm , then F is called an affine IFS. If each fn is a contraction (i.e. there exists αn ∈ [0, 1) such that ∥fn (x) − fn (y)∥2 ≤ αn ∥x − y∥2 for all x, y ∈ Rm ), then F is called a contractive IFS. If there exists a metric on Rm Lipschitz equivalent to the Euclidean metric so that each fn is a contraction, then F is called a hyperbolic IFS. For a positive integer N, the symbol Ω = {1, 2, . . . , N }∞ denotes the set of all infinite sequences of symbols {σk }∞ k=1 belonging to the alphabet {1, 2, . . . , N }. The set Ω is endowed with the product topology. An element σ ∈ Ω will also be denoted by the concatenation σ = σ1 σ2 σ3 . . . , where σk denotes the kth component of σ . The IFS F is called point-fibred if for each σ = σ1 σ2 σ3 . . . ∈ Ω , the limit on the right hand side of π (σ ) := limk→∞ fσ1 ◦ fσ2 ◦ · · · ◦ fσk (x) exists, is independent
∗
Corresponding author. E-mail addresses: [email protected] (R. Miculescu), [email protected] (A. Mihail).
0022-247X/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jmaa.2013.05.007
R. Miculescu, A. Mihail / J. Math. Anal. Appl. 407 (2013) 56–68
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of x ∈ Rm for fixed σ , the map π : Ω → Rm is continuous and for each n ∈ {1, 2, . . . , N }, the following diagram commutes sn
Ω ↓π
→
Rm
→
fn
Ω ↓ π , where sn
: Ω → Ω is given by sn (σ ) = nσ . Ω is called a coding map.
Rm
The symbol H denotes the non-empty compact subsets of Rm . A set A ∈ H is called an attractor of the IFS F if A = F (A) and A = limk→∞ F ◦k (B) (with respect to the Hausdorff–Pompeiu metric) for all B ∈ H, where F : H → H is given by F (B ) = ∪Nn=1 fn (B) and F ◦k denotes the k-fold composition of F . A convex body is a compact convex subsets of Rm with non-empty interior. An IFS F is called topologically contractive if ◦
there exists a convex body K such that F (K ) ⊆ K . Part of the above mentioned result can be stated in the following way (see Theorem 1.1. from [1]): Theorem. If F = (Rm ; f1 , f2 , . . . , fN ) is an affine iterated function system, then the following statements are equivalent: (1) (2) (3) (4)
F F F F
is hyperbolic; is point-fibred; has an attractor; is topologically contractive.
IFSs were introduced in their present form by John Hutchinson (see [9]) and popularized by Michael Barnsley (see [2]). They are one of the most common and most general ways to generate fractals. Because of the variety of their applications (for example they are used for the construction of deterministic fractals [2] and have found numerous applications, in particular to image compression and image processing [3]), there is a current effort to extend the classical Hutchinson’s framework to infinite iterated function systems (see, for example, the works of H. Fernau [6], R.D. Mauldin [15], R.D. Mauldin, M. Urbański [16], F. Mendivil [18], K. Leśniak [11,12], G. Gwóźdź-Łukawska, J. Jachymski [7], R. Miculescu, A. Mihail [20–23], M. Hille [8], R. Miculescu, L. Ioana [19] N.A. Secelean [26] and F. Strobin, J. Swaczyna [27]). The aim of this paper is to generalize the result mentioned above. More precisely, we define the notion of affine infinite iterated function system in a Banach space (which can be infinite dimensional) (see Definition 3.2), hyperbolic affine infinite iterated function system (see Definition 3.4), comparison function (see Definition 3.5), ϕ -hyperbolic affine infinite iterated function system (see Definition 3.8), uniformly point-fibred affine infinite iterated function system (see Definition 3.11), attractor of an affine infinite iterated function system (see Definition 3.13) and strictly topologically contractive affine infinite iterated function system (see Definition 3.16). Our main result states that if S = ((X , ∥·∥), (fi )i∈I ) is an affine infinite iterated function system, then the following statements are equivalent: 1. 2. 3. 4. 5.
S is hyperbolic;
there exists a comparison function ϕ0 such that S is ϕ0 -hyperbolic; S has an attractor; S is strictly topologically contractive; S is uniformly point-fibred.
The paper is organized as follows. In Section 2 we present some preliminaries facts concerning convex subsets of a normed space, affine functions and bounded linear operators from a normed space into itself and Hausdorff–Pompeiu pseudo-metric. Section 3 contains the notation, terminology, and definitions that are used throughout the paper. Section 4 is devoted to our main result and Section 5 presents some examples. 2. Preliminaries Given a subset A of a normed space (X , ∥·∥), we denote by co(A) the convex hull of A and by co(A) the closed convex hull of A. n n It is well known that co(A) = { j=1 λj xj | n ∈ N, x1 , . . . , xn ∈ A, λ1 , . . . , λn ≥ 0 and j=1 λj = 1} and co(A) = co(A) (see [4, p. 105] and [17, p. 3]). Lemma 2.1. Let (X , ∥·∥) be a normed space. Then f (co(A)) = co(f (A)), for each affine function f : X → X and each subset A of X . Proof. Since f is affine, there exists a linear function F : X → X and x0 ∈ X such that f = F + x0 . First let us note that co(x + B) = x + co(B)
(∗)
for each x ∈ X and each subset B of X . (*) Then f (co(A)) = F (co(A)) + x0 and co(f (A)) = co(F (A) + x0 ) = co(F (A)) + x0 and therefore it suffices to prove that F (co(A)) = co(F (A)). Since F (co(A)) is a convex set, as F (A) ⊆ F (co(A)), we infer that co(F (A)) ⊆ F (co(A)).
(∗∗)
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It is clear that F (co(A)) ⊆ co(F (A)). From (∗∗) and (∗ ∗ ∗), we get that F (co(A)) = co(F (A)).
(∗ ∗ ∗)
Lemma 2.2. Let A be a subset of a normed space (X , ∥·∥). For ρ > 0, let us consider the sets Aρ = B∥·∥ [A, ρ] = {x ∈ X |
d(x, A) ≤ ρ} = {x ∈ X | infa∈A ∥x − a∥ ≤ ρ} and Kρ = co(Aρ ) = co(Aρ ). Then B∥·∥ [Kρ , r ] ⊆ Kρ+r +ε , for each r > 0 and ε > 0. Proof. If we consider x ∈ B∥·∥ [Kρ , r ] then there exists y ∈ Kρ such that ∥x − y∥ < r + 3ε , i.e., with the notation x1 = x − y,
we have x = y + x1 , where ∥x1 ∥ < r + 3ε . Since y ∈ Kρ = co(Aρ ), there exists y1 ∈ co(Aρ ) such that ∥y − y1 ∥ < 3ε , i.e., with the notation y2 = y − y1 , we have y = y1 + y2 , where ∥y2 ∥ < 3ε . As y1 ∈ co(Aρ ) there exist n ∈ N, λ1 , λ2 , . . . , λn ∈ [0, ∞) n n and u1 , u2 , . . . , un ∈ B∥·∥ [A, ρ] such that i=1 λi = 1 and y1 = i=1 λi ui . Because ui ∈ B∥·∥ [A, ρ] there exists vi ∈ A such that ∥ui − vi ∥ < ρ + 3ε , i.e., with the notation wi = ui − vi , we have ui = vi + wi , where ∥wi ∥ < ρ + 3ε , for n n each i ∈ {1, 2, . . . , n}. Hence x = y + x1 = y1 + y2 + x1 = i=1 λi (vi + wi + y2 + x1 ) i=1 λi ui + y2 + x1 = ε ε ε and since ∥wi + y2 + x1 ∥ ≤ ∥wi ∥ + ∥y2 ∥ + ∥x1 ∥ < ρ + 3 + 3 + r + 3 = ρ + r + ε and vi ∈ A, we infer that vi + wi + y2 + x1 ∈ B∥·∥ [A, ρ + r + ε] for all i ∈ {1, 2, . . . , n}, so x ∈ co(B∥·∥ [A, ρ + r + ε]) ⊆ co(Aρ+r +ε ) = Kρ+r +ε and the proof is done. Definition 2.1. Given a normed space (X , ∥·∥) and U and V subsets of X , we define δ∥·∥ (U , V ) = infu∈U ,v∈V ∥u − v∥. ◦
Lemma 2.3. If δ∥·∥ (A, B) > 0, where A and B are subsets of the normed space (X , ∥·∥), then A ⊆ X \ B. ◦
Proof. If, by reductio ad absurdum, the conclusion is false, then there exists x0 ∈ A such that x0 ̸∈ X \ B = X \ B. Then for each ε > 0 there exists xε ∈ B such that ∥x0 − xε ∥ < ε . Hence δ∥·∥ (A, B) ≤ ∥x0 − xε ∥ < ε , for each ε > 0 and consequently we get the contradiction δ∥·∥ (A, B) = 0. Let us recall (see, for example [24, p. 30]) that given a normed space (X , ∥·∥) and A a bounded, balanced (i.e. α A ⊆ A for each α such that |α| ≤ 1), convex neighbourhood of 0, the application ∥·∥A : X → R given by ∥x∥A = inf{λ > 0 | x ∈ λA} for each x ∈ X , is a norm, called Minkowski norm, the norms ∥·∥ and ∥·∥A are equivalent and {x ∈ X | ∥x∥A < 1} ⊆ A ⊆ {x ∈ X | ∥x∥A ≤ 1}. Lemma 2.4. If δ∥·∥A (A, X \ B) ≥ α , where α > 0, A and B are subsets of the normed space (X , ∥·∥) such that A is a bounded, α . balanced, convex neighbourhood of 0, then A ⊆ (1 − θ )B, where θ = 2(α+ 1) 1 Proof. If the conclusion is not valid, then there exists x0 ∈ A such that x0 ̸∈ (1 −θ )B, i.e. 1−θ x0 ̸∈ B. Then α ≤ δ∥·∥A (A, X \ B) ≤ x0 − 1 x0 = ∥x∥A θ < θ = α and therefore we get the contradiction 0 < α ≤ −1. 1−θ 1−θ 1−θ α+2 A
Lemma 2.5. Let (X , ∥·∥) be a normed space, ε > 0, and A and A1 subsets of X such that ε < δ∥·∥ (A1 , X \ A). Then B∥·∥ (A1 , ε) = {x ∈ X | d(x, A1 ) < ε} = {x ∈ X | infa∈A1 ∥x − a∥ < ε} ⊆ A. Proof. If, by reductio ad absurdum, the conclusion is false, then there exists x0 ∈ B∥·∥ (A1 , ε) such that x0 ̸∈ A i.e. x0 ∈ X \ A. Since x0 ∈ B∥·∥ (A1 , ε) there exists x1 ∈ A1 such that ∥x0 − x1 ∥ < ε . Hence we obtain the following contradiction ε < δ∥·∥ (A1 , X \ A) ≤ ∥x1 − x0 ∥ < ε. Lemma 2.6. Let (X , ∥·∥) be a normed space and A, B, A1 and B1 subsets of X such that δ∥·∥ (A1 , X \ A) > 0 and δ∥·∥ (B1 , X \ B) > 0. Then δ∥·∥ (A1 − B1 , X \ (A − B)) > 0. Proof. Let us consider ε ∈ (0, min{δ∥·∥ (A1 , X \ A), δ∥·∥ (B1 , X \ B)}). Then, according with Lemma 2.5, we get B∥·∥ (A1 , ε) ⊆ A
(∗)
B∥·∥ (B1 , ε) ⊆ B.
(∗∗)
and
It is clear that B∥·∥ (A1 − B1 , ε) ⊆ B∥·∥ (A1 , ε) − B∥·∥ (B1 , ε).
(∗ ∗ ∗)
Hence, using (∗), (∗∗) and (∗ ∗ ∗), we get B∥·∥ (A1 − B1 , ε) ⊆ A − B, i.e. B∥·∥ (A1 − B1 , ε)∩(X \(A − B)) = ∅ and consequently
δ∥·∥ (A1 − B1 , X \ (A − B)) ≥ ε > 0. Lemma 2.7. If (X , ∥·∥) is a normed space, A is a bounded, balanced, convex neighbourhood of 0 and f : X → X is a bounded linear operator having the property that f (A) ⊆ µA, where µ > 0, then ∥f ∥A ≤ µ.
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Proof. If x ∈ X has the property that ∥x∥A ≤ 1, then for each ε > 0 there exists λε ∈ (0, 1 + ε) such that x ∈ λε A, λε λε 1 1 1 i.e. λ1 x ∈ A. Then, since 0 < 1+λ < 1 and A is balanced, we obtain that 1+λ x ∈ A, i.e. 1+ε x ∈ A and hence 1+ε f (x) = λ ε
ε
ε
ε
1 1 1 f ( 1+ε x) ∈ f (A) ⊆ µA, so (1+ε)µ f (x) ∈ A, which implies that (1+ε)µ f (x) ≤ 1, i.e. ∥f (x)∥A ≤ (1 + ε)µ for each ε > 0. A Therefore ∥f (x)∥A ≤ µ for each x ∈ X having the property that ∥x∥A ≤ 1. Consequently ∥f ∥A = sup∥x∥A ≤1 ∥f (x)∥A ≤ µ.
Given a metric space (X , d), we consider MX = {N ⊆ X | N is non-empty and bounded}, and BX = {N ⊆ X | def
def
N is non-empty, closed and bounded}. For x ∈ X , ε > 0 and A, B ∈ MX , we consider d(x, A) = infa∈A d(x, a), B(A, ε) ={x ∈ def
def
X | d(x, A) < ε} = ∪a∈A B(a, ε), B[A, ε] ={x ∈ X | d(x, A) ≤ ε} and d(A, B) = supx∈A d(x, B). The map h∗ : MX × MX → [0, ∞), given by h∗ (A, B) = max{d(A, B), d(B, A)}, for all A, B ∈ MX , satisfies the following properties: (i) h∗ (A, A) = 0, for all A ∈ MX ; (ii) h∗ (A, B) = h∗ (B, A), for all A, B ∈ MX ; (iii) h∗ (A, B) ≤ h∗ (A, C ) + h∗ (C , B), for all A, B, C ∈ MX . Therefore h∗ is a pseudo-metric in MX which is called the Hausdorff–Pompeiu pseudo-metric. Proposition 2.1. For all A1 , A2 ∈ MX , the following properties are valid: (i) (ii) (iii) (iv)
for all ε > 0, we have: h∗ (A1 , A2 ) ≤ ε if and only if A1 ⊆ B[A2 , ε] and A2 ⊆ B[A1 , ε]; h∗ (A1 , A2 ) = inf{r > 0 : A1 ⊆ B(A2 , r ) and A2 ⊆ B(A1 , r )}; h∗ (A1 , A2 ) = h∗ (A1 , A2 ); h∗ (A1 , A2 ) = 0 if and only if A1 = A2 ;
(v) if {Ai1 }i∈I and {Ai2 }i∈I are two families of elements of MX such that ∪i∈I Ai1 h (∪ ∗
i i∈I A1
,∪
i i∈I A2
) = h (∪ ∗
i i∈I A1
,∪
i i∈I A2
) ≤ supi∈I h ( , ∗
Ai1
Ai2
∈ MX and ∪i∈I Ai2 ∈ MX , then
).
