Mediterr. J. Math. 9 (2012), 515–524 DOI 10.1007/s00009-011-0133-9 1660-5446/12/030515-10, published online February 17, 2011 © 2011 Springer Basel AG

Mediterranean Journal of Mathematics

Lipscomb’s L(A) Space Fractalized in lp(A) Alexandru Mihail and Radu Miculescu∗

Abstract. In this paper, by using some of our new results concerning the shift space for an infinite IFS (see A. Mihail and R. Miculescu, The shift space for an infinite iterated function system, Math. Rep. Bucur. 11 (2009), 21-32), we show that, for an infinite set A, the embedded version of the Lipscomb space L(A) in lp (A), p ∈ [1, ∞), with the metric induced from lp (A), denoted by ωpA , is the attractor of an infinite iterated function system comprising affine transformations of lp (A). In this way we provide a generalization of the positive answer that we gave to an open problem of J.C. Perry (see Lipscomb’s universal space is the attractor of an infinite iterated function system, Proc. Amer. Math. Soc. 124 (1996), 2479-2489) in one of our previous works (see R. Miculescu and A. Mihail, Lipscomb space ω A is the attractor of an infinite IFS containing affine transformations of l2 (A), Proc. Amer. Math. Soc. 136 (2008), 587–592). Moreover, as a byproduct, we provide a generalization of Corollary 15 from Perry’s paper by proving that ωpA is a closed subset of lp (A). Mathematics Subject Classification (2010). Primary 28A80, 37C70; Secondary 54H05. Keywords. Lipscomb’s space, infinite iterated function system, attractor of an infinite IFS.

1. Introduction Lipscomb space L(A) was introduced in order to solve the long standing problem of finding an analogue to N¨obeling’s ”classical universal space theorem” which yields the fact that all finite-dimensional separable metric spaces are modeled as subspaces of finite products of the unit interval. Actually L(A) yielded the correspondingly analogous result - all finite-dimensional metric spaces of weight |A| ≥ ℵ0 are modeled as subspaces of finite products of ∗ Corresponding

author.

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L(A) (see [3]). With the growth of fractal geometry (and its mathematical framework of attractors of finite IFSs), L(A) arguably provided the first notable example where an infinite IFS played a key role - Could it be that L(A) was the attractor (a generalized fractal) of an infinite IFS, just as classical fractals, e.g., Sierpinski’s triangle, are attractors of finite IFSs? S. L. Lipscomb and J. C. Perry (see [2]) showed that L(A) can be embedded in l2 (A) with the metric induced from l2 (A). J. C. Perry (see [6]) embedded L(A) in the Tikhonov cube I A and showed that this set with the (different) topology induced by I A is the attractor of an iterated function system containing an infinite number of affine transformations of I A . R. Miculescu and A. Mihail (see [4]) showed that the imbedded version of L(A) endowed with the l2 (A)-induced topology is the attractor of an infinite iterated function system comprising affine transformations of l2 (A). In the proof of this result we essentially used Corollary 15 from [6]. In this paper, by using a different technique, namely using some results concerning the shift space for an infinite IFS (see [5]), we show that, for an infinite set A, the embedded version of L(A) in lp (A), p ∈ [1, ∞), with the metric induced from lp (A), denoted by ωpA , is the attractor of an infinite iterated function system comprising affine transformations of lp (A). In this way we provide not only a generalization of the positive answer that we gave to an open problem of J.C. Perry (see [6]) in one of our previous works (see [4]), but also a generalization of Corollary 15 from the above mentioned Perry’s paper.

2. Preliminaries In this section we present the notations and the main results concerning the Hausdorff-Pompeiu semidistance and infinite iterated function systems. spaces. A family of Definition 2.1. Let (X, dX ) and (Y, dY ) be two metric  fi (A) is bounded, functions (fi )i∈I from X to Y is called bounded if the set for every bounded subset A of X.

i∈I

Definition 2.2. Let (X, d) be a metric space. For a function f : X → X, we consider the Lipschitz constant associated to f , given by Lip(f ) =

sup x,y∈X;x=y

d(f (x), f (y)) ∈ [0, +∞]. d(x, y)

A function f : X → X is called Lipschitz function if Lip(f ) < +∞ and it is called contraction if Lip(f ) < 1. Notation 2.3. For a set X, P(X) denotes the set of all subsets of X and by P ∗ (X) we mean P(X) − {∅} and, given a subset A ⊆ P(X), by A∗ we mean A − {∅}. Given a metric space (X, d), the set of bounded closed subsets of X is denoted by B(X).

