Attractors of Iterated Function Systems

proceedings of the american mathematical society Volume 116, Number I, September 1992

P. F. DUVALL, JR. AND L. S. HUSCH

(Communicated by James E. West)

Abstract. In this paper, the question of which compact metric spaces can be attractors of hyperbolic iterated function systems on Euclidean space is studied. It is shown that given any finite-dimensional compact metric X , there is a Cantor set C such that the disjoint union of C and X is an attractor. In the process, it is proved that every such X is the Lipschitz image of a Cantor set.

1. Introduction If X is a complete metric space and F — {f , f2, ... , fk} is a collection of contraction mappings of X to itself, then F is said to be a (hyperbolic) Iterated Function System. It is well known [4] that for such an F there is a unique compact set A such that A = |J,=1 f(A). A is called the attractor for F. In this paper, we are concerned with the question of which compact sets in Euclidean space can be realized as the attractors of some IFS. Williams [6] initially investigated the topological structure of attractors of IFS's. Hata [3] generalized Williams' work and showed that not every compactum can be realized as the attractor of an IFS, since, for example, a connected attractor must be locally connected. Hata posed the question whether every finite-dimensional locally connected continuum is the attractor of some IFS. Barnsley [ 1] showed to what extent every compactum can be approximated by an attractor of an IFS. As a special case, we note that it is not difficult to show that every compact subpolyhedron in R" is an attractor. We show (Theorem 4.1 ) that for each zz > 0 there is a Cantor set C" c R " with the property that if X is any compact set in R" , the set C" x {0} U {0} x X c R3" is the attractor of an IFS. It follows that every compact, finite dimensional metric space that contains a closed and open Cantor subset can be embedded in some Euclidean space as an attractor.

Received by the editors February 11, 1991. 1991 Mathematics Subject Classification. Primary 54H15; Secondary 54H20, 54E40. Key words and phrases. Iterated function systems, attractors, fractal geometry, contractions, Lipschitz mappings. This research was done while L. Husch was a Distinguished Visiting Professor at the University of North Carolina, Greensboro; support is gratefully acknowledged.

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Most of our notation is standard, although we shall use || • || to denote the sum norm on R" . That is, if x = (x, , x2, ... , xn) then ||x|| = |x, | + |x2| + ■■•+ 1*." n i

2. Lipschitz mappings of Cantor sets onto compacta A classical theorem of topology states that every compact metric space is the continuous image of the Cantor set. In this section, we construct, for each zz > 0, a Cantor set V" with the property that for every compactum IcR" there is a Lipschitz mapping of V" onto X . To begin, fix an zz > 1 and let B be the unit cube in R" spanned by the standard basis vectors. Subdivide B into cubes B¡, i £ Y, where Y = {0, 1, ... , 2" - 1} and F, = n"=0[e, > 1/2 + Ej] with ej beinß one half the y'th binary digit of i. Subdivide the F('s in a similar manner and continue the process so that at the zcth stage, B is the union of 2 nk cubes F, , . . For '\ll'"'k each zc > 1, give the space of zc-tuples of integers the discrete topology, let

vk= U B'tli£..tk*V\>h>—-,ik) it,i2,...,iker

and define pk : Vk+l -> Vk by pk(x ,ix,i2, ... , ik+l) = (x, z, , z'2, ... , ik). If i = {ik} is a sequence of integers with ik e T, let ^o=n^,z,.,t k=i In the next section, we will need the following observation, whose proof is left to the reader.

Proposition 2.1. If ik = xki+ xkfl -\-1- xkn2n~ , with xki £ {0, 1}, then the mth component of x(i) has the base 2 expansion -xlmx2mxim

Let V" be the inverse limit of the sequence

1 2

Note that we may view a point in V" as a pair (x, z), where i is a sequence of integers from Y and x = x(z'). Define a metric p on V" by

p((x, Î), {y, /)) = || x - y || + 1/2 ,

where || • || is the sum norm described in the introduction and zc + 1 is the first index in which i and j disagree (note that if z = j then x = y).