Definition 2.2. The restriction of h∗ to BX × BX is a metric. We call it the Hausdorff–Pompeiu metric and denote it by h. We use the terminology Hausdorff–Pompeiu distance for the metric h which is also called the Hausdorff metric or the Pompeiu–Hausdorff metric (as in the book of R. Tyrrell Rockafellar and Roger Wets entitled ‘‘Variational Analysis’’, 3rd printing, Springer-Verlag, 2009, p. 144). For a nice history of this notion one can consult the work entitled ‘‘One hundred years since the introduction of the set distance by Dimitrie Pompeiu’’, appeared in System modeling and optimization, IFIP International Federation for Information Processing, Volume 199, 2006, pp. 35–39, Springer-Verlag, by T. Bârsan and D. Tiba. Proposition 2.2. If the metric space (X , d) is complete, then the metric space (BX , h) is complete. For the proof of the above theorem one can consult Theorem 26.1, p. 44, from [13]. Let (X , ∥·∥) be a normed space and |||·||| a norm on X equivalent to ∥·∥. Let us denote by d∥·∥ the metric induced by the norm ∥·∥ and by d||·|| the metric induced by the norm |||·|||. Then the set of non-empty, bounded, closed subsets of X with respect to d∥·∥ (denoted by B(X ,∥·∥) ) coincides with the set of non-empty, bounded, closed subsets of X with respect to d||·|| (denoted by B(X ,|| ·|| ) ) and we simply denote it by B (X ). Let us denote by h∥·∥ the Hausdorff–Pompeiu metric on B (X ) = B(X ,∥·∥) associated with d∥·∥ and by h||·|| the Hausdorff–Pompeiu metric on B (X ) = B(X ,|| ·|| ) associated with d||·|| . Lemma 2.8. The metrics h∥·∥ and h||·|| , defined above, are equivalent. Proof. Since the norms ∥·∥ and |||·||| are equivalent, there exist α ∈ (0, ∞) and β ∈ (0, ∞) such that α ∥x∥ ≤ |||x||| ≤ β ∥x∥ for any x ∈ X . Then we have infy∈B |||x − y||| ≤ |||x − y||| ≤ β ∥x − y∥, for any A, B ∈ B (X ), x ∈ A and y ∈ B, hence infy∈B |||x − y||| ≤ β infy∈B ∥x − y∥ ≤ β supx∈A infy∈B ∥x − y∥ and consequently supx∈A infy∈B |||x − y||| ≤ β supx∈A infy∈B ∥x − y∥. In a similar way one can prove that supx∈B infy∈A |||x − y||| ≤ β supx∈B infy∈A ∥x − y∥. Therefore, we have max{supx∈A infy∈B |||x − y|||, supx∈B infy∈A |||x − y|||} ≤ β max{supx∈A infy∈B ∥x − y∥ , supx∈B infy∈A ∥x − y∥}, i.e. h||·|| (A, B) ≤ β h∥·∥ (A, B). Similarly one can obtain α h∥·∥ (A, B) ≤ h||·|| (A, B). Therefore α h∥·∥ (A, B) ≤ h||·|| (A, B) ≤ β h∥·∥ (A, B) for all A, B ∈ B (X ) which means that the metrics h∥·∥ and h||·|| are equivalent. Remark 2.1. Let us note that if (X , ∥·∥) is a Banach space and the norm |||·||| on X is equivalent to ∥·∥, then (X , |||·|||) is also a Banach space and therefore (B(X ), h||·|| ) is a complete metric space. 3. Definitions and notations Definition 3.1 (Bounded Family of Functions). Given a Banach space (X , ∥·∥), a family of function (fi )i∈I , where fi : X → X , is called bounded if for each bounded subset A of X , the set ∪i∈I fi (A) is bounded. Definition 3.2 (Affine Infinite Iterated Function System—AIIFS). Given a Banach space (X , ∥·∥), an affine infinite iterated function system is a pair S = ((X , ∥·∥), (fi )i∈I ), where (fi )i∈I is a bounded family of functions (fi )i∈I having the property that, for each i ∈ I, there exists a bounded linear operator Ai : X → X and an element bi of X such that fi = Ai + bi .
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Remark 3.1. The family (fi )i∈I is bounded if and only if the sets {bi | i ∈ I } and {∥Ai ∥ | i ∈ I } are bounded. Remark 3.2. Just in order to emphasize that, in contrast with the classical theory, the set I could be infinite, we call S an affine infinite iterated function system, even though I could also be finite (as in our second example from Section 5, where I has only one element). We also point out that the Banach space (X , ∥·∥) could be infinite dimensional, by contrast with the framework from [1], where only finite dimensional Banach spaces are considered. In our second example from Section 5 we consider the infinite dimensional Banach space c0 endowed with the standard norm. Definition 3.3 (Contractive AIIFS). An affine infinite iterated function system S = ((X , ∥·∥), (fi )i∈I ) is called contractive if there exists C ∈ [0, 1) such that ∥Ai ∥ = lip(fi ) ≤ C for each i ∈ I. Definition 3.4 (Hyperbolic AIIFS). An affine infinite iterated function system S = ((X , ∥·∥), (fi )i∈I ) is called hyperbolic if there exists a norm |||·||| on X equivalent to ∥·∥ having the property that the affine infinite iterated function system ((X , |||·|||), (fi )i∈I ) is contractive. Remark 3.3. Each affine infinite iterated function system which is contractive is hyperbolic. Definition 3.5 (Comparison Function). A function ϕ : [0, ∞) → [0, ∞) is called a comparison function if it has the following properties: (i) ϕ is increasing; (ii) ϕ(t ) < t for any t > 0; (iii) ϕ is right-continuous. Remark 3.4 (See Remark 1 from [14]). Any function ϕ : [0, ∞) → [0, ∞) satisfying (ii) and (iii) from the above definition has the following property: limn→∞ ϕ n (t ) = 0 for any t > 0. Remark 3.5 (See [25, p. 25]). Any function ϕ : [0, ∞) → [0, ∞) satisfying (i) and (ii) from the above definition has the property that ϕ(0) = 0. Remark 3.6 (See Remark 2 from [5]). For any comparison function ϕ : [0, ∞) → [0, ∞) and any non-empty bounded subset A of [0, ∞) the following relations are valid: 1. sup ϕ(A) ≤ ϕ(sup A); 2. inf ϕ(A) = ϕ(inf A). Definition 3.6 (ϕ -Contraction). Let (X , d) be a metric space and a function ϕ : [0, ∞) → [0, ∞). A function f : X → X is called an ϕ -contraction if d(f (x), f (y)) ≤ ϕ(d(x, y)), for all x, y ∈ X . Matkowski’s theorem (See [25, p. 31]). Let (X , d) be a complete metric and ϕ : [0, ∞) → [0, ∞) a function such that: (i) (ii) (iii) (iv)
ϕ is increasing; ϕ(t ) < t for all t > 0; ϕ(0) = 0; limn→∞ ϕ [n] (t ) = 0 for all t > 0.
Then an ϕ -contraction f : X → X has a fixed point x∗ ∈ X and, for each x ∈ X , the sequence (f n (x))n∈N converges to x∗ . From the above Theorem, taking into account the Remarks 3.4 and 3.5, we obtain the following: Corollary 3.1. Let (X , d) be a complete metric space and f : X → X be an ϕ -contraction, where ϕ is a comparison function. Then f has a fixed point x∗ ∈ X and, for each x ∈ X , the sequence (f n (x))n∈N converges to x∗ . Definition 3.7 (ϕ -Contractive AIIFS). Given a comparison function ϕ : [0, ∞) → [0, ∞), an affine infinite iterated function system S = ((X , ∥·∥), (fi )i∈I ) is called ϕ -contractive if fi is an ϕ -contraction for each i ∈ I. Remark 3.7. Each affine infinite iterated function system S = ((X , ∥·∥), (fi )i∈I ) which is contractive (so there exists C ∈ [0, 1) such that ∥Ai ∥ = lip(fi ) ≤ C for each i ∈ I) is ϕ0 -contractive, where ϕ0 : [0, ∞) → [0, ∞) is given by ϕ0 (t ) = Ct for each t ∈ [0, ∞). Definition 3.8 (ϕ -Hyperbolic AIIFS). Given a comparison function ϕ : [0, ∞) → [0, ∞), an affine infinite iterated function system S = ((X , ∥·∥), (fi )i∈I ) is called ϕ -hyperbolic if there exists a metric |||·||| on X equivalent to ∥·∥ such that the affine infinite iterated function system ((X , |||·|||), (fi )i∈I ) is ϕ -contractive.
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Remark 3.8. Each affine infinite iterated function system which is ϕ -contractive is ϕ -hyperbolic. In the following N denotes the natural numbers, N∗ = N − {0} and N∗n = {1, 2, . . . , n}, where n ∈ N∗ . Given two ∗ sets A and B, by BA we mean the set of functions from A to B. By Λ = Λ(B) we mean the set BN and by Λn = Λn (B) we ∗ ∗ mean the set BNn . The elements of Λ = Λ(B) = BN are written as words ω = ω1 ω2 . . . ωm ωm+1 . . . and the elements of ∗ Λn = Λn (B) = BNn are written as words ω = ω1 ω2 . . . ωn . Hence Λ(B) is the set of infinite words with letters from the alphabet B and Λn (B) is the set of words of length n with letters from the alphabet B. If ω = ω1 ω2 . . . ωm ωm+1 . . . ∈ Λ(B) or if ω = ω1 ω2 . . . ωn ∈ Λn (B), where m, n ∈ N∗ , n ≥ m, then the word ω1 ω2 . . . ωm is denoted by [ω]m . For a non-empty ∗
set I, on Λ = Λ(I ) = I N , we consider the metric dΛ (α, β) =
∞
k=1
the function Fi : Λ → Λ given by Fi (ω) = iω for all ω ∈ Λ(I ).
β 1−δα k k
3k
y
, where δx =
1, if x = y . 0, if x ̸= y
For i ∈ I, let us consider
Definition 3.9 (The Shift Space Associated to an Affine Infinite Iterated Function System). Given an affine infinite iterated function system S = ((X , ∥·∥), (fi )i∈I ), the metric space (Λ(I ), dΛ ) is called the shift space associated to S . Definition 3.10 (Point-Fibred AIIFS). An affine infinite iterated function system S = ((X , ∥·∥), (fi )i∈I ) is called point-fibred ∗ if for each α = α1 α2 . . . αn . . . ∈ I N = Λ there exists xα ∈ X such that limn→∞ fα1 α2 ...αn (x) = xα , for each x ∈ X , where fα1 α2 ...αn stands for fα1 ◦ fα2 ◦ · · · ◦ fαm . Remark 3.9. We obtain, in the frame of the above definition, that fi (xα ) = xiα for any α ∈ Λ and i ∈ I. Λ In other words, the following diagram commutes ↓ π X
Fi
→ fi
→
Λ ↓ π , where π
: Λ → X is given by π (α) = xα for each α ∈ Λ
X
and it is called the coding map for S . Definition 3.11 (Uniformly Point-Fibred AIIFS). An affine infinite iterated function system S = ((X , ∥·∥), (fi )i∈I ) is called uniformly point-fibred if it is point-fibred and for each bounded subset B of X and exists nB,ε ∈ N such for each ε > 0 there ∗ that for each n ∈ N, n ≥ nB,ε , x ∈ B and α = α1 α2 . . . αn . . . ∈ I N = Λ we have fα1 α2 ...αn (x) − xα < ε . Remark 3.10. If the affine infinite iterated function system S = ((X , ∥·∥), (fi )i∈I ) is uniformly point-fibred, then π is continuous. Proof. We shall prove that π is continuous at an arbitrary element β = β1 β2 . . . βn . . . ∈ Λ. It suffices to show that for each sequence (α n )n∈N of elements from Λ such that limn→∞ α n = β , we have limn→∞ π (α n ) = π (β), where α n = α1n α2n . . . αkn . . . , for each n ∈ N. To this aim let us consider a fixed element x0 of X and let us note that, since S is uniformly point-fibred, for each ε > 0 there exists mε ∈ N such that for each n ∈ N, n ≥ mε , α1 , α2 , . . . , αn ∈ I and γ ∈ Λ, we have
fα
1 α2 ...αn
(x0 ) − π (α1 α2 . . . αn γ ) < ε.
(∗)
As limn→∞ α n = β there exists nε ∈ N such that for each n ∈ N, n ≥ nε we have dΛ (α n , β) <
1
3mε
, so
(∗∗)
[α n ]mε = [β]mε
because if this is not the case then there exists k ∈ {1, 2, . . . , mε } such that αkn ̸= βk and consequently we obtain the following contradiction: 31k ≤ dΛ (α n , β) < 3m1 ε . Hence, taking into account (∗∗), we conclude that for each ε > 0 there exists nε ∈
(*)
N such that for each n ∈ N, n ≥ nε we have ∥π (α n ) − π (β)∥ ≤ π (α n ) − f[α n ]mε (x0 ) + f[α n ]mε (x0 ) − π (β) ≤ ε +ε = 2ε , i.e. limn→∞ π (α n ) = π (β).
Definition 3.12 (The Function FS Associated with an AIIFS). For an affine infinite iterated function system S ((X , ∥·∥), (fi )i∈I ) one defines FS : B (X ) → B (X ) given by FS (B) = ∪i∈I fi (B) for each B ∈ B (X ).
=
[n]
FS designates the n-fold composition of FS . Definition 3.13 (Attractor of an AIIFS). A set A ∈ B (X ) is called an attractor of the affine infinite iterated function system S [n] if FS (A) = A and limn→∞ FS (B) = A, for all B ∈ B (X ), where the limit is considered with respect to h∥·∥ . Definition 3.14 (Convex Body). A set A ∈ B (X ), where (X , ∥·∥) is a normed space, is called a convex body of X if it is convex and has non-empty interior.
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R. Miculescu, A. Mihail / J. Math. Anal. Appl. 407 (2013) 56–68
Definition 3.15 (Strictly Topologically Contractive AIIFS). An affine infinite iterated function system S is called strictly topologically contractive if there exists a convex body K of X such that δ∥·∥ (FS (K ), X \ K ) > 0. Definition 3.16 (Topologically Contractive AIIFS). An affine infinite iterated function system S is called topologically ◦
contractive if there exists a convex body K of X such that FS (K ) ⊆ K . Remark 3.11. It is obvious that an affine infinite iterated function system which is uniformly point-fibred is point-fibred. Taking into account Lemma 2.3, with A = FS (K ) and B = X \ K , we infer that if an affine infinite iterated function system S ◦
is strictly topologically contractive, then it is topologically contractive. In particular, we have FS (K ) ⊆ K ⊆ K . The second example presented in Section 5 provides a point-fibred affine infinite iterated function system which is not uniformly point-fibred and a topologically contractive affine infinite iterated function system which is not strictly topologically contractive. However, in case that #(I ) < ∞ and dim(X ) < ∞, by comparing the result of R. Atkins, M. Barnsley, A. Vince and D. Wilson with our main result (see Theorem 4.1), one can see that an affine infinite iterated function system is uniformly point-fibred if and only if it is point-fibred and it is strictly topologically contractive if and only if it topologically contractive. 4. The main result Theorem 4.1. If S = ((X , ∥·∥), (fi )i∈I ) is an affine infinite iterated function system, then the following statements are equivalent: 1. 2. 3. 4. 5.
S is hyperbolic;
there exists a comparison function ϕ0 such that S is ϕ0 -hyperbolic; S has an attractor; S is strictly topologically contractive; S is uniformly point-fibred.