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Definition 2.4. For a metric space (X, d), we consider on P ∗ (X) the generalized Hausdorff-Pompeiu pseudometric h : P ∗ (X) × P ∗ (X) → [0, +∞] defined by h(A, B) = max(d(A, B), d(B, A)), where   d(A, B) = sup d(x, B) = sup x∈A

x∈A

inf d(x, y) .

y∈B

For the properties of Hausdorff-Pompeiu semidistance one can consult the book [7]. Theorem 2.5. (see [1] and [4]) For a metric space (X, d) let h : P ∗ (X) × P ∗ (X) → [0, ∞] be the Hausdorff-Pompeiu semidistance. If (X, d) is complete, then (B ∗ (X), h) is a complete metric space. Definition 2.6. An infinite iterated function system (IIFS) on X consists of a bounded family of contractions (fi )i∈I on X such that supLip(fi ) < 1 and i∈I

it is denoted by S = (X, (fi )i∈I ). One can associate to an infinite iterated function system S = (X, (fi )i∈I ) the function FS : B ∗ (X) → B ∗ (X) given by  FS (B) = fi (B), i∈I

for all B ∈ B ∗ (X). Therefore, since Lip(FS ) ≤ supLip(fi ), FS is a contraction and using i∈I

the Banach’s contraction theorem, one can prove the following: Theorem 2.7. (see [4] and [5]) Given a complete metric space (X, d) and an IIFS S = (X, (fi )i∈I ), there exists a unique A(S) ∈ B ∗ (X) such that FS (A(S)) = A(S). Definition 2.8. In the frame of Theorem 2.7, A(S) is called the attractor associated to S. Remark 2.9. (see [4]) If T ∈ B ∗ (X) has the property that FS (T ) ⊆ T , then A(S) ⊆ T .

3. The shift space for an IIFS In this section we present the shift (or the code space) of an IIFS which generalizes the notion of the shift space for an iterated function system (see [5]). Terminology and notations. In the following N denotes the natural numbers and N∗ = N − {0}. Given two sets A and B, by B A we mean the set ∗ of functions from A to B. By Λ = Λ(B) we mean the set B N and by ∗ Λn = Λn (B) we mean the set B {1,2,...,n} . The elements of Λ = Λ(B) = B N are written as infinite words ω = ω1 ω2 . . . ωm ωm+1 . . . and the elements of

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Λn = Λn (B) = B {1,2,...,n} 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.By Λ∗ = Λ∗ (B) Λn (B) ∪ {λ}, we denote the set of all finite words Λ∗ = Λ∗ (B) = n∈N∗

where by λ we mean the empty word. 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 ∗ Λ = Λ(I) = (I)N , ω1 ω2 . . . ωm is denoted by [ω]m . For a nonvoid set I, on  β ∞  1, if x = y 1−δαkk we consider the metric dΛ (α, β) = , where δxy = . 3k 0, if x = y k=1 Definition 3.1. The complete metric space (Λ(I), dΛ ) is called the shift space associated to the infinite iterated function system S = (X, (fi )i∈I ). Notation 3.2. Let (X, d) be a metric space, S = (X, (fi )i∈I ) be an IIFS on not X and A = A(S) its attractor. For ω = ω1 ω2 . . . ωm ∈ Λm (I), we consider not not fω = fω1 ◦ fω2 ◦ . . . ◦ fωm and, for a subset H of X, Hω = fω (H). In particular Aω = fω (A). We also consider fλ = Id and Aλ = A. The following result is part of Theorem 4.1 from [5]. Theorem 3.3. Let S = (X, (fi )i∈I ) be an IIFS, where (X, d) is a complete not metric space and let A = A(S) be the attractor of S. Then: 1) For m ∈ N, we have A[ω]m+1 ⊆ A[ω]m , for all ω ∈ Λ = Λ(I), where by A[ω]0 we mean the set A, and lim d(A[ω]m ) = 0.

