A 1-dimensional Peano continuum which is not an IFS attractor

A 1-DIMENSIONAL PEANO CONTINUUM WHICH IS NOT AN IFS A TTRA CTOR T A R AS BANAKH AND MAG DALENA NO W AK Abstract. Answe ri ng an old question of M.Hata, we construct an example of a 1-dimensional P eano con tin uum which is not homeomorphic to an attractor of IFS. 1. Introduction A compact metric space X is called an IFS-attr actor if X = S n i =1 f i ( X ) for some contracting self-maps f 1 , . . . , f n : X → X . In this case the family { f 1 , . . . , f n } is called an iter ate d function system (briefly , an IFS), see [2]. W e recall that a map f : X → X is c ontr acting if its Lipsc hitz constant Lip( f ) = sup x 6 = y d ( f i ( x ) , f i ( y )) d ( x, y ) is less than 1. A ttracto rs of IFS app ear natura lly in the Theory of F r actals, see [2], [3]. T opo- logical pro p erties of IFS-attra ctors were studied by M.Hata in [4]. In par ticular, he observed that ea ch connected IFS-a ttractor X is loca lly connec ted. The reason is that X has pr op erty S. W e recall [6, 8.2] that a metr ic space X has pr op erty S if for every ε > 0 the space X can be cov ered by finite num be r of co nnected subsets of diameter < ε . It is well-known [6, 8 .4] that a connected compact metric spa ce X is lo cally co nnected if a nd only if it has prop erty S if and o nly if X is a Pe ano c ontinuum (whic h means that X is the co n tinuous image of the int er v al [0 , 1]). Therefore, a compac t space X is not homeomorphic to a n IFS-attractor whenever X is connected but not locally connected. Now it is natural t o ask if there is a P eano contin uum homeomorphic to no IFS- attractor. An eas y answ er is “Y es” as every IFS-attractor has finite to po logical dimension, see [3]. Consequently , no infinite- dimensional compact top ologica l space is homeomorphic to an IFS-attrac tor. In such a wa y w e arr ive to the following q uestion p osed by M. Hata in [4]. Problem 1.1. Is e ach finite-dimensional Pe ano c ontinuum home omorphi c to an IFS-attr actor? In this pap er we shall give a negativ e answer to this question. Our counterexam- ple is a rim-finite plane P eano con tinuum . A top ologica l space X is called rim-finite if it has a base of the top olo gy co nsisting o f op en sets with finite b oundaries . I t follows tha t each co mpact rim-finite space X ha s dimension dim( X ) ≤ 1. 2000 M athematics Subje ct Classific ation. Primary 28A8 0; 54D05; 54F50; 54F45. Key wor ds and phr ases. F ractal, P eano contin uum, Iterated F unction System, IFS-attract or. The second author was supp orted in part by PHD f ellowships i mpor tan t for regional dev elopment . 1 2 T ARAS BANAKH AND MAG DALENA NO W AK Theorem 1.2. Ther e is a rim-finite plane Pe ano c ontinu um home omorphic to no IFS-attr actor. It should b e mentioned that an ex ample of a Peano contin uum K ⊂ R 2 , which is not isometric to a n IFS-attractor was constructed by M.Kwieci ´ nski in [5]. Ho wev er the contin uum of K wieci ´ nski is home omorphic to an IFS-attractor , so it do es not give a n answer to Pro blem 1.1. 2. S-dimension of IFS -a ttractors In order to prov e Theorem 1.2 w e shall observe that eac h c onnected IFS-attrac tor has finite S -dimension. This dimension w as introduced and studied in [1]. The S -dimensio n S-Dim( X ) is defined for each metric space X with prop erty S . F or each ε > 0 denote by S ε ( X ) the smallest n umber of connected subsets of diameter < ε that cover the spac e X and le t S-Dim( X ) = lim ε → +0 − ln S ε ( X ) ln ε . F or each Peano con tinuum X we can also consider a top o logical in v ariant S-dim( X ) = inf { S-Dim( X , d ) : d is a metric generating the topolog y of X } . By [1, 5.1], S- dim( X ) ≥ dim( X ), where dim( X ) sta nds for the cov ering top olo gical dimension of X . Theorem 2.1. Ass u me that a c onne cte d c omp act metric sp ac e X is an attr actor of a n IFS f 1 , f 2 , . . . , f n : X → X with c ontr acting c onst ant λ = max i ≤ n Lip( f i ) < 1 . Then X has fi nite S - dimensions S-dim( X ) ≤ S-Dim( X ) ≤ − ln( n ) ln( λ ) . Pr o