Proof. (1) ⇒ (2) Since S = ((X , ∥·∥), (fi )i∈I ) is hyperbolic, there exists a norm |||·||| on X equivalent to ∥·∥ having the property that the affine infinite iterated function system ((X , |||·|||), (fi )i∈I ) is contractive. Hence there exist α, β > 0 such that α ∥x∥ ≤ |||x||| ≤ β ∥x∥ for all x ∈ X and C ∈ [0, 1) such that |||Ai ||| ≤ C for each i ∈ I. Then the norm |||·|||1 given by 1 |||·||| is equivalent to |||·||| and therefore equivalent to ∥·∥. β
The function ϕ0 : [0, ∞) → [0, ∞) given by ϕ0 (t ) = Ct for all t ∈ [0, ∞) is a comparison function and since |||fi (x) − fi (y)|||1 = |||Ai (x) − Ai (y)|||1 = β1 |||Ai (x) − Ai (y)||| = β1 |||Ai (x − y)||| ≤ C β1 |||x − y||| = C |||x − y|||1 = ϕ0 (|||x − y|||1 ) for all x, y ∈ X and i ∈ I , ((X , |||·|||1 ), (fi )i∈I ) is ϕ0 -contractive, so S = ((X , ∥·∥), (fi )i∈I ) is ϕ0 -hyperbolic. (2) ⇒ (3) As S = ((X , ∥·∥), (fi )i∈I ) is ϕ0 -hyperbolic there exists a metric |||·||| on X equivalent to ∥·∥ such that the affine infinite iterated function system ((X , |||·|||), (fi )i∈I ) is ϕ0 -contractive, i.e.
|||fi (x) − fi (y)||| ≤ ϕ0 (|||x − y|||)
(∗)
for all x, y ∈ X and i ∈ I. We claim that h||·|| (FS (A), FS (B)) ≤ ϕ0 (h∥·∥ (A, B)), for all A, B ∈ B (X ). Indeed, using Proposition 2.1(v), for any A, B ∈ B (X ), we have h||·|| (FS (A), FS (B)) = h||·|| (∪i∈I fi (A), ∪i∈I fi (B)) ≤ sup h||·|| (fi (A), fi (B)) i∈I
= sup max{d||·|| (fi (A), fi (B)), d||·|| (fi (B), fi (A))} i∈I
= sup max sup inf |||fi (x) − fi (y)|||, sup inf |||fi (x) − fi (y)||| x∈A y∈B
i∈I
x∈B y∈A
and, taking into account (∗), we get
h||·|| (FS (A), FS (B)) ≤ max sup inf ϕ0 (|||x − y|||), sup inf ϕ0 (|||x − y|||) . x∈A y∈B
x∈B y∈A
Then, using Remark 3.6, we obtain
h||·|| (FS (A), FS (B)) ≤ max ϕ0
sup inf |||x − y||| , ϕ0 x∈A y∈B
sup inf ∥x − y∥ x∈B y∈A
R. Miculescu, A. Mihail / J. Math. Anal. Appl. 407 (2013) 56–68
63
and, since ϕ0 is increasing, we conclude that h||·|| (FS (A), FS (B)) ≤ ϕ0
max sup inf |||x − y|||, sup inf ∥x − y∥ x∈B y∈A
x∈A y∈B
,
i.e. h||·|| (FS (A), FS (B)) ≤ ϕ0 (h∥·∥ (A, B)). Consequently the function FS : B||·|| (X ) → B||·|| (X ) is a ϕ0 -contraction and since (B||·|| (X ), h||·|| ) is a complete metric space, according to Corollary 3.1, we infer that there exists A ∈ B||·|| (X ) = B (X ) such that FS (A) = A and limn→∞ FS (B) = A, for all B ∈ B (X ), where the limit is considered with respect to h||·|| . As h∥·∥ and h||·|| are equivalent (see Lemma 2.8) we also have [n]
limn→∞ FS (B) = A, for all B ∈ B (X ), where the limit is considered with respect to h∥·∥ . [n]
(3) ⇒ (4) In accordance with the hypothesis, there exists a set A ∈ B (X ) such that FS (A) = A and limn→∞ FS[n] (B) = A, for all B ∈ B (X ), where the limit is considered with respect to h∥·∥ . In the sequel, we use the notations from Lemma 2.2. [n] Since limn→∞ FS (A1 ) = A, where the limit is considered with respect to h∥·∥ , there exists n0 ∈ N such that [n0 ]
h∥·∥ (FS
[n ]
(A1 ), A) < 41 . Then FS 0 (A1 ) ⊆ B∥·∥ [A, 41 ] = A 1 and therefore 4
fi1 ,i2 ,...,in0 (A1 ) ⊆ A 1 ,
(i)
4
for each i1 , i2 , . . . , in0 ∈ I. Then, using Lemma 2.1 and the continuity of the functions fi , we have [n0 ]
FS
(K1 ) ⊆ ∪i1 ,i2 ,...,in0 ∈I fi1 ,i2 ,...,in0 (K1 ) = ∪i1 ,i2 ,...,in0 ∈I fi1 ,i2 ,...,in0 (co(A1 )) ⊆ ∪i1 ,i2 ,...,in0 ∈I fi1 ,i2 ,...,in0 (co(A1 )) = ∪i1 ,i2 ,...,in0 ∈I co(fi1 ,i2 ,...,in0 (A1 )) (i)
⊆ ∪i1 ,i2 ,...,in0 ∈I co(A 1 ) = ∪i1 ,i2 ,...,in0 ∈I K 1 = K 1 4
4
4
and consequently [n0 ]
co(FS
(K1 )) ⊆ K 1 .
(∗)
4
We can consider the sets n 0 −1
M =
(co(FS[k] (K1 )) − co(FS[k] (K1 )))
k=0
and n 0 −1
N = K1 − K1 + 4
4
(co(FS[k] (K1 )) − co(FS[k] (K1 ))).
k=1
Let us note that M and M are bounded, balanced, convex neighbourhoods of 0,
B∥·∥
N,
1
4
n 0 −1 1 1 ⊆ B∥·∥ K 1 , − B∥·∥ K 1 , + (co(FS[k] (K1 )) − co(FS[k] (K1 ))) 4
4
4
4
(ii)
k =1
and
B∥·∥
K1 , 4
1 4
⊆ K1 .
(iii)
Here is the justification for (ii): if x ∈ B∥·∥ (N , 14 ) then there exist u ∈ N and y ∈ X such that x = u + y and ∥y∥ < Therefore there exist u1 , u2 ∈ K 1 and w ∈
n 0 −1 k=1
4
(co(FS[k] (K1 )) − co(FS[k] (K1 ))) such that x = u1 − u2 + w + y = s1 − s2 + w ,
where s1 = u1 + 12 y and s2 = u2 − 12 y. Since ∥s1 − u1 ∥ = ∥s2 − u2 ∥ = and consequently x ∈ B∥·∥ (K 1 , ) − B∥·∥ (K 1 , ) + 1 4
4
4
1 4
n0 −1 k=1 1 4
1 2
∥y ∥ <
1 8
< 14 , we infer that s1 , s2 ∈ B∥·∥ (K 1 , 14 )
4
Ai (M ) ⊆ N ⊆ B∥·∥ for each i ∈ I.
N,
1 4
⊆ M,
4
(co(FS (K1 )) − co(FS (K1 ))) and the proof of (ii) is done. [k]
[k]
For the justification of (iii), just note that B∥·∥ (K 1 , ) ⊆ B∥·∥ [K 1 , 14 ] and take, in Lemma 2.2, ρ = We claim that
1 . 4
4
1 4
,r =
1 4
and ε =
1 . 2
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R. Miculescu, A. Mihail / J. Math. Anal. Appl. 407 (2013) 56–68
Indeed, using the linearity and continuity of Ai and the fact that B + x = B + x, for each B ⊆ X and x ∈ X , we have
n −1 0 [k] [k] Ai (co(FS (K1 )) − co(FS (K1 )))
Ai (M ) =
k=0 n0 −1
=
(Ai (co(FS[k] (K1 ))) − Ai (co(FS[k] (K1 ))))
k=0 n0 −1
⊆
(Ai (co(FS[k] (K1 ))) − Ai (co(FS[k] (K1 ))))
k=0 n0 −1
⊆
(Ai (co(FS[k] (K1 )) + bi ) − Ai (co(FS[k] (K1 )) + bi ))
k=0 n0 −1
= Lemma 2.1
=
(fi (co(FS[k] (K1 ))) − fi (co(FS[k] (K1 ))))
k=0 n0 −1
(co(fi (FS[k] (K1 ))) − co(fi (FS[k] (K1 ))))
k=0 n0 −1
⊆
(co(FS[k+1] (K1 )) − co(FS[k+1] (K1 )))
k=0
=
[n0 ]
co(FS
n0 −1
[n ]
(K1 )) − co(FS 0 (K1 )) +
(co(FS[k] (K1 )) − co(FS[k] (K1 )))
k=1 n 0 −1
(*)
⊆
K1 − K1 + 4
4
(ii)
⊆
B∥·∥
K1 , 4
1 (co(FS[k] (K1 )) − co(FS[k] (K1 ))) = N ⊆ B∥·∥ N , 4
k=1
1 4
− B∥·∥ K 1 , 4
1
n 0 −1
4
+
(co(FS[k] (K1 )) − co(FS[k] (K1 )))
k =1
n0 −1
(iii)
⊆
K1 − K1 +
(co(FS[k] (K1 )) − co(FS[k] (K1 )))
k=1 n0 −1
=
co(FS (K1 )) − co(FS (K1 )) + [0]
[0]
(co(FS[k] (K1 )) − co(FS[k] (K1 )))
k=1 n0 −1
=
(co(FS (K1 )) − co(FS (K1 ))) = M [k]
[k]
k=0
for each i ∈ I and the proof of our claim is done. Since ∥·∥M , the Minkowski norm associated with the M, is equivalent to ∥·∥, there exist a, b > 0 such that a ∥·∥ ≤ ∥·∥M ≤ b ∥·∥ and therefore, taking into account the above claim, we infer that Ai (M ) ⊆ N ⊆ B∥·∥M (N , 4a ) ⊆ B∥·∥ (N , 14 ) ⊆ M and hence δ∥·∥M (Ai (M ), X \ M ) ≥ 4a for each i ∈ I. Using Lemma 2.4, for A = Ai (M ), B = M and α = 4a , we get that
α Ai (M ) ⊆ (1 − θ )M, for each i ∈ I, where θ = 2(α+ = 2aa+8 . Now, using Lemma 2.7, for f = Ai , µ = 1 − θ = α0 < 1 1) and A = M, we get ∥Ai ∥M ≤ α0 for each i ∈ I. Let us consider ε > 0, r = supi∈I ∥bi ∥ ∈ R (see Remark 3.1) and c such that r +ε < c. 1−α0 Then not
fi (cM ) ⊆ (c − ε)M , for each i ∈ I. Then fi (cM ) ⊆ fi (cM ) ⊆ (c − ε)M = (c − ε)M ,
(iv)
for each i ∈ I. We claim that B∥·∥
M
fi (cM ),
for each i ∈ I.
ε 2
⊆ cM
(v)
R. Miculescu, A. Mihail / J. Math. Anal. Appl. 407 (2013) 56–68
65
Indeed, if x ∈ B∥·∥M (fi (cM ), 2ε ), then there exist y ∈ fi (cM ) and u ∈ X such that x = y + u and ∥u∥M < 2ε . Then, using (iv),
we have ∥x∥M ≤ ∥y∥M + ∥u∥M < c − ε + 2ε = c − 2ε < c, which implies that x ∈ cM = cM. Now we claim that B∥·∥
M
FS (cM ),
ε 4
⊆ cM .
(vi)
Indeed, if x ∈ B∥·∥ (FS (cM ), 4ε ), then there exist y ∈ FS (cM ) = ∪i∈I fi (cM ) and u ∈ X such that x = y + u and ∥u∥M < 4ε . M
As y ∈ ∪i∈I fi (cM ), there exists y1 ∈ ∪i∈I fi (cM ) such that ∥y − y1 ∥M < 4ε . Since y1 ∈ ∪i∈I fi (cM ), there exists i0 ∈ I such that y1 ∈ fi0 (cM ) and, with the notation y − y1 = w , we have x = y1 + w + u. As ∥w + u∥M ≤ ∥w∥M + ∥u∥M < 4ε + 4ε = 2ε , we infer that x ∈ B∥·∥ (fi0 (cM ), 2ε ). Taking into account (v), we conclude that x ∈ cM. M
Finally, we claim that
δ∥·∥ (FS (cM ), X \ cM ) > 0. Indeed, if it is not the case, then, as ∥·∥ and ∥·∥M are equivalent, we obtain that δ∥·∥M (FS (cM ), X \ cM ) = 0 < 4ε , i.e. infu∈FS (cM ),v∈X \cM ∥u − v∥M < 4ε which implies that there exist u0 ∈ FS (cM ) and v0 ∈ X \ cM such that ∥u0 − v0 ∥M < 4ε . Hence v0 ∈ B∥·∥ (FS (cM ), 4ε ) and, using (vi), we obtain the contradiction v0 ∈ cM. M
In conclusion, as cM is a convex body, S is strictly topologically contractive. (4) ⇒ (1) Since S is strictly topologically contractive, there exists a convex body K such that δ∥·∥ (FS (K ), X \ K ) > 0. def
If C = K − K , then C is a bounded, balanced, convex neighbourhood of 0 and ∥·∥C , the Minkowski norm associated with C , is equivalent to ∥·∥. (∗) Taking into account Lemma 2.6, for A1 = B1 = FS (K ) and A = B = K , we obtain that δ∥·∥C (FS (K ) − FS (K ), X \ C ) > 0. Since Ai (C ) = Ai (K ) − Ai (K ) = fi (K ) − fi (K ) ⊆ FS (K ) − FS (K ), with the notation α = δ∥·∥C (FS (K ) − FS (K ), X \ C ), we have δ∥·∥C (Ai (C ), X \ C ) ≥ α for each i ∈ I. Then, using Lemma 2.4, for A = Ai (C ) and B = C , we get that Ai (C ) ⊆ (1 − θ )C , for α . Therefore, taking into account Lemma 2.7, for f = Ai , A = C and µ = 1 − θ , we obtain that each i ∈ I, where θ = 2(α+ 1)
∥Ai ∥C ≤ 1 − θ ,
(∗∗)
for each i ∈ I. From (∗) and (∗∗) we conclude that S is hyperbolic. (1) ⇒ (5) Since S = ((X , ∥·∥), (fi )i∈I ) is hyperbolic, there exists a norm |||·||| on X equivalent to ∥·∥ having the property that the affine infinite iterated function system ((X , |||·|||), (fi )i∈I ) is contractive. Therefore there exist C ∈ [0, 1) and m > 0 such that |||Ai ||| = lip(fi ) ≤ C and |||bi ||| ≤ m for each i ∈ I. Then, given an arbitrary α = α1 α2 . . . αn . . . ∈ Λ, we have
||fα α ...α α (x) − fα α ...α (x)|| ≤ C n ||fα (x) − x|| n n+1 n 1 2 1 2 n+1 ≤ C n (||fαn+1 (x)|| + |||x|||) = C n (||Aαn+1 (x) + bαn+1 || + |||x|||) ≤ C n (||Aαn+1 (x)|| + ||bαn+1 || + |||x|||) ≤ C n ((C + 1) |||x||| + ||bαn+1 ||) ≤ C n ((C + 1) |||x||| + m), n for each n ∈ N and x ∈ X . Since the series C ((C + 1) |||x||| + m) is convergent, using Weierstrass M-Test, we infer that the series (fα1 α2 ...αn αn+1 (x) − fα1 α2 ...αn (x)) is |||·||| convergent (and therefore ∥·∥ convergent), hence there exists limn→∞ fα1 α2 ...αn (x) and let us denote it by xα . Hence limn→∞ fα1 α2 ...αn (x) = xα . As we have ||fα1 α2 ...αn (x) − fα1 α2 ...αn (y)|| ≤ C n |||x − y||| for each n ∈ N, x, y ∈ X , we infer that limn→∞ fα1 α2 ...αn (x) = limn→∞ fα1 α2 ...αn (y) for each x, y ∈ X . Therefore S = ((X , ∥·∥), (fi )i∈I ) is point-fibred. Moreover, given a ∥·∥ bounded subset B of X , then ∪i∈I fi (B) is ∥·∥ bounded, so B ∪ ∪i∈I fi (B) is ∥·∥ bounded and therefore, as |||·||| and ∥·∥ are equivalent, there exists δB > 0 such that |||y||| ≤ δB , for each y ∈ B ∪ ∪i∈I fi (B). Then, as above, for an arbitrary α = α1 α2 . . . αn . . . ∈ Λ, we have ||fα1 α2 ...αn αn+1 (x) − fα1 α2 ...αn (x)|| ≤ 2δB C n , for each n ∈ N and x ∈ B and, consequently, S = ((X , ∥·∥), (fi )i∈I ) is uniformly point-fibred. (5) ⇒ (3) Let us consider the set A = π (Λ). First, let us note that since fi (xα ) = xiα we get fi (π (Λ)) = π (iΛ) for any i ∈ I. Taking into account the fact that fi is continuous, we have fi (π (Λ)) ⊆ fi (π (Λ)) ⊆ π (iΛ) ⊆ π (Λ) for any i ∈ I, so ∪i∈I fi (π (Λ)) ⊆ π (Λ), and therefore ∪i∈I fi (A) ⊆ A.