Proposition 2.2. V" is a Cantor set and p is a metric consistent with the inverse limit topology on V" .

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Proof. Recall that if {Xk} is a sequence of metric spaces with bounded metrics dk, then ¿(KM-vJ) = E%^ fc=i z2" is a metric on the product of the Xfs that induces the product topology. Let dk be the metric on Vk given by

dk((x, i,, i2,..., ik) ,(y,jx,j2,.--, jk)) = || x - y || + e ; where e = 0 if {ils i2, ... , ik) = {j{, j2, ... , jk) , 1 otherwise. Then, if (x, i) and (y, j) are in V" and k + 1 is the first index for which i and j disagree, the construction of d above gives

OO m m oo . diix,î),iy,])) = Y. ~m + E p m=\ m=k+\

-\\x-y\\ -r-^r2k = piix, Î), (y,/)).

Thus /> agrees with the product metric. To see that V" is a Cantor set, note that the map (x, i) —>i is a homeomorphism of V" onto the space of sequences of integers from Y. a Now let X be a compact subset of B, and define a mapping 0 : F" —».Y as follows. For each k-tuple (z, ,/,,..., i.) such that F, . , intersects X, 1 Z K' i|/2 ••■lt choose a point x, , , e F, , ,. ill. Given (x, z) € V" , define 'l'2'"'jt 'l'2'"'t - fx, , , if x 4. X and zc is the last index with X r\B¡ ¡ ,, / 0 , 0(x,z) = |;1^ if;fe^_

Theorem 2.1. 0 ¿j a Lipschitz map of V" onto X. Proof. Given a - (x, i), b = (y, j) e V" , there are three cases to consider. Case 1. Both x and y are in X. In this case, \\(a)- 4>(b)\\= \\ x - y \\< p(a, b). Case 2. x £ X , but y & X. Let rc + l be the first index in which i and j disagree, and let (b)= x, , . . Then k < I and hh'"Jt \\y-xJiJ2...jJib) = x, . ,. , '\'2'"'l hh'Jm and let k + 1 be the first index in which i and j disagree. We may assume that I < m, and we would like to have k < I. If m < k, then tf>{a)— tf>{b)

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and there is nothing to prove. By the definition of tp, the case I < k < m leads to a contradiction, so we assume k < I. Then ll^)-^)|| = ||x¡i;2...;;-xV2...yJ| <||x-y|| + ||x-x,iÍ2..,.J| + ||y-x,i;2...yJ|

< ||x-y ||+2zz/2A <2npia,b). Thus tp is a Lipschitz map with Lipschitz constant at most 2zz. g Since every compact set in R" can be moved into B by similarity transfor- mations, we have Corollary 2.1. If X is any compact set in R", there is a Lipschitz map of V" onto X.

3. Standard Cantor sets Let C1 be the set of points in R of the form (x, y), where x, y e [0, 1] have base 2 expansions of the form x = .x,x2x3 • • • and y = .yxy2y3 ■■■ , with x; = 0 for z even and y( = 0 for z odd. Define contractions h0 and hx on R2 by tyx.jO-g.f) and h,ix,y)=(f + ^f).

Proposition 3.1. C ¿5 a Cantor set and is the attractor for the IFS F =

Proof It is easy to check that h0iCl) and hfC1) are disjoint and that their union is C . Thus C is the attractor and the disjointness implies that it is a Cantor set [3, Theorem 4.4]. d If F = {f, f2, ... , fk} and G = {g{, g2, ... , g,} are IFSs on the complete metric spaces X and Y with attractors A and B , respectively, define the IFS F x G on X x Y to be the set of all products f x g.. The reader can easily verify the following proposition. Proposition 3.2. The attractor of F x G is X x Y.