m→∞

2) For every a ∈ A and every ω ∈ Λ, we have  A[ω]m and lim f[ω]m (a) = aω . {aω } = m∈N∗

m→∞

3) The function π : Λ → A, defined by π(ω) = aω , for every ω ∈ Λ, has the following properties: i) π(Λ) = A;  fi (A), then π is onto. ii) if A = i∈I

Definition 3.4. The function π : Λ → A from the above theorem is called the canonical projection from the shift space on the attractor of the IIFS. Definition 3.5. A family of functions (fi )i∈I , where (X, d) is a metric space and fi : X → X, is called bi-Lipschitz if there exist α, β ∈ (0, ∞) such that α ≤ β and αd(x, y) ≤ d(fi (x), fi (y)) ≤ βd(x, y) for every x, y ∈ X and every i ∈ I.

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The following result is Corollary 6.1 from [5]. Theorem 3.6. Let (X, d) be a complete metric space, S = (X, (fi )i∈I ) an IIFS such that the family (fi )i∈I is bi-Lipschitz and A = A(S) the attractor of S. Assume that there exist ε > 0 and n ∈ N∗ such that for every different indexes i1 , i2 , . . . , in and every y1 ∈ Ai1 , y2 ∈ Ai2 , . . . , yn ∈ Ain , we have max {d(yk , yl ) | k, l ∈ {1, 2, . . . , n}} ≥ ε. It follows that A=



fi (A).

i∈I

4. The embedded version of the Lipscomb’s space in lp (A) Let A be an arbitrary infinite set and single out a point z of A. Let us  consider the set A = A − {z}. For a given p ∈ [1, ∞), the points of lp (A)  are collections of real numbers indexed by points of A . If E is the set of real  numbers, then x ∈ lp (A) means x = {xa } ∈ E A such that xa = 0 for all   p but countable many a ∈ A and |xa | converges. The topology of lp (A) is a   p1  p |xa − ya | , where we think xa as induced from the metric d(x, y) = a

the a-th coordinate of x. Our notation for lp (A) is inspired by the notation used in [6]. For the relation of this lp (A) with the ”standard” lp (A) one can consult A9.6 Proposition, page 211, from Stephen Leon Lipscomb, Fractals and Universal Spaces in Dimension Theory, Springer’s Monographs in Mathematics Series, 2009. 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 1   , if k is the first index where ai = ai . d(v,v ) = k  0, if v=v Let us note that N (A) = Λ(A) and that the metrics d and dΛ are equivalent. 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). For p ∈ [1, ∞), let us consider the function pp : N (A) → lp (A) given by pp (α) = (αb )b∈A ,

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where α = a1 a2 . . . ∈ N (A) and ⎧

1 ⎨ , if there exists k such that ak = b 2k . αb = k with a =b k ⎩ 0, if there exists no k such that ak = b The above function is well defined since taking into account the inequality

 xpn

≤

n

b∈A

k with ak =b

xn

,

n

where p ∈ [1, ∞) and xn ≥ 0, we obtain  p ⎛ 

p

1 ≤⎝ αb = 2k    b∈A

p

b∈A

k with ak =b

⎞p  p

1 1 ⎠ ≤ = 1. 2k 2k k

Let us note that the function sp : L(A) → lp (A) given by α) = pp (α), sp ( for every α  ∈ L(A) is well defined. The set pp (N (A)) = sp (L(A)) it denoted by ωpA . p Lemma 4.1. ΔA p = {x = (xb )b∈A ∈ l (A) : each xb ≥ 0 and

convex set.

 A

xb ≤ 1} is a

A Proof. If x = (xb )b∈A ∈ ΔA p , y = (yb )b∈A ∈ Δp and λ ∈ [0, 1], then

λxb + (1 − λ)yb ≥ 0 and

(xb + (1 − λ)yb ) = λ

A

xb + (1 − λ)

A

yb ≤ 1.

A

This means that z = λx + (1 − λ)y = (λxb + (1 − λ)yb )b∈A ∈ ΔA p, 

so ΔA p is a convex set. p Lemma 4.2. ΔA p is a closed subset of l (A). 