(i)
66
R. Miculescu, A. Mihail / J. Math. Anal. Appl. 407 (2013) 56–68 n If x ∈ A = π (Λ) then there exists xn ∈ π (Λ) such that x = limn→∞ xn . Since there exists αn = α1n α2n . . . αm . . . such that
n xn = π (αn ) we have xn = π (α1n α2n . . . αm . . .) = fα1n (xα2n ...αmn ... ) ∈ fα1n (π (Λ)) ⊆ fα1n (π (Λ)) ⊆ ∪i∈I fi (A), so x ∈ ∪i∈I fi (A), i.e.
A ⊆ ∪i∈I fi (A).
(ii)
From (i) and (ii) we obtain A = ∪i∈I fi (A), i.e. FS (A) = A.
(1)
For a fixed B ∈ B (X ), according to the hypothesis, for each ε > 0 there exists nB,ε ∈ N such that for each n ∈ N, n ≥ nB,ε , x ∈ B and α = α1 α2 . . . αn . . . ∈ Λ we have fα1 α2 ...αn (x) − xα < 2ε , i.e.
fα
1 α2 ...αn
ε (x) − π (α) < .
(iii)
2
For each a ∈ A = π (Λ) there exists aε ∈ π (Λ) such that
ε
∥ a − aε ∥ <
(iv)
2
ε ε ε and sincethere exists αε = α1α2 . . . αn . . . ∈ Λ suchthat aε = π (αε ), using (iii) and (iv), for each n ∈ N, n ≥ nB,ε and x ∈ B ε ε ε ε ε ε we have fα1 α2 ...αn (x) − a ≤ fα1 α2 ...αn (x) − π (αε ) + ∥a − aε ∥ ≤ 2ε + 2ε = ε .
Hence for each ε > 0 there exists nB,ε ∈ N such that for each n ∈ N, n ≥ nB,ε we have A ⊆ B(FS (B), ε) and [n]
FS (B) ⊆ B(A, ε), so h∥·∥ (FS (B), A) ≤ ε , i.e. [n]
[n]
lim FS (B) = A [n]
(2)
n→∞
for each B ∈ B (X ). The relations (1) and (2) close the proof.
5. Examples Example 1. In [20] we constructed an affine infinite iterated function system defined on l2 (A) which has an attractor (namely Lipscomb’s space ωA ). Let us present the details of this construction. ′ Let A be an arbitrary set. Single out a point z of A and let us consider the set A = A \ {z }. The points of l2 (A) are ′
′
collections of real numbers indexed by points of A . If E is the set of real numbers, then x ∈ l2 (A) means x = {xa } ∈ E A 2 ′ such that xa = 0 for all but countable many a ∈ A and xa converges. The topology of l2 (A) is induced from the metric 2 d(x, y) = a (xa − ya ) , where we think xa as the a-th coordinate of x. Let us also consider, for the case when A is an arbitrary set with the discrete topology, the Baire space N (A) which is the topological product of countably many copies An of A. Hence the points of N (A) consist of all sequences v = a1 a2 . . . an . . . , with an ∈ A. Moreover N (A) is a metric space with the metric ′
d(v,v ) =
1, k
0,
′
if k is the first index where ai ̸= ai ′
if v = v .
Lipscomb’s space L(A) is a quotient space of N (A) such that each equivalence class consists of either a single point or two points. Those classes with two points come from identifying the point a1 a2 . . . ak−2 ak−1 ak ak . . . with the point a1 a2 . . . ak−2 ak ak−1 ak−1 . . ., where ak−1 ̸= ak . Therefore Lipscomb’s space L(A) is obtained via a projection or identification map p : N (A) → L(A). It is known that Lipscomb’s space L(A) can be embedded in Hilbert’s space l2 (A) by considering the function s : L(A) → l2 (A) given by s(α) = (αb )b∈A′ , where α = p(v), v = a1 a2 . . . . ∈ N (A) and
αb =
1
2k k with ak =b 0,
,
if there exists k such that ak = b if there exists no k such that ak = b.
Let ωA = s(L(A)) be the embedded version of L(A) with the l2 (A)-induced topology. For each a ∈ A \{z }, let fa : l2 (A) → l2 (A) be the function given by fa (x) = 21 (x + ua ), for all x ∈ l2 (A), and let fz : l2 (A) → l2 (A) be the function given by fz (x) = 21 x, for all x ∈ l2 (A), where ua = (αj )j∈A′ ∈ l2 (A) is described by αj = 0, for j ̸= a and αa = 1. Our result states that ωA , the
embedded version of L(A) endowed with the l2 (A)-induced topology, is the attractor of the affine infinite iterated function system S = (l2 (A), {fa }a∈A ). Therefore, according to our main result, S represents an uniformly point-fibred and strictly topologically contractive affine infinite iterated function system for which #(I ) = ∞ and dim(X ) = ∞.
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Example 2. Let us consider the affine iterated function system S = ((c0 , ∥·∥∞ ), T ), where T : c0 → c0 is the linear operator 1 given by T ((xn )n≥1 ) = ((1 − n+ )xn )n≥1 for each (xn )n≥1 ∈ c0 . Note that T is bounded with ∥T ∥ ≤ 1 and 1 T
[m]
((xn )n≥1 ) =
1−
1
m
n+1
(∗)
xn n ≥1
for each m ∈ N and each (xn )n≥1 ∈ c0 . Let us also note that T is bijective and T −1 : c0 → c0 , which is given by 1 T −1 ((xn )n≥1 ) = (( n+ )xn )n≥1 for each (xn )n≥1 ∈ c0 , is continuous and therefore n T (A) = T (A)
(∗∗)
for each subset A of c0 . Claim 1. S is point-fibred and π : Λ → c0 is 0. Proof. It is enough to prove that limm→∞ T [m] ((xn )n≥1 ) = 0 for each (xn )n≥1 ∈ c0 . Indeed, given a fixed (xn )n≥1 ∈ c0 , since limn→∞ that for each n ∈ N, n > xn = 0, for each ε > 0 there exists nε ∈ N1 such 1 m m nε we have |xn | < ε and therefore (1 − n+ ) x < ε for each m ∈ N . Because lim ( 1 − ) x = limm→∞ (1 − 31 )m x2 = n m→∞ 1 1 2
· · · = limm→∞ (1 − nε1+1 )m xnε = 0 there exists mε ∈ N such that for each m ∈ N, m > mε we have (1 − 12 )m x1 < ε, (1 − 31 )m x2 < ε, . . . , (1 − nε1+1 )m xnε < ε. Consequently for each ε > 0 there exists mε ∈ N such that for each [m] 1 )m xn < ε, which means that limm→∞ T [m] ((xn )n≥1 ) = 0. m ∈ N, m > mε , we have T ((xn )n≥1 ) = supn≥1 (1 − n+ 1
Claim 2. S is not uniformly point-fibred. Proof. Indeed, if this is not the case, then, based on the proof of the implication (5) ⇒ (3) of our main result, we infer that limn→∞ h(T [n] (B[0, 1]), {0}) = 0, hence limn→∞ d(T [n] (B[0, 1]), {0}) = 0 and consequently limn→∞ supx∈B[0,1] d(T [n] (x), {0}) = 0. Therefore there exists n0 ∈ N such that supx∈B[0,1] d(T [n0 ] (x), {0}) < 12 . In particular, for em = (0, 0, . . . , 0, 1, 0, . . .) ∈ B[0, 1], where 1 is on the m-th position, we have d(T [n0 ] (em ), {0}) <
1
,
2 1 n0 = (1 − 1 )n0 < 1 , for each m ∈ N. By passing to limit as m goes to ∞ in the above ) e m m+1 m+1 2 inequality we obtain the contradiction 1 ≤ 21 .
(*) i.e. T [n0 ] (em ) = (1 −
Claim 3. S is topologically contractive. ◦
Proof. Indeed, since B[0, 1] is a convex body, it suffices to prove that T (B[0, 1]) ⊆ B[0, 1]. Taking into account (∗∗), the ◦
◦
above inclusion is the same with T (B[0, 1]) ⊆ B[0, 1], i.e. with T (B[0, 1]) ⊆ B[0, 1]. In order to prove the last inclusion, let us take an arbitrary element x = (xn )n≥1 from B[0, 1]. Then, as x ∈ c0 , there exists n0 ∈ N such that |xn | < 21 , for 1 each n ∈ N, n > n0 . Since (1 − n+ )xn ≤ (1 − n+1 1 ) ∥x∥ ≤ 1 − 1
1 , n 0 +1
1 for each n ∈ N, n ≤ n0 , and (1 − n+ )xn ≤ 1
not (1 − n+1 1 ) |xn | < 21 , for each n ∈ N, n > n0 , we infer that ∥T (x)∥ = ((1 − n+1 1 )xn )n≥1 ≤ max{1 − n01+1 , 12 } = εx . Then B(T (x), 1 − εx ) ⊆ B[0, 1] since if u ∈ B(T (x), 1 − εx ), then ∥u∥ ≤ ∥u − T (x)∥ + ∥T (x)∥ < 1 − εx + εx = 1, i.e. u ∈ B(0, 1). ◦ ◦ Consequently T (x) ∈ B[0, 1] for each x ∈ B[0, 1], i.e. T (B[0, 1]) ⊆ B[0, 1]. Claim 4. S is not strictly topologically contractive. Proof. Let us suppose, by contrary, that there exists a convex body K such that δ(FS (K ), X \ K ) > 0, i.e., based on (∗∗), ◦
such that δ(T (K ), X \ K ) > 0. Then, using Lemma 2.3, we get that T (K ) ⊆ K ⊆ K , and, taking into account Lemma 2.6, for A1 = B1 = T (K ) and A = B = K , we get that δ(T (K ) − T (K ), X \ (K − K )) > 0. Considering the set C = K − K , which is not
a convex, bounded, balanced neighbourhood of 0, the last inequality can be written as δ(T (C ), X \ C ) = ε0 > 0. Since C is ε bounded there exists M > 0 such that C ⊆ B[0, M ]. Let us consider η ∈ (0, 80 ) and n0 ∈ N such that n0 > 4M − 1. Note that ε 0
δ en0 ̸∈ C
(∗ ∗ ∗) for each δ > M since if this is not the case, then δ en0 ∈ C ⊆ B[0, M ] and we obtain the contradiction M < δ = δ en0 ≤ M. Therefore the non-empty set {γ > 0 | γ en0 ̸∈ C } is bounded from below by 0 and we can consider θ = inf{η > 0 | ηen0 ̸∈ C }. Let us note that (θ − η)en0 ∈ C ;
(i)
(θ + η)en0 ̸∈ C
(ii)
and
θ ≤ M.
(iii)
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Indeed, if (i) would not be valid, then (θ − η)en0 ̸∈ C which implies the contradiction θ ≤ θ − η, i.e. η ≤ 0. If (ii) would not be valid, then (θ + η)en0 ∈ C . Since there exists λ ∈ (0, θ + η) such that λen0 ̸∈ C we obtain a contradiction in the λ λ following way: because 0 < θ+η < 1 and C is balanced, we infer that θ+η (θ + η)en0 ∈ C , i.e. λen0 ∈ C . For the justification of (iii), let us note that, taking into account (∗ ∗ ∗), we have θ ≤ δ for each δ > M. Therefore θ ≤ M. Finally we get the following contradiction
ε0
(i) and (ii)
≤ = = = (iii)
≤ i.e. ε0 ≤ 0.
T ((θ − η)en ) − (θ + η)en 0 0 (θ − η) 1 − 1 − (θ + η) e e n n 0 0 n0 + 1 (θ − η) 1 − 1 − (θ + η) en0 n0 + 1 η θ θ 1 n + 1 − n + 1 − 2η < η 2 − n + 1 + n + 1 0 0 0 0 ε0 ε0 ε0 M < + = , 2η + n0 + 1 4 4 2
Example 3. Let (en )n≥1 be the canonical basis of the Hilbert space l2 = {(xn )n≥1 |
n≥1
x2n < ∞} and let us consider the
function πm : l → l defined by πm ((xn )n≥1 ) = (yn )n≥1 , where ym = xm and yn = 0 if n ̸= m, for each (xn )n≥1 ∈ l2 . π (x)+e We claim that S = (l2 , (fn )n≥0 ), where f0 (x) = 2x and fn (x) = n 2 n , for each x ∈ l2 and each n ∈ N, is an affine infinite iterated function system having an attractor, namely the set ∪n≥1 [0, en ]. Indeed, fn is a contraction for each n, the family of function (fn )n≥0 is bounded and since ∪n≥1 [0, en ] is a non-empty, closed, bounded set, f0 (∪n≥1 [0, en ]) = ∪n≥1 [0, e2n ] and fn (∪n≥1 [0, en ]) = [ e2n , en ] for each n ≥ 1, we infer that FS (∪n≥1 [0, en ]) = ∪n≥1 [0, en ]. Hence, according with Theorem 2.2 from [23], ∪n≥1 [0, en ] is an attractor of S . 2
2
Acknowledgment We want to thank the referee whose generous and valuable remarks and comments brought improvements to the paper and enhanced clarity. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27]
R. Atkins, M. Barnsley, A. Vince, D. Wilson, A characterization of hyperbolic affine iterated function systems, Topology Proc. 36 (2010) 189–211. M.F. Barnsley, Fractals Everywhere, Academic Press, Boston, MA, 1988. M.F. Barnsley, Fractal image compression, Notices Amer. Math. Soc. 43 (1996) 657–662. J.B. Conway, A Course in Functional Analysis, Springer-Verlag, 1985. D. Dumitru, A. Mihail, The shift space of an iterated function system containing Meir–Keeler functions, An. Univ. Bucureşti Mat. 57 (2008) 75–88. H. Fernau, Infinite iterated function systems, Math. Nachr. 170 (1994) 79–91. G. Gwóźdź-Łukawska, J. Jachymski, The Hutchinson–Barnsley theory for infinite iterated function systems, Bull. Aust. Math. Soc. 72 (2005) 441–454. M.R. Hille, Remarks on limits sets of infinite iterated functions systems, Monatsh. Math. 168 (2012) 215–237. J.E. Hutchinson, Fractals and self similarity, Indiana Univ. Math. J. 30 (1981) 713–747. A. Kameyama, Distances on topological self-similar sets, in: Michel L. Lapidus, Machiel van Frankenhuijsen (Eds.), Fractal Geometry and Applications: A Jubilee of Benoit Mandelbrot, in: Proceedings of Symposia in Pure Mathematics, vol. 72, Amer. Math. Soc., Providence, RI, 2004, pp. 117–129. Part 1. K. Leśniak, Infinite iterated function systems: a multivalued approach, Bull. Pol. Acad. Sci. Math. 52 (2004) 1–8. K. Leśniak, Invariant sets and Knaster–Tarski principle, Cent. Eur. J. Math. 10 (6) (2012) 2077–2087. S.L. Lipscomb, Fractals and Universal Spaces in Dimension Theory, Springer-Verlag, 2009. J. Matkowski, Fixed point theorems for mappings with a contractive iterate at a point, Proc. Amer. Math. Soc. 62 (1977) 344–348. R.D. Mauldin, Infinite Iterated Function Systems: Theory and Applications, in: Progr. Probab., vol. 37, Birkhäuser-Verlag, Basel, Switzerland, 1995, pp. 91–110. R.D. Mauldin, M. Urbański, Dimensions and measures in infinite iterated function systems, Proc. Lond. Math. Soc. 73 (1996) 105–154. R.E. Megginson, An Introduction to Banach Space Theory, Springer-Verlag, New York, 1998. F. Mendivil, A generalization of IFS with probabilities to infinitely many maps, Rocky Mountain J. Math. 28 (1998) 1043–1051. R. Miculescu, L. Ioana, Some connections between the attractors of an IIFS S and the attractors of the sub-IFSs of S , Fixed Point Theory Appl. 2012 (2012) 141. R. Miculescu, A. Mihail, A, Lipscomb’s space ωA is the attractor of an infinite IFS containing affine transformations of l2 (A), Proc. Amer. Math. Soc. 136 (2008) 587–592. R. Miculescu, A. Mihail, Lipscomb’s L(A) space fractalized in lp (A), Mediterr. J. Math. 9 (2012) 515–524. R. Miculescu, A. Mihail, The independence of p of the Lipscomb’s L(A) space fractalized in lp (A), Topology Appl. 160 (2013) 241–250. A. Mihail, R. Miculescu, The shift space for an infinite iterated function system, Math. Rep. (Bucur.) 11 (61) (2009) 21–32. W. Rudin, Functional Analysis, second ed., McGraw-Hill, New York, 1991. I.A. Rus, A. Petruşel, G. Petruşel, Fixed Point Theory, Cluj University Press, Cluj-Napoca, 2008. N.A. Secelean, The existence of the attractor of countable iterated function systems, Mediterr. J. Math. 9 (2012) 65–84. F. Strobin, J. Swaczyna, On a certain generalisation of the iterated function system, Bull. Aust. Math. Soc. 87 (2013) 37–54.