It follows that for each zz > 1 the zz-fold product C" = C x C x ■■ ■ x C is a Cantor set in R " that is the attractor of the IFS F" = F xF x ■■ ■ x F . We note in passing that since F" consists of 2" similitudes with similarity constant \ , the Hausdorff dimension of C" is zz [2]. In the previous section, we showed that each compactum in a finite-dimensional Euclidean space is the Lipschitz image of an abstractly defined Cantor set. The purpose of the present construction is to be able to replace V" by the more geometrically appealing C" . Again, we need some preliminary notation. If u £ Cn, we can represent u as a 2z?-tuple u = (x , y , x , y , ... , x" , y") where each xm has the binary expansion xm — .x™x2 ■■ ■ and xk = 0 if k is even and the ym have similar expansions with yk = 0 for k odd. Then we

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can define a mapping ip: C" —>V" by y/{ü) = (x(z'), i), where

n m , m i ._ v Xk +yk m=\ L

Theorem 3.1. The map ip: C" —►Vn is a Lipschitz equivalence.

Proof. It follows from the uniqueness of representation of integers and the special properties of the points in C" that \p is one-to-one and onto. Let u be as above, let v = (w , z , w , z , ... , wn , z") be a second point, and let y/(u) = (a, i), ip{v) = ib, j). By Proposition 2.1 the mth coordinate of a is xm + ym and the znth coordinate of b is wm + zm , so we have

Il LU V^ I m , m,; m m. IIa-b\\ = i^\x +y W - Z \ m=\ n <2^|x, v^ | m -w m, | + , |yi m -z m, | = ||m-ü||.,. .. m=\

Suppose that k + 1 is the first index in which i and j disagree. Since a and b must have some coordinates that disagree in the zc+ 1st position of their binary expansions, ||a - Zz||> 1/2 +1 . Therefore,

piwiu), y/{v)) = \\a - b\\ + ~2k < 3\\a - b\\ < 3\\u - v\\.

On the other hand,

m V^ i rn m, , , m m, . ¿-ri2zz '—'' " '2 m=l

since xm, wm and ym, zm agree in at least their first zc places. Therefore 2zz ||w - tz|| < 2zz||a - /3|| + —r2k = 2np(ipiu), y/{v)). G

Corollary 3.1. If X is a compact set in R" , there is a Lipschitz map of Cn onto X.

4. Constructing IFSs Recall the contractions h0 and /z, on R constructed in §3. For an zz > 1 we index the members of the IFS Fn as follows. For an integer i £ Y, let z = y^tln x, 2 be its binary representation, and let /, = /z, x A x ■• ■x A

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Proposition 4.1. Each /,,..., is a contraction of R " with contractivity constant 1/2* and c= U fl]l2-,k(c") i,,i2,...,ik€T for each k. Given a compact set X cRn with 0 £ X, let C + X denote the union C + X = {Cn x {0}) U ({0} x X) c R3".

Theorem 4.1. There is an IFS on R " whose attractor is C + X. Proof. Write R3" as R2" x R" and let A : C" -> X be a Lipschitz map with Lipschitz constant L. By the "Lipschitz Tietze Theorem" [5], we may assume that X is defined on all of R " . For each i £ Y let fl be defined by fi{x,y) = {fi{x),0). Choose a k such that L/2 < 1 and for each zc-tuple i{, i2, ... , ik £ Y, let

gll...l(x,y) = (0,Afii...i(x)). 'l'2 'k 'l'2 'k Then the collection Ulzenu^^Jz,,/,,... ,ik} is an IFS whose attractor is C + X. a

References 1. M. Barnsley, Fractals everywhere, Academic Press, Boston, MA, 1988. 2. K. Falconer, Fractal geometry, Wiley, New York, 1990. 3. M. Hata, On the structure of self-similar sets, Japan J. Appl. Math. 2 (1985), 381-414. 4. J. E. Hutchinson, Fractals and self similarity, Indiana Univ. Math. J. 30 (1981), 713-747. 5. J. Luukkainen and J. Väisälä, Elements of Lipschitz topology, Ann. Acad. Sei. Fenn. Ser. A 3 (1977), 85-122. 6. R. F. Williams, Composition of contractions, Bol. Soc. Brasil. Mat. 2 (1971), 55-59.

Department of Mathematics, University of North Carolina at Greensboro, Greens- boro, North Carolina 27412 E-mail address: [email protected]

Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996 E-mail address: [email protected]

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