Proof. For a finite set J ⊆ A , the linear functional fJ defined by

fJ (x) = xb , J

for any x = (xb )b∈A ∈ l (A) is continuous since p

1

||fJ || ≤ |J| q , where |J| denotes the number of elements of J and p1 + 1q = 1. Let (xn = (xnb )b∈A )n be a sequence of elements from ΔA p convergent to a x = (xb )b∈A .

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For J = {b}, where b is an arbitrary element from A , we have xnb = f{b} (xn ) → f{b} (x) = xb . Because xnb ≥ 0 it results that xb ≥ 0. In a similar way it results that

xb ≤ 1

J 

for every finiteset J ⊆ A . Because xb = sup A







J⊆A ;J finite J

A

Therefore x ∈

ΔA p,

so

ΔA p

xb it results that xb ≤ 1.



is a closed subset of lp (A).



5. An iterated function system Let us consider, for an infinite set A, the complete metric space (lp (A), d) that was described in the previous section. For each a ∈ A, let fa : lp (A) → lp (A) be the function given by fa (x) = 12 (x + ua ), for all x ∈ lp (A), where A uz = 0lp (A) ∈ ΔA p and, for a ∈ A \ {z}, ua = (αj )j∈A ∈ Δp is described by αj = 0, for j = a and αa = 1. Let us note that, for each a ∈ A, fa is a 12 -contraction and that the family {fa }a∈A is bi-Lipschitz. Moreover, the family (fa )a∈A is bounded. Indeed, if K is a bounded subset of lp (A), then there existsr > 0 such that K ⊆ Br (0). Then fa (Br (0)) ⊆ B r+1 (0), for each a ∈ A, so fa (K) ⊆ 2 a∈A  fa (K) is bounded. B r+1 (0), i.e. 2

a∈A

Let us consider the IIFS S = (lp (A), (fa )a∈A ). According to Theorem 2.7, there exists a bounded closed non-empty subset M of lp (A), called the attractor of S, such that  fa (M ). M = FS (M ) = a∈A

Let us note that N (A) can be seen as the shift space for the IIFS S. Lemma 5.1. In the above frame we have 1 1 A A fa (ΔA p ) = u a + Δ p ⊆ Δp , 2 2 for all a ∈ A. Proof. It results from the definition of fa and from the convexity of ΔA p (see Lemma 4.1). 

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Lemma 5.2. In the above frame we have M ⊆ ΔA p. Proof. Taking into account Lemma 5.1, we have  A fa (ΔA p ) ⊆ Δp a∈A p and therefore since, according to Lemma 4.2, ΔA p is a closed subset of l (A), it results that  A FS (ΔA fa (ΔA p ) ⊆ Δp . p)= a∈A

From Remark 2.9, we obtain that 

M ⊆ ΔA p.

Lemma 5.3. In the above frame, for a, b and c distinct elements from A, if A A x ∈ fa (ΔA p ), y ∈ fb (Δp ) and z ∈ fc (Δp ), then max{d(x, y), d(x, z), d(y, z)} ≥

1 . 4

Proof. Let us consider that a, b, c ∈ A . 1 1 A If x = (xd )d∈A ∈ fa (ΔA p ) = 2 ua + 2 Δp , then xb + x c ≤ 1 − xa ≤ and therefore xb ≤ or

1 2

1 4

1 . 4 Without losing the generality, we can suppose that 1 xb ≤ . 4 1 1 A ) = Then if y = (yd )d∈A ∈ fb (ΔA p 2 ub + 2 Δp , let us note that xc ≤

d(x, y) ≥ |xb − yb | = yb − xb ≥ so

1 , 4

1 . 4 If z ∈ {a, b, c}, without losing the generality, we can suppose that a = z. 1 A If x = (xd )d∈A ∈ fa (ΔA p ) = 2 Δp , we have

1 xb + xc ≤ xd ≤ . 2  max{d(x, y), d(x, z), d(y, z)} ≥ d(x, y) ≥

A

As above, we conclude that max{d(x, y), d(x, z), d(y, z)} ≥

1 . 4



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Lemma 5.4. In the above frame M=