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Alternative characterization of hyperbolic affine infinite iterated function systems Radu Miculescu, Alexandru Mihail ∗ Bucharest University, Faculty of Mathematics and Computer Science, Str. Academiei 14, 010014 Bucharest, Romania
article
info
Article history: Received 11 February 2013 Available online 9 May 2013 Submitted by R.M. Aron Keywords: Affine infinite iterated function system Comparison function Attractor of an affine infinite iterated function system ϕ -hyperbolic affine infinite iterated function system Uniformly point-fibre affine infinite iterated function system Convex body Strictly topologically contractive affine infinite iterated function system
abstract In this paper we present a characterization of hyperbolic affine infinite iterated function systems defined on an arbitrary normed space. Our result is a generalization of Theorem 1.1 from the paper ‘‘A characterization of hyperbolic affine iterated function systems’’, Topology Proceedings, 36 (2010), 189–211, by R. Atkins, M. Barnsley, A. Vince and D. Wilson. Some examples are presented. © 2013 Elsevier Inc. All rights reserved.
1. Introduction Motivated by the following question of A. Kameyama (see [10]): Given a topological self-similar set, does there exist an associated system of contraction mappings?, in a recent paper, R. Atkins, M. Barnsley, A. Vince and D.C. Wilson [1] presented a theorem that characterizes hyperbolic affine iterated function systems defined on Rm . More precisely, Rm being endowed with the Euclidean metric, if N > 0 is an integer and fn : Rm → Rm , n ∈ {1, 2, . . . , N }, are continuous mappings, then F = (Rm ; f1 , f2 , . . . , fN ) is called an iterated function system (IFS). If each fn is an affine map on Rm , then F is called an affine IFS. If each fn is a contraction (i.e. there exists αn ∈ [0, 1) such that ∥fn (x) − fn (y)∥2 ≤ αn ∥x − y∥2 for all x, y ∈ Rm ), then F is called a contractive IFS. If there exists a metric on Rm Lipschitz equivalent to the Euclidean metric so that each fn is a contraction, then F is called a hyperbolic IFS. For a positive integer N, the symbol Ω = {1, 2, . . . , N }∞ denotes the set of all infinite sequences of symbols {σk }∞ k=1 belonging to the alphabet {1, 2, . . . , N }. The set Ω is endowed with the product topology. An element σ ∈ Ω will also be denoted by the concatenation σ = σ1 σ2 σ3 . . . , where σk denotes the kth component of σ . The IFS F is called point-fibred if for each σ = σ1 σ2 σ3 . . . ∈ Ω , the limit on the right hand side of π (σ ) := limk→∞ fσ1 ◦ fσ2 ◦ · · · ◦ fσk (x) exists, is independent
∗
Corresponding author. E-mail addresses: [email protected] (R. Miculescu), [email protected] (A. Mihail).
0022-247X/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jmaa.2013.05.007
R. Miculescu, A. Mihail / J. Math. Anal. Appl. 407 (2013) 56–68
57
of x ∈ Rm for fixed σ , the map π : Ω → Rm is continuous and for each n ∈ {1, 2, . . . , N }, the following diagram commutes sn
Ω ↓π
→
Rm
→
fn
Ω ↓ π , where sn
: Ω → Ω is given by sn (σ ) = nσ . Ω is called a coding map.
Rm
The symbol H denotes the non-empty compact subsets of Rm . A set A ∈ H is called an attractor of the IFS F if A = F (A) and A = limk→∞ F ◦k (B) (with respect to the Hausdorff–Pompeiu metric) for all B ∈ H, where F : H → H is given by F (B ) = ∪Nn=1 fn (B) and F ◦k denotes the k-fold composition of F . A convex body is a compact convex subsets of Rm with non-empty interior. An IFS F is called topologically contractive if ◦
there exists a convex body K such that F (K ) ⊆ K . Part of the above mentioned result can be stated in the following way (see Theorem 1.1. from [1]): Theorem. If F = (Rm ; f1 , f2 , . . . , fN ) is an affine iterated function system, then the following statements are equivalent: (1) (2) (3) (4)
F F F F
is hyperbolic; is point-fibred; has an attractor; is topologically contractive.
IFSs were introduced in their present form by John Hutchinson (see [9]) and popularized by Michael Barnsley (see [2]). They are one of the most common and most general ways to generate fractals. Because of the variety of their applications (for example they are used for the construction of deterministic fractals [2] and have found numerous applications, in particular to image compression and image processing [3]), there is a current effort to extend the classical Hutchinson’s framework to infinite iterated function systems (see, for example, the works of H. Fernau [6], R.D. Mauldin [15], R.D. Mauldin, M. Urbański [16], F. Mendivil [18], K. Leśniak [11,12], G. Gwóźdź-Łukawska, J. Jachymski [7], R. Miculescu, A. Mihail [20–23], M. Hille [8], R. Miculescu, L. Ioana [19] N.A. Secelean [26] and F. Strobin, J. Swaczyna [27]). The aim of this paper is to generalize the result mentioned above. More precisely, we define the notion of affine infinite iterated function system in a Banach space (which can be infinite dimensional) (see Definition 3.2), hyperbolic affine infinite iterated function system (see Definition 3.4), comparison function (see Definition 3.5), ϕ -hyperbolic affine infinite iterated function system (see Definition 3.8), uniformly point-fibred affine infinite iterated function system (see Definition 3.11), attractor of an affine infinite iterated function system (see Definition 3.13) and strictly topologically contractive affine infinite iterated function system (see Definition 3.16). Our main result states that if S = ((X , ∥·∥), (fi )i∈I ) is an affine infinite iterated function system, then the following statements are equivalent: 1. 2. 3. 4. 5.
S is hyperbolic;
there exists a comparison function ϕ0 such that S is ϕ0 -hyperbolic; S has an attractor; S is strictly topologically contractive; S is uniformly point-fibred.
The paper is organized as follows. In Section 2 we present some preliminaries facts concerning convex subsets of a normed space, affine functions and bounded linear operators from a normed space into itself and Hausdorff–Pompeiu pseudo-metric. Section 3 contains the notation, terminology, and definitions that are used throughout the paper. Section 4 is devoted to our main result and Section 5 presents some examples. 2. Preliminaries Given a subset A of a normed space (X , ∥·∥), we denote by co(A) the convex hull of A and by co(A) the closed convex hull of A. n n It is well known that co(A) = { j=1 λj xj | n ∈ N, x1 , . . . , xn ∈ A, λ1 , . . . , λn ≥ 0 and j=1 λj = 1} and co(A) = co(A) (see [4, p. 105] and [17, p. 3]). Lemma 2.1. Let (X , ∥·∥) be a normed space. Then f (co(A)) = co(f (A)), for each affine function f : X → X and each subset A of X . Proof. Since f is affine, there exists a linear function F : X → X and x0 ∈ X such that f = F + x0 . First let us note that co(x + B) = x + co(B)
(∗)
for each x ∈ X and each subset B of X . (*) Then f (co(A)) = F (co(A)) + x0 and co(f (A)) = co(F (A) + x0 ) = co(F (A)) + x0 and therefore it suffices to prove that F (co(A)) = co(F (A)). Since F (co(A)) is a convex set, as F (A) ⊆ F (co(A)), we infer that co(F (A)) ⊆ F (co(A)).
(∗∗)
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It is clear that F (co(A)) ⊆ co(F (A)). From (∗∗) and (∗ ∗ ∗), we get that F (co(A)) = co(F (A)).
(∗ ∗ ∗)
Lemma 2.2. Let A be a subset of a normed space (X , ∥·∥). For ρ > 0, let us consider the sets Aρ = B∥·∥ [A, ρ] = {x ∈ X |
d(x, A) ≤ ρ} = {x ∈ X | infa∈A ∥x − a∥ ≤ ρ} and Kρ = co(Aρ ) = co(Aρ ). Then B∥·∥ [Kρ , r ] ⊆ Kρ+r +ε , for each r > 0 and ε > 0. Proof. If we consider x ∈ B∥·∥ [Kρ , r ] then there exists y ∈ Kρ such that ∥x − y∥ < r + 3ε , i.e., with the notation x1 = x − y,
we have x = y + x1 , where ∥x1 ∥ < r + 3ε . Since y ∈ Kρ = co(Aρ ), there exists y1 ∈ co(Aρ ) such that ∥y − y1 ∥ < 3ε , i.e., with the notation y2 = y − y1 , we have y = y1 + y2 , where ∥y2 ∥ < 3ε . As y1 ∈ co(Aρ ) there exist n ∈ N, λ1 , λ2 , . . . , λn ∈ [0, ∞) n n and u1 , u2 , . . . , un ∈ B∥·∥ [A, ρ] such that i=1 λi = 1 and y1 = i=1 λi ui . Because ui ∈ B∥·∥ [A, ρ] there exists vi ∈ A such that ∥ui − vi ∥ < ρ + 3ε , i.e., with the notation wi = ui − vi , we have ui = vi + wi , where ∥wi ∥ < ρ + 3ε , for n n each i ∈ {1, 2, . . . , n}. Hence x = y + x1 = y1 + y2 + x1 = i=1 λi (vi + wi + y2 + x1 ) i=1 λi ui + y2 + x1 = ε ε ε and since ∥wi + y2 + x1 ∥ ≤ ∥wi ∥ + ∥y2 ∥ + ∥x1 ∥ < ρ + 3 + 3 + r + 3 = ρ + r + ε and vi ∈ A, we infer that vi + wi + y2 + x1 ∈ B∥·∥ [A, ρ + r + ε] for all i ∈ {1, 2, . . . , n}, so x ∈ co(B∥·∥ [A, ρ + r + ε]) ⊆ co(Aρ+r +ε ) = Kρ+r +ε and the proof is done. Definition 2.1. Given a normed space (X , ∥·∥) and U and V subsets of X , we define δ∥·∥ (U , V ) = infu∈U ,v∈V ∥u − v∥. ◦
Lemma 2.3. If δ∥·∥ (A, B) > 0, where A and B are subsets of the normed space (X , ∥·∥), then A ⊆ X \ B. ◦
Proof. If, by reductio ad absurdum, the conclusion is false, then there exists x0 ∈ A such that x0 ̸∈ X \ B = X \ B. Then for each ε > 0 there exists xε ∈ B such that ∥x0 − xε ∥ < ε . Hence δ∥·∥ (A, B) ≤ ∥x0 − xε ∥ < ε , for each ε > 0 and consequently we get the contradiction δ∥·∥ (A, B) = 0. Let us recall (see, for example [24, p. 30]) that given a normed space (X , ∥·∥) and A a bounded, balanced (i.e. α A ⊆ A for each α such that |α| ≤ 1), convex neighbourhood of 0, the application ∥·∥A : X → R given by ∥x∥A = inf{λ > 0 | x ∈ λA} for each x ∈ X , is a norm, called Minkowski norm, the norms ∥·∥ and ∥·∥A are equivalent and {x ∈ X | ∥x∥A < 1} ⊆ A ⊆ {x ∈ X | ∥x∥A ≤ 1}. Lemma 2.4. If δ∥·∥A (A, X \ B) ≥ α , where α > 0, A and B are subsets of the normed space (X , ∥·∥) such that A is a bounded, α . balanced, convex neighbourhood of 0, then A ⊆ (1 − θ )B, where θ = 2(α+ 1) 1 Proof. If the conclusion is not valid, then there exists x0 ∈ A such that x0 ̸∈ (1 −θ )B, i.e. 1−θ x0 ̸∈ B. Then α ≤ δ∥·∥A (A, X \ B) ≤ x0 − 1 x0 = ∥x∥A θ < θ = α and therefore we get the contradiction 0 < α ≤ −1. 1−θ 1−θ 1−θ α+2 A
Lemma 2.5. Let (X , ∥·∥) be a normed space, ε > 0, and A and A1 subsets of X such that ε < δ∥·∥ (A1 , X \ A). Then B∥·∥ (A1 , ε) = {x ∈ X | d(x, A1 ) < ε} = {x ∈ X | infa∈A1 ∥x − a∥ < ε} ⊆ A. Proof. If, by reductio ad absurdum, the conclusion is false, then there exists x0 ∈ B∥·∥ (A1 , ε) such that x0 ̸∈ A i.e. x0 ∈ X \ A. Since x0 ∈ B∥·∥ (A1 , ε) there exists x1 ∈ A1 such that ∥x0 − x1 ∥ < ε . Hence we obtain the following contradiction ε < δ∥·∥ (A1 , X \ A) ≤ ∥x1 − x0 ∥ < ε. Lemma 2.6. Let (X , ∥·∥) be a normed space and A, B, A1 and B1 subsets of X such that δ∥·∥ (A1 , X \ A) > 0 and δ∥·∥ (B1 , X \ B) > 0. Then δ∥·∥ (A1 − B1 , X \ (A − B)) > 0. Proof. Let us consider ε ∈ (0, min{δ∥·∥ (A1 , X \ A), δ∥·∥ (B1 , X \ B)}). Then, according with Lemma 2.5, we get B∥·∥ (A1 , ε) ⊆ A
(∗)
B∥·∥ (B1 , ε) ⊆ B.
(∗∗)
and
It is clear that B∥·∥ (A1 − B1 , ε) ⊆ B∥·∥ (A1 , ε) − B∥·∥ (B1 , ε).
(∗ ∗ ∗)
Hence, using (∗), (∗∗) and (∗ ∗ ∗), we get B∥·∥ (A1 − B1 , ε) ⊆ A − B, i.e. B∥·∥ (A1 − B1 , ε)∩(X \(A − B)) = ∅ and consequently
δ∥·∥ (A1 − B1 , X \ (A − B)) ≥ ε > 0. Lemma 2.7. If (X , ∥·∥) is a normed space, A is a bounded, balanced, convex neighbourhood of 0 and f : X → X is a bounded linear operator having the property that f (A) ⊆ µA, where µ > 0, then ∥f ∥A ≤ µ.
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Proof. If x ∈ X has the property that ∥x∥A ≤ 1, then for each ε > 0 there exists λε ∈ (0, 1 + ε) such that x ∈ λε A, λε λε 1 1 1 i.e. λ1 x ∈ A. Then, since 0 < 1+λ < 1 and A is balanced, we obtain that 1+λ x ∈ A, i.e. 1+ε x ∈ A and hence 1+ε f (x) = λ ε
ε
ε
ε
1 1 1 f ( 1+ε x) ∈ f (A) ⊆ µA, so (1+ε)µ f (x) ∈ A, which implies that (1+ε)µ f (x) ≤ 1, i.e. ∥f (x)∥A ≤ (1 + ε)µ for each ε > 0. A Therefore ∥f (x)∥A ≤ µ for each x ∈ X having the property that ∥x∥A ≤ 1. Consequently ∥f ∥A = sup∥x∥A ≤1 ∥f (x)∥A ≤ µ.
