This paper presents a sufficient condition for a continuum in $R^n$ to be
embeddable in $R^n$ in such a way that its image is not an attractor of any
iterated function system. An example of a continuum in $R^2$ that is not an
attractor of any weakly contracting iterated function system is also given

Kulczycki, Marcin

Nowak, Magdalena

Open Mathematics

English

arXiv

Jagiellonian Univeristy Repository

Cent. Eur. J. Math. • 10(6) • 2012 • 2073-2076DOI: 10.2478/s11533-012-0108-5Central European Journal of MathematicsA class of continua that are not attractors of any IFSResearch ArticleMarcin Kulczycki1∗, Magdalena Nowak1,2†1 Institute of Mathematics, Faculty of Mathematics and Computer Science, Jagiellonian University, Łojasiewicza 6,30-348 Kraków, Poland2 Institute of Mathematics, Jan Kochanowski University, Świętokrzyska 15, 25-406 Kielce, PolandReceived 15 February 2012; accepted 22 May 2012Abstract: This paper presents a sufficient condition for a continuum in Rn to be embeddable in Rn in such a way that itsimage is not an attractor of any iterated function system. An example of a continuum in R2 that is not an attractorof any weak iterated function system is also given.MSC: 28A80, 54F15, 37B25, 54H20Keywords: Fractal • Continuum • Iterated function system • Attractor© Versita Sp. z o.o.1. IntroductionThe notion of an iterated function system (abbrev. IFS), introduced by John Hutchinson in 1981 [4], has proven to be afertile field of research as well as a versatile and useful tool in lossy data compression (especially where image data isconcerned). This paper is a study in one specific aspect of the theory  the possibility of encoding a particular set asan attractor of an IFS. We now recall some basic terminology.Let (X, d) be a complete metric space. A map f : X → X is called a contraction if there exists a constant λ ∈ [0, 1) suchthat for every x, y ∈ X we have d(f(x), f(y)) ≤ λd(x, y). A map f : X → X is called contractive if for every x, y ∈ X ,x 6= y, we have d(f(x), f(y)) < d(x, y), see [2]. A family F = {f1, . . . , fn} of (contractive maps) contractions fi : X → X iscalled a (weak) iterated function system, see [1]. Given a compact B ⊂ X , defineF (B) = n⋃i=1 fi(B).∗ E-mail: Marcin.Kulczycki@im.uj.edu.pl† E-mail: magdalena.nowak805@gmail.com2073Brought to you by | Uniwersytet JagiellonskiAuthenticatedDownload Date | 10/16/18 10:01 AMA class of continua that are not attractors of any IFSThis transformation, acting on the space of nonempty compact subsets of X with the Hausdorff metric, is called theBarnsley–Hutchinson operator. A fixed point of F is called the attractor of (weak) IFS.It is shown in [4] that every IFS has a unique attractor (an analogous fact may not be true for a weak IFS, unlesscompactness of X is assumed). M. Hata proved in [3] that if the attractor of some IFS is connected, then it is also locallyconnected. M.J. Sanders showed in [7] that every arc of finite length is an attractor for some IFS. Additionally, he hasproven that if a is an endpoint of some arc A ⊂ Rn which has the properties:(1) for all x, y ∈ A \ {a} the length of the subarc of A with endpoints x and y is finite,(2) for every x ∈ A \ {a} the length of the subarc of A with endpoints x and a is infinite,then A is not an attractor of any IFS acting on Rn. One example of such an arc is the harmonic spiral [6]. The exampleof M. Kwieciński from [5] may also be easily modified to satisfy these assumptions.It is elementary to check that every continuum in R is an attractor of some IFS. Moreover, any embedding of suchcontinuum in R still is an attractor of some IFS. In dimension two and higher, however, the situation becomes morecomplex. Our results provide a sufficient condition for a continuum to be