523

fa (M ).

a∈A

Proof. Since, according to Lemma 5.2, M ⊆ ΔA p , we have fa (M ) ⊆ fa (ΔA p ), for every a ∈ A. According to the previous Lemma, we have 1 , 4 for every x ∈ fa (M ), y ∈ fb (M ) and z ∈ fc (M ), where a, b, c are arbitrary disjoint elements of A. The conclusion results using Theorem 3.6.  max{d(x, y), d(x, z), d(y, z)} ≥

6. ωpA is the attractor of (lp (A), (fa )a∈A ) Theorem 6.1. With the above notations ωpA = pp (N (A)) = sp (L(A)) is the attractor of the IIFS S = (lp (A), (fa )a∈A ) and pp : N (A) → ωpA is the canonical projection from the shift space on the attractor of the IIFS S. Proof. We claim that sp ( α) = pp (α) = lim (fa1 ◦ fa2 ◦ . . . ◦ fan )(0lp (A) ), n→∞

for all α = a1 a2 . . . ∈ N (A). Indeed, for each n ∈ N, we have   d (fa1 ◦ fa2 ◦ . . . ◦ fan )(0lp (A) ), pp (α)   uan + 2uan−1 + . . . + 2n−1 ua1  =d , (α ) b b∈A 2n 1 p    p1

1 1 1 = n−1 . ≤ 2 1 − 2−p 2p(k−1) k>n

1 1 p n−1 ( 1−2−p ) n→∞ 2

Since lim

1

= 0, we obtain that

  pp (α) = lim (fa1 ◦ fa2 ◦ . . . ◦ fan ) 0lp (A) . n→∞

(6.1)

Let us note that (6.2) 0lp (A) ∈ M. 1 Indeed the 2 -contraction fz has the property that fz (M ) ⊆ M = ∅ and 0lp (A) is the fixed point of fz . Then, since 0lp (A) can be presented as the limit

of a sequence of points of the closed set M , we infer that 0lp (A) ∈ M . From (6.1) and (6.2), taking into account Theorem 3.3, 2), we get that pp (α) = π(α)

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and so pp (N (A)) = sp (L(A)) = π(N (A)). Using Lemma 5.4 and Theorem 3.3, 3), ii), we infer that π is onto and therefore M = π(N (A)). From the last two equalities, we obtain M = sp (L(A)).



Corollary 6.2. In the above framework, sp (L(A)) is a closed subset of lp (A). Remark 6.3. In the above proof of Theorem 6.1 we essentially use Theorem 3.3 (which is part of one of our results from [5]), in contrast with the proof of the particular case p = 2 that one can found in [4], where the particular case p = 2 of Corollary 6.1 (which is Corollary 15 from [6]) plays a crucial role.

References [1] M. F. Barnsley, Fractals everywhere, Academic Press, 1988. [2] S.L. Lipscomb and J.C. Perry, Lipscomb’s L(A) space fractalized in Hilbert’s l2 (A) space, Proc. Amer. Math. Soc. 115 (1992), 1157-1165. [3] S.L. Lipscomb, On imbedding finite-dimensional metric spaces, Trans. Amer. Math. Soc. 211 (1975), 143-160. [4] R. Miculescu and A. Mihail, Lipscomb space ω A is the attractor of an infinite IFS containing affine transformations of l2 (A), Proc. Amer. Math. Soc. 136 (2008), no. 2, 587–592. [5] A. Mihail and R. Miculescu, The shift space for an infinite iterated function system, Math. Rep. Bucur. 11(61) (2009), No. 1, 21-32. [6] J.C. Perry, Lipscomb’s universal space is the attractor of an infinite iterated function system, Proc. Amer. Math. Soc. 124 (1996), 2479-2489. [7] N.A. Secelean, M˘ asur˘ a ¸si Fractali, Editura Universit˘ a¸tii ”Lucian Blaga” din Sibiu, 2002. Alexandru Mihail and Radu Miculescu Department of Mathematics and Informatics Bucharest University Academiei Street, No. 14 Bucharest Romania e-mail: mihail [email protected] [email protected] Received: October 22, 2010. Accepted: January 10, 2011.