Given a metric space (X , d), we consider MX = {N ⊆ X | N is non-empty and bounded}, and BX = {N ⊆ X | def
def
N is non-empty, closed and bounded}. For x ∈ X , ε > 0 and A, B ∈ MX , we consider d(x, A) = infa∈A d(x, a), B(A, ε) ={x ∈ def
def
X | d(x, A) < ε} = ∪a∈A B(a, ε), B[A, ε] ={x ∈ X | d(x, A) ≤ ε} and d(A, B) = supx∈A d(x, B). The map h∗ : MX × MX → [0, ∞), given by h∗ (A, B) = max{d(A, B), d(B, A)}, for all A, B ∈ MX , satisfies the following properties: (i) h∗ (A, A) = 0, for all A ∈ MX ; (ii) h∗ (A, B) = h∗ (B, A), for all A, B ∈ MX ; (iii) h∗ (A, B) ≤ h∗ (A, C ) + h∗ (C , B), for all A, B, C ∈ MX . Therefore h∗ is a pseudo-metric in MX which is called the Hausdorff–Pompeiu pseudo-metric. Proposition 2.1. For all A1 , A2 ∈ MX , the following properties are valid: (i) (ii) (iii) (iv)
for all ε > 0, we have: h∗ (A1 , A2 ) ≤ ε if and only if A1 ⊆ B[A2 , ε] and A2 ⊆ B[A1 , ε]; h∗ (A1 , A2 ) = inf{r > 0 : A1 ⊆ B(A2 , r ) and A2 ⊆ B(A1 , r )}; h∗ (A1 , A2 ) = h∗ (A1 , A2 ); h∗ (A1 , A2 ) = 0 if and only if A1 = A2 ;
(v) if {Ai1 }i∈I and {Ai2 }i∈I are two families of elements of MX such that ∪i∈I Ai1 h (∪ ∗
i i∈I A1
,∪
i i∈I A2
) = h (∪ ∗
i i∈I A1
,∪
i i∈I A2
) ≤ supi∈I h ( , ∗
Ai1
Ai2
∈ MX and ∪i∈I Ai2 ∈ MX , then
).
Definition 2.2. The restriction of h∗ to BX × BX is a metric. We call it the Hausdorff–Pompeiu metric and denote it by h. We use the terminology Hausdorff–Pompeiu distance for the metric h which is also called the Hausdorff metric or the Pompeiu–Hausdorff metric (as in the book of R. Tyrrell Rockafellar and Roger Wets entitled ‘‘Variational Analysis’’, 3rd printing, Springer-Verlag, 2009, p. 144). For a nice history of this notion one can consult the work entitled ‘‘One hundred years since the introduction of the set distance by Dimitrie Pompeiu’’, appeared in System modeling and optimization, IFIP International Federation for Information Processing, Volume 199, 2006, pp. 35–39, Springer-Verlag, by T. Bârsan and D. Tiba. Proposition 2.2. If the metric space (X , d) is complete, then the metric space (BX , h) is complete. For the proof of the above theorem one can consult Theorem 26.1, p. 44, from [13]. Let (X , ∥·∥) be a normed space and |||·||| a norm on X equivalent to ∥·∥. Let us denote by d∥·∥ the metric induced by the norm ∥·∥ and by d||·|| the metric induced by the norm |||·|||. Then the set of non-empty, bounded, closed subsets of X with respect to d∥·∥ (denoted by B(X ,∥·∥) ) coincides with the set of non-empty, bounded, closed subsets of X with respect to d||·|| (denoted by B(X ,|| ·|| ) ) and we simply denote it by B (X ). Let us denote by h∥·∥ the Hausdorff–Pompeiu metric on B (X ) = B(X ,∥·∥) associated with d∥·∥ and by h||·|| the Hausdorff–Pompeiu metric on B (X ) = B(X ,|| ·|| ) associated with d||·|| . Lemma 2.8. The metrics h∥·∥ and h||·|| , defined above, are equivalent. Proof. Since the norms ∥·∥ and |||·||| are equivalent, there exist α ∈ (0, ∞) and β ∈ (0, ∞) such that α ∥x∥ ≤ |||x||| ≤ β ∥x∥ for any x ∈ X . Then we have infy∈B |||x − y||| ≤ |||x − y||| ≤ β ∥x − y∥, for any A, B ∈ B (X ), x ∈ A and y ∈ B, hence infy∈B |||x − y||| ≤ β infy∈B ∥x − y∥ ≤ β supx∈A infy∈B ∥x − y∥ and consequently supx∈A infy∈B |||x − y||| ≤ β supx∈A infy∈B ∥x − y∥. In a similar way one can prove that supx∈B infy∈A |||x − y||| ≤ β supx∈B infy∈A ∥x − y∥. Therefore, we have max{supx∈A infy∈B |||x − y|||, supx∈B infy∈A |||x − y|||} ≤ β max{supx∈A infy∈B ∥x − y∥ , supx∈B infy∈A ∥x − y∥}, i.e. h||·|| (A, B) ≤ β h∥·∥ (A, B). Similarly one can obtain α h∥·∥ (A, B) ≤ h||·|| (A, B). Therefore α h∥·∥ (A, B) ≤ h||·|| (A, B) ≤ β h∥·∥ (A, B) for all A, B ∈ B (X ) which means that the metrics h∥·∥ and h||·|| are equivalent. Remark 2.1. Let us note that if (X , ∥·∥) is a Banach space and the norm |||·||| on X is equivalent to ∥·∥, then (X , |||·|||) is also a Banach space and therefore (B(X ), h||·|| ) is a complete metric space. 3. Definitions and notations Definition 3.1 (Bounded Family of Functions). Given a Banach space (X , ∥·∥), a family of function (fi )i∈I , where fi : X → X , is called bounded if for each bounded subset A of X , the set ∪i∈I fi (A) is bounded. Definition 3.2 (Affine Infinite Iterated Function System—AIIFS). Given a Banach space (X , ∥·∥), an affine infinite iterated function system is a pair S = ((X , ∥·∥), (fi )i∈I ), where (fi )i∈I is a bounded family of functions (fi )i∈I having the property that, for each i ∈ I, there exists a bounded linear operator Ai : X → X and an element bi of X such that fi = Ai + bi .
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Remark 3.1. The family (fi )i∈I is bounded if and only if the sets {bi | i ∈ I } and {∥Ai ∥ | i ∈ I } are bounded. Remark 3.2. Just in order to emphasize that, in contrast with the classical theory, the set I could be infinite, we call S an affine infinite iterated function system, even though I could also be finite (as in our second example from Section 5, where I has only one element). We also point out that the Banach space (X , ∥·∥) could be infinite dimensional, by contrast with the framework from [1], where only finite dimensional Banach spaces are considered. In our second example from Section 5 we consider the infinite dimensional Banach space c0 endowed with the standard norm. Definition 3.3 (Contractive AIIFS). An affine infinite iterated function system S = ((X , ∥·∥), (fi )i∈I ) is called contractive if there exists C ∈ [0, 1) such that ∥Ai ∥ = lip(fi ) ≤ C for each i ∈ I. Definition 3.4 (Hyperbolic AIIFS). An affine infinite iterated function system S = ((X , ∥·∥), (fi )i∈I ) is called hyperbolic if there exists a norm |||·||| on X equivalent to ∥·∥ having the property that the affine infinite iterated function system ((X , |||·|||), (fi )i∈I ) is contractive. Remark 3.3. Each affine infinite iterated function system which is contractive is hyperbolic. Definition 3.5 (Comparison Function). A function ϕ : [0, ∞) → [0, ∞) is called a comparison function if it has the following properties: (i) ϕ is increasing; (ii) ϕ(t ) < t for any t > 0; (iii) ϕ is right-continuous. Remark 3.4 (See Remark 1 from [14]). Any function ϕ : [0, ∞) → [0, ∞) satisfying (ii) and (iii) from the above definition has the following property: limn→∞ ϕ n (t ) = 0 for any t > 0. Remark 3.5 (See [25, p. 25]). Any function ϕ : [0, ∞) → [0, ∞) satisfying (i) and (ii) from the above definition has the property that ϕ(0) = 0. Remark 3.6 (See Remark 2 from [5]). For any comparison function ϕ : [0, ∞) → [0, ∞) and any non-empty bounded subset A of [0, ∞) the following relations are valid: 1. sup ϕ(A) ≤ ϕ(sup A); 2. inf ϕ(A) = ϕ(inf A). Definition 3.6 (ϕ -Contraction). Let (X , d) be a metric space and a function ϕ : [0, ∞) → [0, ∞). A function f : X → X is called an ϕ -contraction if d(f (x), f (y)) ≤ ϕ(d(x, y)), for all x, y ∈ X . Matkowski’s theorem (See [25, p. 31]). Let (X , d) be a complete metric and ϕ : [0, ∞) → [0, ∞) a function such that: (i) (ii) (iii) (iv)
ϕ is increasing; ϕ(t ) < t for all t > 0; ϕ(0) = 0; limn→∞ ϕ [n] (t ) = 0 for all t > 0.
Then an ϕ -contraction f : X → X has a fixed point x∗ ∈ X and, for each x ∈ X , the sequence (f n (x))n∈N converges to x∗ . From the above Theorem, taking into account the Remarks 3.4 and 3.5, we obtain the following: Corollary 3.1. Let (X , d) be a complete metric space and f : X → X be an ϕ -contraction, where ϕ is a comparison function. Then f has a fixed point x∗ ∈ X and, for each x ∈ X , the sequence (f n (x))n∈N converges to x∗ . Definition 3.7 (ϕ -Contractive AIIFS). Given a comparison function ϕ : [0, ∞) → [0, ∞), an affine infinite iterated function system S = ((X , ∥·∥), (fi )i∈I ) is called ϕ -contractive if fi is an ϕ -contraction for each i ∈ I. Remark 3.7. Each affine infinite iterated function system S = ((X , ∥·∥), (fi )i∈I ) which is contractive (so there exists C ∈ [0, 1) such that ∥Ai ∥ = lip(fi ) ≤ C for each i ∈ I) is ϕ0 -contractive, where ϕ0 : [0, ∞) → [0, ∞) is given by ϕ0 (t ) = Ct for each t ∈ [0, ∞). Definition 3.8 (ϕ -Hyperbolic AIIFS). Given a comparison function ϕ : [0, ∞) → [0, ∞), an affine infinite iterated function system S = ((X , ∥·∥), (fi )i∈I ) is called ϕ -hyperbolic if there exists a metric |||·||| on X equivalent to ∥·∥ such that the affine infinite iterated function system ((X , |||·|||), (fi )i∈I ) is ϕ -contractive.
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Remark 3.8. Each affine infinite iterated function system which is ϕ -contractive is ϕ -hyperbolic. In the following N denotes the natural numbers, N∗ = N − {0} and N∗n = {1, 2, . . . , n}, where n ∈ N∗ . Given two ∗ sets A and B, by BA we mean the set of functions from A to B. By Λ = Λ(B) we mean the set BN and by Λn = Λn (B) we ∗ ∗ mean the set BNn . The elements of Λ = Λ(B) = BN are written as words ω = ω1 ω2 . . . ωm ωm+1 . . . and the elements of ∗ Λn = Λn (B) = BNn are written as words ω = ω1 ω2 . . . ωn . Hence Λ(B) is the set of infinite words with letters from the alphabet B and Λn (B) is the set of words of length n with letters from the alphabet B. If ω = ω1 ω2 . . . ωm ωm+1 . . . ∈ Λ(B) or if ω = ω1 ω2 . . . ωn ∈ Λn (B), where m, n ∈ N∗ , n ≥ m, then the word ω1 ω2 . . . ωm is denoted by [ω]m . For a non-empty ∗
set I, on Λ = Λ(I ) = I N , we consider the metric dΛ (α, β) =
∞
k=1
the function Fi : Λ → Λ given by Fi (ω) = iω for all ω ∈ Λ(I ).
β 1−δα k k
3k
y
, where δx =
1, if x = y . 0, if x ̸= y
For i ∈ I, let us consider
Definition 3.9 (The Shift Space Associated to an Affine Infinite Iterated Function System). Given an affine infinite iterated function system S = ((X , ∥·∥), (fi )i∈I ), the metric space (Λ(I ), dΛ ) is called the shift space associated to S . Definition 3.10 (Point-Fibred AIIFS). An affine infinite iterated function system S = ((X , ∥·∥), (fi )i∈I ) is called point-fibred ∗ if for each α = α1 α2 . . . αn . . . ∈ I N = Λ there exists xα ∈ X such that limn→∞ fα1 α2 ...αn (x) = xα , for each x ∈ X , where fα1 α2 ...αn stands for fα1 ◦ fα2 ◦ · · · ◦ fαm . Remark 3.9. We obtain, in the frame of the above definition, that fi (xα ) = xiα for any α ∈ Λ and i ∈ I. Λ In other words, the following diagram commutes ↓ π X
Fi
→ fi
→
Λ ↓ π , where π
: Λ → X is given by π (α) = xα for each α ∈ Λ
X
and it is called the coding map for S . Definition 3.11 (Uniformly Point-Fibred AIIFS). An affine infinite iterated function system S = ((X , ∥·∥), (fi )i∈I ) is called uniformly point-fibred if it is point-fibred and for each bounded subset B of X and exists nB,ε ∈ N such for each ε > 0 there ∗ that for each n ∈ N, n ≥ nB,ε , x ∈ B and α = α1 α2 . . . αn . . . ∈ I N = Λ we have fα1 α2 ...αn (x) − xα < ε . Remark 3.10. If the affine infinite iterated function system S = ((X , ∥·∥), (fi )i∈I ) is uniformly point-fibred, then π is continuous. Proof. We shall prove that π is continuous at an arbitrary element β = β1 β2 . . . βn . . . ∈ Λ. It suffices to show that for each sequence (α n )n∈N of elements from Λ such that limn→∞ α n = β , we have limn→∞ π (α n ) = π (β), where α n = α1n α2n . . . αkn . . . , for each n ∈ N. To this aim let us consider a fixed element x0 of X and let us note that, since S is uniformly point-fibred, for each ε > 0 there exists mε ∈ N such that for each n ∈ N, n ≥ mε , α1 , α2 , . . . , αn ∈ I and γ ∈ Λ, we have
fα
1 α2 ...αn
(x0 ) − π (α1 α2 . . . αn γ ) < ε.
(∗)
As limn→∞ α n = β there exists nε ∈ N such that for each n ∈ N, n ≥ nε we have dΛ (α n , β) <
1
3mε
, so
(∗∗)
[α n ]mε = [β]mε
because if this is not the case then there exists k ∈ {1, 2, . . . , mε } such that αkn ̸= βk and consequently we obtain the following contradiction: 31k ≤ dΛ (α n , β) < 3m1 ε . Hence, taking into account (∗∗), we conclude that for each ε > 0 there exists nε ∈
(*)
N such that for each n ∈ N, n ≥ nε we have ∥π (α n ) − π (β)∥ ≤ π (α n ) − f[α n ]mε (x0 ) + f[α n ]mε (x0 ) − π (β) ≤ ε +ε = 2ε , i.e. limn→∞ π (α n ) = π (β).
Definition 3.12 (The Function FS Associated with an AIIFS). For an affine infinite iterated function system S ((X , ∥·∥), (fi )i∈I ) one defines FS : B (X ) → B (X ) given by FS (B) = ∪i∈I fi (B) for each B ∈ B (X ).
=
[n]
FS designates the n-fold composition of FS . Definition 3.13 (Attractor of an AIIFS). A set A ∈ B (X ) is called an attractor of the affine infinite iterated function system S [n] if FS (A) = A and limn→∞ FS (B) = A, for all B ∈ B (X ), where the limit is considered with respect to h∥·∥ . Definition 3.14 (Convex Body). A set A ∈ B (X ), where (X , ∥·∥) is a normed space, is called a convex body of X if it is convex and has non-empty interior.