embeddable in Rn so that its image is not anattractor of any IFS.2. Main resultsDefinition 2.1.Let (X, d) be a metric space, A ⊂ X , x, y ∈ A, and ε > 0. Consider all the sequences x1, . . . , xk such that k ∈ N, x1 = x,xk = y, xi ∈ A, d(xi, xi+1) < ε. Denote by d̃(x, y, A, ε) the infimum of the sums ∑k−1i=1 d(xi, xi+1) for these sequences.Define d̃(x, y, A) = limε↘0 d̃(x, y, A, ε). This limit may be infinite.It is elementary that if A ⊂ B then d̃(x, y, A) ≥ d̃(x, y, B).Theorem 2.2.Let n ≥ 2. Let C ⊂ Rn be a continuum. Assume that there exists an (n−1)-dimensional hyperplane B ⊂ Rn such thatB ∩ C = {p} and C \ {p} is connected. Assume additionally that for every x, y ∈ C \ {p} there exists Uxy which is aneighbourhood of p such that d̃(x, y, C \Uxy) < +∞. Then there exists an embedding h : C → Rn such that h(C ) is notan attractor of any IFS.Proof. By applying an affine transformation we may assume without loss of generality that B = {0}×Rn−1,p = (0, . . . , 0), and C ⊂ [0, 1]× [−1, 1]n−1. Next define h1, h2 : Rn → Rn ash1(x1, . . . , xn) = (x1, x1100 x2, . . . , x1100 xn), h2(x1, . . . , xn) = (x1, √x1 sin x−11 + x2, x3, . . . , xn).Then define the embedding h : C → Rn as the composition h2 ◦ h1.Roughly speaking, h1 transforms C into a sharp needle, while h2 bends that needle to fit into a thickened-up graph ofthe function √x sin x−1. As a result of the second transformation the needle becomes, speaking imprecisely, of infinitelength. Figure 1 illustrates the process for n = 2.The map h1 does not increase distance, and therefore for every x, y ∈ h1(C \{p}) there exists U1xy which is a neighbour-hood of h1(p) such that d̃(x, y, h1(C )\U1xy) < +∞. Note that, outside of any neighbourhood U of h1(p), the expansivityconstant of h2h1(C )\U is bounded from above. This implies that for every x, y ∈ h(C \{p}) there exists U2xy which is aneighbourhood of h(p) such that d̃(x, y, h(C )\U2xy) < +∞.2074Brought to you by | Uniwersytet JagiellonskiAuthenticatedDownload Date | 10/16/18 10:01 AMM. Kulczycki, M. NowakFigure 1. The map h for n = 2Consider now a contraction f : h(C )→ h(C ) with a Lipschitz constant λ < 1. We would like to prove that if h(p) ∈ f(h(C ))then f(h(C )) = {h(p)}. To this end, conjecture that f is not constant and h(p) ∈ f(h(C )).Assume first that f(h(p)) = h(p). Fix any x ∈ h(C ) such that f(x) 6= h(p). Note that the sequence x, f(x), f2(x), . . . isconvergent to h(p). Also note that, by the assumptions, d̃(x, f(x), h(C )) is finite and additionally d̃(fi(x), f i+1(x), h(C )) ≤λid̃(x, f(x), h(C )). But this would imply that d̃(x, h(p), h(C )) is also finite, while it is not, since it can be seen from thedefinition of h2 that d̃(x, h(p), h([0, 1]× [−1, 1]n−1)) is infinite.If, on the other hand, f(h(p)) 6= h(p) then there exist x ∈ h(C ) such that f(x) = h(p) and y ∈ h(C ) \ {h(p)} suchthat f(y) 6= h(p). Then d̃(x, y, h(C )) would be finite and d̃(f(x), f(y), h(C )) would be infinite, which contradicts thecontractiveness of f , completing the proof that if f takes value h(p) on at least one argument then it has to be constant.Consequently, if F is the Barnsley–Hutchinson operator for some IFS and F (h(C )) ⊂ h(C ), then F (h(C )) may compriseof {h(p)} and possibly also finitely many other closed sets not containing h(p). But then F (h(C )) 6= h(C ), proving thath(C ) is not an attractor of F .Remark 2.3.The assumptions of Theorem 2.2 are technical and may seem very restrictive. Its assertion, however, is true not onlyfor the continua that satisfy them directly, but also for the continua that are homeomorphic to subsets of Rn satisfyingthese assumptions. This significantly widens the class of sets the theorem is useful for. For example, if any two pointsin the continuum A ⊂ Rn can