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Definition 3.15 (Strictly Topologically Contractive AIIFS). An affine infinite iterated function system S is called strictly topologically contractive if there exists a convex body K of X such that δ∥·∥ (FS (K ), X \ K ) > 0. Definition 3.16 (Topologically Contractive AIIFS). An affine infinite iterated function system S is called topologically ◦
contractive if there exists a convex body K of X such that FS (K ) ⊆ K . Remark 3.11. It is obvious that an affine infinite iterated function system which is uniformly point-fibred is point-fibred. Taking into account Lemma 2.3, with A = FS (K ) and B = X \ K , we infer that if an affine infinite iterated function system S ◦
is strictly topologically contractive, then it is topologically contractive. In particular, we have FS (K ) ⊆ K ⊆ K . The second example presented in Section 5 provides a point-fibred affine infinite iterated function system which is not uniformly point-fibred and a topologically contractive affine infinite iterated function system which is not strictly topologically contractive. However, in case that #(I ) < ∞ and dim(X ) < ∞, by comparing the result of R. Atkins, M. Barnsley, A. Vince and D. Wilson with our main result (see Theorem 4.1), one can see that an affine infinite iterated function system is uniformly point-fibred if and only if it is point-fibred and it is strictly topologically contractive if and only if it topologically contractive. 4. The main result Theorem 4.1. If S = ((X , ∥·∥), (fi )i∈I ) is an affine infinite iterated function system, then the following statements are equivalent: 1. 2. 3. 4. 5.
S is hyperbolic;
there exists a comparison function ϕ0 such that S is ϕ0 -hyperbolic; S has an attractor; S is strictly topologically contractive; S is uniformly point-fibred.
Proof. (1) ⇒ (2) Since S = ((X , ∥·∥), (fi )i∈I ) is hyperbolic, there exists a norm |||·||| on X equivalent to ∥·∥ having the property that the affine infinite iterated function system ((X , |||·|||), (fi )i∈I ) is contractive. Hence there exist α, β > 0 such that α ∥x∥ ≤ |||x||| ≤ β ∥x∥ for all x ∈ X and C ∈ [0, 1) such that |||Ai ||| ≤ C for each i ∈ I. Then the norm |||·|||1 given by 1 |||·||| is equivalent to |||·||| and therefore equivalent to ∥·∥. β
The function ϕ0 : [0, ∞) → [0, ∞) given by ϕ0 (t ) = Ct for all t ∈ [0, ∞) is a comparison function and since |||fi (x) − fi (y)|||1 = |||Ai (x) − Ai (y)|||1 = β1 |||Ai (x) − Ai (y)||| = β1 |||Ai (x − y)||| ≤ C β1 |||x − y||| = C |||x − y|||1 = ϕ0 (|||x − y|||1 ) for all x, y ∈ X and i ∈ I , ((X , |||·|||1 ), (fi )i∈I ) is ϕ0 -contractive, so S = ((X , ∥·∥), (fi )i∈I ) is ϕ0 -hyperbolic. (2) ⇒ (3) As S = ((X , ∥·∥), (fi )i∈I ) is ϕ0 -hyperbolic there exists a metric |||·||| on X equivalent to ∥·∥ such that the affine infinite iterated function system ((X , |||·|||), (fi )i∈I ) is ϕ0 -contractive, i.e.
|||fi (x) − fi (y)||| ≤ ϕ0 (|||x − y|||)
(∗)
for all x, y ∈ X and i ∈ I. We claim that h||·|| (FS (A), FS (B)) ≤ ϕ0 (h∥·∥ (A, B)), for all A, B ∈ B (X ). Indeed, using Proposition 2.1(v), for any A, B ∈ B (X ), we have h||·|| (FS (A), FS (B)) = h||·|| (∪i∈I fi (A), ∪i∈I fi (B)) ≤ sup h||·|| (fi (A), fi (B)) i∈I
= sup max{d||·|| (fi (A), fi (B)), d||·|| (fi (B), fi (A))} i∈I
= sup max sup inf |||fi (x) − fi (y)|||, sup inf |||fi (x) − fi (y)||| x∈A y∈B
i∈I
x∈B y∈A
and, taking into account (∗), we get
h||·|| (FS (A), FS (B)) ≤ max sup inf ϕ0 (|||x − y|||), sup inf ϕ0 (|||x − y|||) . x∈A y∈B
x∈B y∈A
Then, using Remark 3.6, we obtain
h||·|| (FS (A), FS (B)) ≤ max ϕ0
sup inf |||x − y||| , ϕ0 x∈A y∈B
sup inf ∥x − y∥ x∈B y∈A
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63
and, since ϕ0 is increasing, we conclude that h||·|| (FS (A), FS (B)) ≤ ϕ0
max sup inf |||x − y|||, sup inf ∥x − y∥ x∈B y∈A
x∈A y∈B
,
i.e. h||·|| (FS (A), FS (B)) ≤ ϕ0 (h∥·∥ (A, B)). Consequently the function FS : B||·|| (X ) → B||·|| (X ) is a ϕ0 -contraction and since (B||·|| (X ), h||·|| ) is a complete metric space, according to Corollary 3.1, we infer that there exists A ∈ B||·|| (X ) = B (X ) such that FS (A) = A and limn→∞ FS (B) = A, for all B ∈ B (X ), where the limit is considered with respect to h||·|| . As h∥·∥ and h||·|| are equivalent (see Lemma 2.8) we also have [n]
limn→∞ FS (B) = A, for all B ∈ B (X ), where the limit is considered with respect to h∥·∥ . [n]
(3) ⇒ (4) In accordance with the hypothesis, there exists a set A ∈ B (X ) such that FS (A) = A and limn→∞ FS[n] (B) = A, for all B ∈ B (X ), where the limit is considered with respect to h∥·∥ . In the sequel, we use the notations from Lemma 2.2. [n] Since limn→∞ FS (A1 ) = A, where the limit is considered with respect to h∥·∥ , there exists n0 ∈ N such that [n0 ]
h∥·∥ (FS
[n ]
(A1 ), A) < 41 . Then FS 0 (A1 ) ⊆ B∥·∥ [A, 41 ] = A 1 and therefore 4
fi1 ,i2 ,...,in0 (A1 ) ⊆ A 1 ,
(i)
4
for each i1 , i2 , . . . , in0 ∈ I. Then, using Lemma 2.1 and the continuity of the functions fi , we have [n0 ]
FS
(K1 ) ⊆ ∪i1 ,i2 ,...,in0 ∈I fi1 ,i2 ,...,in0 (K1 ) = ∪i1 ,i2 ,...,in0 ∈I fi1 ,i2 ,...,in0 (co(A1 )) ⊆ ∪i1 ,i2 ,...,in0 ∈I fi1 ,i2 ,...,in0 (co(A1 )) = ∪i1 ,i2 ,...,in0 ∈I co(fi1 ,i2 ,...,in0 (A1 )) (i)
⊆ ∪i1 ,i2 ,...,in0 ∈I co(A 1 ) = ∪i1 ,i2 ,...,in0 ∈I K 1 = K 1 4
4
4
and consequently [n0 ]
co(FS
(K1 )) ⊆ K 1 .
(∗)
4
We can consider the sets n 0 −1
M =
(co(FS[k] (K1 )) − co(FS[k] (K1 )))
k=0
and n 0 −1
N = K1 − K1 + 4
4
(co(FS[k] (K1 )) − co(FS[k] (K1 ))).
k=1
Let us note that M and M are bounded, balanced, convex neighbourhoods of 0,
B∥·∥
N,
1
4
n 0 −1 1 1 ⊆ B∥·∥ K 1 , − B∥·∥ K 1 , + (co(FS[k] (K1 )) − co(FS[k] (K1 ))) 4
4
4
4
(ii)
k =1
and
B∥·∥
K1 , 4
1 4
⊆ K1 .
(iii)
Here is the justification for (ii): if x ∈ B∥·∥ (N , 14 ) then there exist u ∈ N and y ∈ X such that x = u + y and ∥y∥ < Therefore there exist u1 , u2 ∈ K 1 and w ∈
n 0 −1 k=1
4
(co(FS[k] (K1 )) − co(FS[k] (K1 ))) such that x = u1 − u2 + w + y = s1 − s2 + w ,
where s1 = u1 + 12 y and s2 = u2 − 12 y. Since ∥s1 − u1 ∥ = ∥s2 − u2 ∥ = and consequently x ∈ B∥·∥ (K 1 , ) − B∥·∥ (K 1 , ) + 1 4
4
4
1 4
n0 −1 k=1 1 4
1 2
∥y ∥ <
1 8
< 14 , we infer that s1 , s2 ∈ B∥·∥ (K 1 , 14 )
4
Ai (M ) ⊆ N ⊆ B∥·∥ for each i ∈ I.
N,
1 4
⊆ M,
4
(co(FS (K1 )) − co(FS (K1 ))) and the proof of (ii) is done. [k]
[k]
For the justification of (iii), just note that B∥·∥ (K 1 , ) ⊆ B∥·∥ [K 1 , 14 ] and take, in Lemma 2.2, ρ = We claim that
1 . 4
4
1 4
,r =
1 4
and ε =
1 . 2
64
R. Miculescu, A. Mihail / J. Math. Anal. Appl. 407 (2013) 56–68
Indeed, using the linearity and continuity of Ai and the fact that B + x = B + x, for each B ⊆ X and x ∈ X , we have
n −1 0 [k] [k] Ai (co(FS (K1 )) − co(FS (K1 )))
Ai (M ) =
k=0 n0 −1
=
(Ai (co(FS[k] (K1 ))) − Ai (co(FS[k] (K1 ))))
k=0 n0 −1
⊆
(Ai (co(FS[k] (K1 ))) − Ai (co(FS[k] (K1 ))))
k=0 n0 −1
⊆
(Ai (co(FS[k] (K1 )) + bi ) − Ai (co(FS[k] (K1 )) + bi ))
k=0 n0 −1
= Lemma 2.1
=
(fi (co(FS[k] (K1 ))) − fi (co(FS[k] (K1 ))))
k=0 n0 −1
(co(fi (FS[k] (K1 ))) − co(fi (FS[k] (K1 ))))
k=0 n0 −1
⊆
(co(FS[k+1] (K1 )) − co(FS[k+1] (K1 )))
k=0
=
[n0 ]
co(FS
n0 −1
[n ]
(K1 )) − co(FS 0 (K1 )) +
(co(FS[k] (K1 )) − co(FS[k] (K1 )))
k=1 n 0 −1
(*)
⊆
K1 − K1 + 4
4
(ii)
⊆
B∥·∥
K1 , 4
1 (co(FS[k] (K1 )) − co(FS[k] (K1 ))) = N ⊆ B∥·∥ N , 4
k=1
1 4
− B∥·∥ K 1 , 4
1
n 0 −1
4
+
(co(FS[k] (K1 )) − co(FS[k] (K1 )))
k =1
n0 −1
(iii)
⊆
K1 − K1 +
(co(FS[k] (K1 )) − co(FS[k] (K1 )))
k=1 n0 −1
=
co(FS (K1 )) − co(FS (K1 )) + [0]
[0]
(co(FS[k] (K1 )) − co(FS[k] (K1 )))
k=1 n0 −1
=
(co(FS (K1 )) − co(FS (K1 ))) = M [k]
[k]
k=0
for each i ∈ I and the proof of our claim is done. Since ∥·∥M , the Minkowski norm associated with the M, is equivalent to ∥·∥, there exist a, b > 0 such that a ∥·∥ ≤ ∥·∥M ≤ b ∥·∥ and therefore, taking into account the above claim, we infer that Ai (M ) ⊆ N ⊆ B∥·∥M (N , 4a ) ⊆ B∥·∥ (N , 14 ) ⊆ M and hence δ∥·∥M (Ai (M ), X \ M ) ≥ 4a for each i ∈ I. Using Lemma 2.4, for A = Ai (M ), B = M and α = 4a , we get that
α Ai (M ) ⊆ (1 − θ )M, for each i ∈ I, where θ = 2(α+ = 2aa+8 . Now, using Lemma 2.7, for f = Ai , µ = 1 − θ = α0 < 1 1) and A = M, we get ∥Ai ∥M ≤ α0 for each i ∈ I. Let us consider ε > 0, r = supi∈I ∥bi ∥ ∈ R (see Remark 3.1) and c such that r +ε < c. 1−α0 Then not
fi (cM ) ⊆ (c − ε)M , for each i ∈ I. Then fi (cM ) ⊆ fi (cM ) ⊆ (c − ε)M = (c − ε)M ,
(iv)
for each i ∈ I. We claim that B∥·∥
M
fi (cM ),
for each i ∈ I.
ε 2
⊆ cM
(v)
R. Miculescu, A. Mihail / J. Math. Anal. Appl. 407 (2013) 56–68
65
Indeed, if x ∈ B∥·∥M (fi (cM ), 2ε ), then there exist y ∈ fi (cM ) and u ∈ X such that x = y + u and ∥u∥M < 2ε . Then, using (iv),
we have ∥x∥M ≤ ∥y∥M + ∥u∥M < c − ε + 2ε = c − 2ε < c, which implies that x ∈ cM = cM. Now we claim that B∥·∥
M
FS (cM ),
ε 4
⊆ cM .
(vi)
Indeed, if x ∈ B∥·∥ (FS (cM ), 4ε ), then there exist y ∈ FS (cM ) = ∪i∈I fi (cM ) and u ∈ X such that x = y + u and ∥u∥M < 4ε . M
As y ∈ ∪i∈I fi (cM ), there exists y1 ∈ ∪i∈I fi (cM ) such that ∥y − y1 ∥M < 4ε . Since y1 ∈ ∪i∈I fi (cM ), there exists i0 ∈ I such that y1 ∈ fi0 (cM ) and, with the notation y − y1 = w , we have x = y1 + w + u. As ∥w + u∥M ≤ ∥w∥M + ∥u∥M < 4ε + 4ε = 2ε , we infer that x ∈ B∥·∥ (fi0 (cM ), 2ε ). Taking into account (v), we conclude that x ∈ cM. M
Finally, we claim that
δ∥·∥ (FS (cM ), X \ cM ) > 0. Indeed, if it is not the case, then, as ∥·∥ and ∥·∥M are equivalent, we obtain that δ∥·∥M (FS (cM ), X \ cM ) = 0 < 4ε , i.e. infu∈FS (cM ),v∈X \cM ∥u − v∥M < 4ε which implies that there exist u0 ∈ FS (cM ) and v0 ∈ X \ cM such that ∥u0 − v0 ∥M < 4ε . Hence v0 ∈ B∥·∥ (FS (cM ), 4ε ) and, using (vi), we obtain the contradiction v0 ∈ cM. M
In conclusion, as cM is a convex body, S is strictly topologically contractive. (4) ⇒ (1) Since S is strictly topologically contractive, there exists a convex body K such that δ∥·∥ (FS (K ), X \ K ) > 0. def
If C = K − K , then C is a bounded, balanced, convex neighbourhood of 0 and ∥·∥C , the Minkowski norm associated with C , is equivalent to ∥·∥. (∗) Taking into account Lemma 2.6, for A1 = B1 = FS (K ) and A = B = K , we obtain that δ∥·∥C (FS (K ) − FS (K ), X \ C ) > 0. Since Ai (C ) = Ai (K ) − Ai (K ) = fi (K ) − fi (K ) ⊆ FS (K ) − FS (K ), with the notation α = δ∥·∥C (FS (K ) − FS (K ), X \ C ), we have δ∥·∥C (Ai (C ), X \ C ) ≥ α for each i ∈ I. Then, using Lemma 2.4, for A = Ai (C ) and B = C , we get that Ai (C ) ⊆ (1 − θ )C , for α . Therefore, taking into account Lemma 2.7, for f = Ai , A = C and µ = 1 − θ , we obtain that each i ∈ I, where θ = 2(α+ 1)
∥Ai ∥C ≤ 1 − θ ,
(∗∗)
for each i ∈ I. From (∗) and (∗∗) we conclude that S is hyperbolic. (1) ⇒ (5) Since S = ((X , ∥·∥), (fi )i∈I ) is hyperbolic, there exists a norm |||·||| on X equivalent to ∥·∥ having the property that the affine infinite iterated function system ((X , |||·|||), (fi )i∈I ) is contractive. Therefore there exist C ∈ [0, 1) and m > 0 such that |||Ai ||| = lip(fi ) ≤ C and |||bi ||| ≤ m for each i ∈ I. Then, given an arbitrary α = α1 α2 . . . αn . . . ∈ Λ, we have
||fα α ...α α (x) − fα α ...α (x)|| ≤ C n ||fα (x) − x|| n n+1 n 1 2 1 2 n+1 ≤ C n (||fαn+1 (x)|| + |||x|||) = C n (||Aαn+1 (x) + bαn+1 || + |||x|||) ≤ C n (||Aαn+1 (x)|| + ||bαn+1 || + |||x|||) ≤ C n ((C + 1) |||x||| + ||bαn+1 ||) ≤ C n ((C + 1) |||x||| + m), n for each n ∈ N and x ∈ X . Since the series C ((C + 1) |||x||| + m) is convergent, using Weierstrass M-Test, we infer that the series (fα1 α2 ...αn αn+1 (x) − fα1 α2 ...αn (x)) is |||·||| convergent (and therefore ∥·∥ convergent), hence there exists limn→∞ fα1 α2 ...αn (x) and let us denote it by xα . Hence limn→∞ fα1 α2 ...αn (x) = xα . As we have ||fα1 α2 ...αn (x) − fα1 α2 ...αn (y)|| ≤ C n |||x − y||| for each n ∈ N, x, y ∈ X , we infer that limn→∞ fα1 α2 ...αn (x) = limn→∞ fα1 α2 ...αn (y) for each x, y ∈ X . Therefore S = ((X , ∥·∥), (fi )i∈I ) is point-fibred. Moreover, given a ∥·∥ bounded subset B of X , then ∪i∈I fi (B) is ∥·∥ bounded, so B ∪ ∪i∈I fi (B) is ∥·∥ bounded and therefore, as |||·||| and ∥·∥ are equivalent, there exists δB > 0 such that |||y||| ≤ δB , for each y ∈ B ∪ ∪i∈I fi (B). Then, as above, for an arbitrary α = α1 α2 . . . αn . . . ∈ Λ, we have ||fα1 α2 ...αn αn+1 (x) − fα1 α2 ...αn (x)|| ≤ 2δB C n , for each n ∈ N and x ∈ B and, consequently, S = ((X , ∥·∥), (fi )i∈I ) is uniformly point-fibred. (5) ⇒ (3) Let us consider the set A = π (Λ). First, let us note that since fi (xα ) = xiα we get fi (π (Λ)) = π (iΛ) for any i ∈ I. Taking into account the fact that fi is continuous, we have fi (π (Λ)) ⊆ fi (π (Λ)) ⊆ π (iΛ) ⊆ π (Λ) for any i ∈ I, so ∪i∈I fi (π (Λ)) ⊆ π (Λ), and therefore ∪i∈I fi (A) ⊆ A.
(i)
66
R. Miculescu, A. Mihail / J. Math. Anal. Appl. 407 (2013) 56–68 n If x ∈ A = π (Λ) then there exists xn ∈ π (Λ) such that x = limn→∞ xn . Since there exists αn = α1n α2n . . . αm . . . such that
n xn = π (αn ) we have xn = π (α1n α2n . . . αm . . .) = fα1n (xα2n ...αmn ... ) ∈ fα1n (π (Λ)) ⊆ fα1n (π (Λ)) ⊆ ∪i∈I fi (A), so x ∈ ∪i∈I fi (A), i.e.
A ⊆ ∪i∈I fi (A).
(ii)
From (i) and (ii) we obtain A = ∪i∈I fi (A), i.e. FS (A) = A.
(1)
For a fixed B ∈ B (X ), according to the hypothesis, for each ε > 0 there exists nB,ε ∈ N such that for each n ∈ N, n ≥ nB,ε , x ∈ B and α = α1 α2 . . . αn . . . ∈ Λ we have fα1 α2 ...αn (x) − xα < 2ε , i.e.
fα
1 α2 ...αn
ε (x) − π (α) < .
(iii)
2
For each a ∈ A = π (Λ) there exists aε ∈ π (Λ) such that
ε
∥ a − aε ∥ <
(iv)
2
ε ε ε and sincethere exists αε = α1α2 . . . αn . . . ∈ Λ suchthat aε = π (αε ), using (iii) and (iv), for each n ∈ N, n ≥ nB,ε and x ∈ B ε ε ε ε ε ε we have fα1 α2 ...αn (x) − a ≤ fα1 α2 ...αn (x) − π (αε ) + ∥a − aε ∥ ≤ 2ε + 2ε = ε .
Hence for each ε > 0 there exists nB,ε ∈ N such that for each n ∈ N, n ≥ nB,ε we have A ⊆ B(FS (B), ε) and [n]
FS (B) ⊆ B(A, ε), so h∥·∥ (FS (B), A) ≤ ε , i.e. [n]
[n]
lim FS (B) = A [n]
(2)
n→∞
for each B ∈ B (X ). The relations (1) and (2) close the proof.
5. Examples Example 1. In [20] we constructed an affine infinite iterated function system defined on l2 (A) which has an attractor (namely Lipscomb’s space ωA ). Let us present the details of this construction. ′ Let A be an arbitrary set. Single out a point z of A and let us consider the set A = A \ {z }. The points of l2 (A) are ′
′
collections of real numbers indexed by points of A . If E is the set of real numbers, then x ∈ l2 (A) means x = {xa } ∈ E A 2 ′ such that xa = 0 for all but countable many a ∈ A and xa converges. The topology of l2 (A) is induced from the metric 2 d(x, y) = a (xa − ya ) , where we think xa as the a-th coordinate of x. Let us also consider, for the case when A is an arbitrary set with the discrete topology, the Baire space N (A) which is the topological product of countably many copies An of A. Hence the points of N (A) consist of all sequences v = a1 a2 . . . an . . . , with an ∈ A. Moreover N (A) is a metric space with the metric ′
d(v,v ) =
1, k
0,
′
if k is the first index where ai ̸= ai ′
if v = v .
Lipscomb’s space L(A) is a quotient space of N (A) such that each equivalence class consists of either a single point or two points. Those classes with two points come from identifying the point a1 a2 . . . ak−2 ak−1 ak ak . . . with the point a1 a2 . . . ak−2 ak ak−1 ak−1 . . ., where ak−1 ̸= ak . Therefore Lipscomb’s space L(A) is obtained via a projection or identification map p : N (A) → L(A). It is known that Lipscomb’s space L(A) can be embedded in Hilbert’s space l2 (A) by considering the function s : L(A) → l2 (A) given by s(α) = (αb )b∈A′ , where α = p(v), v = a1 a2 . . . . ∈ N (A) and
αb =
1
2k k with ak =b 0,
,
if there exists k such that ak = b if there exists no k such that ak = b.
Let ωA = s(L(A)) be the embedded version of L(A) with the l2 (A)-induced topology. For each a ∈ A \{z }, let fa : l2 (A) → l2 (A) be the function given by fa (x) = 21 (x + ua ), for all x ∈ l2 (A), and let fz : l2 (A) → l2 (A) be the function given by fz (x) = 21 x, for all x ∈ l2 (A), where ua = (αj )j∈A′ ∈ l2 (A) is described by αj = 0, for j ̸= a and αa = 1. Our result states that ωA , the
embedded version of L(A) endowed with the l2 (A)-induced topology, is the attractor of the affine infinite iterated function system S = (l2 (A), {fa }a∈A ). Therefore, according to our main result, S represents an uniformly point-fibred and strictly topologically contractive affine infinite iterated function system for which #(I ) = ∞ and dim(X ) = ∞.
R. Miculescu, A. Mihail / J. Math. Anal. Appl. 407 (2013) 56–68
67
Example 2. Let us consider the affine iterated function system S = ((c0 , ∥·∥∞ ), T ), where T : c0 → c0 is the linear operator 1 given by T ((xn )n≥1 ) = ((1 − n+ )xn )n≥1 for each (xn )n≥1 ∈ c0 . Note that T is bounded with ∥T ∥ ≤ 1 and 1 T
[m]
((xn )n≥1 ) =
1−
1
m
n+1
(∗)
xn n ≥1
for each m ∈ N and each (xn )n≥1 ∈ c0 . Let us also note that T is bijective and T −1 : c0 → c0 , which is given by 1 T −1 ((xn )n≥1 ) = (( n+ )xn )n≥1 for each (xn )n≥1 ∈ c0 , is continuous and therefore n T (A) = T (A)
(∗∗)
for each subset A of c0 . Claim 1. S is point-fibred and π : Λ → c0 is 0. Proof. It is enough to prove that limm→∞ T [m] ((xn )n≥1 ) = 0 for each (xn )n≥1 ∈ c0 . Indeed, given a fixed (xn )n≥1 ∈ c0 , since limn→∞ that for each n ∈ N, n > xn = 0, for each ε > 0 there exists nε ∈ N1 such 1 m m nε we have |xn | < ε and therefore (1 − n+ ) x < ε for each m ∈ N . Because lim ( 1 − ) x = limm→∞ (1 − 31 )m x2 = n m→∞ 1 1 2
· · · = limm→∞ (1 − nε1+1 )m xnε = 0 there exists mε ∈ N such that for each m ∈ N, m > mε we have (1 − 12 )m x1 < ε, (1 − 31 )m x2 < ε, . . . , (1 − nε1+1 )m xnε < ε. Consequently for each ε > 0 there exists mε ∈ N such that for each [m] 1 )m xn < ε, which means that limm→∞ T [m] ((xn )n≥1 ) = 0. m ∈ N, m > mε , we have T ((xn )n≥1 ) = supn≥1 (1 − n+ 1
Claim 2. S is not uniformly point-fibred. Proof. Indeed, if this is not the case, then, based on the proof of the implication (5) ⇒ (3) of our main result, we infer that limn→∞ h(T [n] (B[0, 1]), {0}) = 0, hence limn→∞ d(T [n] (B[0, 1]), {0}) = 0 and consequently limn→∞ supx∈B[0,1] d(T [n] (x), {0}) = 0. Therefore there exists n0 ∈ N such that supx∈B[0,1] d(T [n0 ] (x), {0}) < 12 . In particular, for em = (0, 0, . . . , 0, 1, 0, . . .) ∈ B[0, 1], where 1 is on the m-th position, we have d(T [n0 ] (em ), {0}) <
1
,
2 1 n0 = (1 − 1 )n0 < 1 , for each m ∈ N. By passing to limit as m goes to ∞ in the above ) e m m+1 m+1 2 inequality we obtain the contradiction 1 ≤ 21 .
(*) i.e. T [n0 ] (em ) = (1 −
Claim 3. S is topologically contractive. ◦
Proof. Indeed, since B[0, 1] is a convex body, it suffices to prove that T (B[0, 1]) ⊆ B[0, 1]. Taking into account (∗∗), the ◦
◦
above inclusion is the same with T (B[0, 1]) ⊆ B[0, 1], i.e. with T (B[0, 1]) ⊆ B[0, 1]. In order to prove the last inclusion, let us take an arbitrary element x = (xn )n≥1 from B[0, 1]. Then, as x ∈ c0 , there exists n0 ∈ N such that |xn | < 21 , for 1 each n ∈ N, n > n0 . Since (1 − n+ )xn ≤ (1 − n+1 1 ) ∥x∥ ≤ 1 − 1
1 , n 0 +1
1 for each n ∈ N, n ≤ n0 , and (1 − n+ )xn ≤ 1
not (1 − n+1 1 ) |xn | < 21 , for each n ∈ N, n > n0 , we infer that ∥T (x)∥ = ((1 − n+1 1 )xn )n≥1 ≤ max{1 − n01+1 , 12 } = εx . Then B(T (x), 1 − εx ) ⊆ B[0, 1] since if u ∈ B(T (x), 1 − εx ), then ∥u∥ ≤ ∥u − T (x)∥ + ∥T (x)∥ < 1 − εx + εx = 1, i.e. u ∈ B(0, 1). ◦ ◦ Consequently T (x) ∈ B[0, 1] for each x ∈ B[0, 1], i.e. T (B[0, 1]) ⊆ B[0, 1]. Claim 4. S is not strictly topologically contractive. Proof. Let us suppose, by contrary, that there exists a convex body K such that δ(FS (K ), X \ K ) > 0, i.e., based on (∗∗), ◦
such that δ(T (K ), X \ K ) > 0. Then, using Lemma 2.3, we get that T (K ) ⊆ K ⊆ K , and, taking into account Lemma 2.6, for A1 = B1 = T (K ) and A = B = K , we get that δ(T (K ) − T (K ), X \ (K − K )) > 0. Considering the set C = K − K , which is not
a convex, bounded, balanced neighbourhood of 0, the last inequality can be written as δ(T (C ), X \ C ) = ε0 > 0. Since C is ε bounded there exists M > 0 such that C ⊆ B[0, M ]. Let us consider η ∈ (0, 80 ) and n0 ∈ N such that n0 > 4M − 1. Note that ε 0
δ en0 ̸∈ C
(∗ ∗ ∗) for each δ > M since if this is not the case, then δ en0 ∈ C ⊆ B[0, M ] and we obtain the contradiction M < δ = δ en0 ≤ M. Therefore the non-empty set {γ > 0 | γ en0 ̸∈ C } is bounded from below by 0 and we can consider θ = inf{η > 0 | ηen0 ̸∈ C }. Let us note that (θ − η)en0 ∈ C ;
(i)
(θ + η)en0 ̸∈ C
(ii)
and
θ ≤ M.
(iii)
68
R. Miculescu, A. Mihail / J. Math. Anal. Appl. 407 (2013) 56–68
Indeed, if (i) would not be valid, then (θ − η)en0 ̸∈ C which implies the contradiction θ ≤ θ − η, i.e. η ≤ 0. If (ii) would not be valid, then (θ + η)en0 ∈ C . Since there exists λ ∈ (0, θ + η) such that λen0 ̸∈ C we obtain a contradiction in the λ λ following way: because 0 < θ+η < 1 and C is balanced, we infer that θ+η (θ + η)en0 ∈ C , i.e. λen0 ∈ C . For the justification of (iii), let us note that, taking into account (∗ ∗ ∗), we have θ ≤ δ for each δ > M. Therefore θ ≤ M. Finally we get the following contradiction
ε0
(i) and (ii)
≤ = = = (iii)
≤ i.e. ε0 ≤ 0.
T ((θ − η)en ) − (θ + η)en 0 0 (θ − η) 1 − 1 − (θ + η) e e n n 0 0 n0 + 1 (θ − η) 1 − 1 − (θ + η) en0 n0 + 1 η θ θ 1 n + 1 − n + 1 − 2η < η 2 − n + 1 + n + 1 0 0 0 0 ε0 ε0 ε0 M < + = , 2η + n0 + 1 4 4 2
Example 3. Let (en )n≥1 be the canonical basis of the Hilbert space l2 = {(xn )n≥1 |
n≥1
x2n < ∞} and let us consider the
function πm : l → l defined by πm ((xn )n≥1 ) = (yn )n≥1 , where ym = xm and yn = 0 if n ̸= m, for each (xn )n≥1 ∈ l2 . π (x)+e We claim that S = (l2 , (fn )n≥0 ), where f0 (x) = 2x and fn (x) = n 2 n , for each x ∈ l2 and each n ∈ N, is an affine infinite iterated function system having an attractor, namely the set ∪n≥1 [0, en ]. Indeed, fn is a contraction for each n, the family of function (fn )n≥0 is bounded and since ∪n≥1 [0, en ] is a non-empty, closed, bounded set, f0 (∪n≥1 [0, en ]) = ∪n≥1 [0, e2n ] and fn (∪n≥1 [0, en ]) = [ e2n , en ] for each n ≥ 1, we infer that FS (∪n≥1 [0, en ]) = ∪n≥1 [0, en ]. Hence, according with Theorem 2.2 from [23], ∪n≥1 [0, en ] is an attractor of S . 2
2
Acknowledgment We want to thank the referee whose generous and valuable remarks and comments brought improvements to the paper and enhanced clarity. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27]
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