be connected in A by a path of finite length, then it can be easily seen that any one-pointunion of A and [0, 1] is homeomorphic to a subset of Rn+1 for which the assumptions of Theorem 2.2 are satisfied.After the result of Hata [3] it has been an open problem whether every locally connected continuum in Rn is an attractorof some IFS. The example of Kwieciński [5] provided a negative answer, but the same question for weak IFS’s remains,to our best knowledge, open. We shall now give an example of a subcontinuum of R2 that is not an attractor of anyweak IFS.Definition 2.4.In this definition we switch to the standard polar coordinate system (r, θ) on R2, that is (x, y) = (r cosθ, r sinθ).Put p0 = (0, 0) and pn = (2−n, 2−n) for n ≥ 1. For any n ≥ 1 choose a broken line segment ln without self-intersections, consisting of finitely many intervals, that starts at p0, ends at pn, has the total length of 2n, and iscontained in [[0, 2−n)× (2−n − 2−n−2, 2−n + 2−n−2)] ∪ {pn}. Define P = ⋃∞i=1 li.Theorem 2.5.The space P as a subset of R2 with the Euclidean metric is not an attractor of any weak IFS.Proof. Suppose that f : P → P is contractive. We shall examine how many of the points pi can belong to f(P). Iff(p0) 6= p0 then there is a neighbourhood U of f(p0) such that the distance d(p0, U) > 0 and U contains finitely manypoints pi and almost all of the sets f(li). Note that only finitely many of the sets f(li) may reach the outside of U .Also observe that each f(li) covers at most finitely many points pi because the lengths of li are not increased by f (thiselementary property of contractions can be proven either by using δ-chains, or, as in [5], by using the fact that f doesnot increase one-dimensional measure). Consequently, only finitely many of the points pi belong to f(P).Brought to you by | Uniwersytet JagiellonskiAuthenticatedDownload Date | 10/16/18 10:01 AMA class of continua that are not attractors of any IFSFigure 2. The space PIf, on the other hand, f(p0) = p0, then, given n ≥ 1, note that pn may not belong to f(li) for i < n, because the lengthsof these sets are too small to traverse the whole ln. But no other point in P can be mapped onto pn by f , because fdecreases the distance between p0 and any other point. Therefore, the only point pi present in f(P) is p0. In conclusion,if F is a weak IFS, then only finitely many of the points pi can belong to F (P), and therefore P is not an attractorof F .AcknowledgementsThe authors would like to thank the referees for their thorough work and suggesting several important clarifications.The second author was supported by the ESF Human Capital Operational Programme grant 6/1/8.2.1/POKL/2009.References[1] Barnsley M., Fractals Everywhere, Academic Press, Boston, 1988[2] Edelstein M., On fixed and periodic points under contractive mappings, J. Lond. Math. Soc., 1962, 37, 74–79[3] Hata M., On the structure of self-similar sets, Japan J. Appl. Math., 1985, 2(2), 381–414[4] Hutchinson J.E., Fractals and self similarity, Indiana Univ. Math. J., 1981, 30(5), 713–747[5] Kwieciński M., A locally connected continuum which is not an IFS attractor, Bull. Pol. Acad. Sci. Math., 1999, 47(2),127–132[6] Sanders M.J., Non-attractors of iterated fuction systems, Texas Project NexT Journal, 2003, 1, 1–9[7] Sanders M.J., An n-cell in Rn+1 that is not the attractor of any IFS on Rn+1, Missouri J. Math. Sci., 2009, 21(1),13–20Brought to you by | Uniwersytet JagiellonskiAuthenticatedDownload Date | 10/16/18 10:01 AM

Central European Journal of Mathematics

A class of continua that are not attractors of any IFS

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A class of continua that are not attractors of any IFS

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