of. The ine quality S- dim( X ) ≤ S-Dim( X ) follows from the definition of the S -dimension S-dim( X ). The inequalit y S-Dim( X ) ≤ − ln( n ) ln( λ ) will follo w as so on as for every δ > 0 we find ε 0 > 0 such that for ev ery ε ∈ (0 , ε 0 ] we get − ln S ε ( X ) ln ε < − ln( n ) ln( λ ) + δ. Let D = dia m( X ) be the diameter of the metric space X . Since lim k →∞ ln( n k ) ln( λ k − 1 D ) = lim k →∞ k ln( n ) ( k − 1 ) ln( λ ) + ln D = ln( n ) ln( λ ) , there is k 0 ∈ N such that for e ach k ≥ k 0 we get − ln( n k ) ln( λ k − 1 D ) < − ln( n ) ln( λ ) + δ. W e claim that the n umber ε 0 = λ k 0 − 1 D has the required prop erty . Indeed, giv en any ε ∈ (0 , ε 0 ] we can find k ≥ k 0 with λ k D < ε ≤ λ k − 1 D and observe that C k =  f i 1 ◦ · · · ◦ f i k ( X ) : i 1 , . . . , i k ∈ { 1 , . . . , n }  is a cov er o f X b y compact connected subsets ha ving diameter ≤ λ k D < ε . Then S ε ( X ) ≤ |C k | ≤ n k and − ln( S ε ( X )) ln( ε ) ≤ − ln( n k ) ln( λ k − 1 D ) < − ln( n ) ln( λ ) + δ. A 1-DIMENSIONAL PEANO CONTINUUM WHICH IS NOT AN IFS A TTRAC TOR 3  In t he next section we shall construct an example of a r im-finite plane Peano con- tin uum M with infinit e S -dimension S-dim( M ). Theorem 2.1 implies that the space M is not homeo morphic to an IFS-attractor and this pro ves Theor em 1.2. 3. The sp ace M Our space M is a partial case of the s paces constructed in [1] and c alled ”shark teeth”. Consider the piecewise linear perio dic function ϕ ( t ) = ( t − n if t ∈ [ n, n + 1 2 ] for some n ∈ Z , n − t if t ∈ [ n − 1 2 , n ] for some n ∈ Z , whose graph loo ks as follows: ✲ t ✻ ❅ ❅   ❅ ❅ ❅ ❅   ❅ ❅ ❅ ❅   ❅ ❅ ❅ ❅  F or every n ∈ N consider the function ϕ n ( t ) = 2 − n ϕ (2 n t ) , which is a homothetic cop y of the function ϕ ( t ). Consider the non-decrea sing s equence n k = ⌊ log 2 log 2 ( k + 1) ⌋ , k ∈ N , where ⌊ x ⌋ is the in teger part of x . Our example is th e co nt inuum M = [0 , 1] × { 0 } ∪ ∞ [ k =1  t, 1 k ϕ n k ( t )  : t ∈ [0 , 1]  in the plane R 2 , which lo oks as follows: Figure 1. The cont inuum M 4 T ARAS BANAKH AND M AG DALENA NOW AK The following theorem describ es some prop erties of the contin uum M and implies Theorem 1.2 stated in the in tro duction. Theorem 3.1. The sp ac e M has the fol lowing pr op erties: (1) M is a rim-finite plane Pe ano c ontinuum ; (2) dim( M ) = 1 and S-dim( X ) = ∞ ; (3) M is not home omorphic to an IFS attr actor. Pr o of. It is easy to see tha t X is a r im-finite plane P eano cont inuum. The rim- finiteness of M implies tha t dim ( M ) = 1 . T o s how that S-dim( M ) = ∞ , co nsider the n umber sequence ~ m = (2 n k ) ∞ k =1 and observe that the space M is homeomorphic to the “shark teeth” space W ~ m considered in [1]. T aking in to acco un t that lim k →∞ 2 n k k α = 0 for an y α > 0 and applying Theor em 7.3(6 ) of [1], we conclude tha t S- dim M = S-dim W ~ m = ∞ . By Theorem 2.1, the space M is not homeomo rphic to a n IFS-attractor .  4. Some Open Questions W e shall say that a c ompact top o logical spa ce X is a top olo gic al IFS -attr actor if X = S n i =1 f i ( X ) for some contin uous maps f 1 , . . . , f n : X → X such for any op en cov er U of X there is m ∈ N such that for any functions g 1 , . . . , g m ∈ { f 1 , . . . , f n } the set g 1 ◦· · · ◦ g m ( X ) lies in some set U ∈ U . It is easy to see that ea ch IFS-attractor is a top ologica l IFS-attractor and ea ch connected top olog ical IFS-attractor is metriza ble and loca lly connected. Problem 4.1 . Is e ach (finite-dimensional) Pe ano c ontinuum a top olo gic al IFS- attr actor? In p articular, is the sp ac e M c onstructe d in The or em 3.1 a top olo gic al IFS-attr actor? 5. Acknow ledgment The a uthors expres s their sincer e thanks to Wies law Kubi´ s for interesting dis- cussions and v aluable comments. References 1. T.Banakh, M . T uncali, Contr ol le d H ahn-Mazurkiewicz The or em and some new dimension func- tions of Pe ano c ontinua , T op ology Appl. 154 : 7 (2007), 1286–1297. 2. M.Barnsley , F ra ctals everywher e , Academic Press, Boston, 1988. 3. G. Edgar, Meas ur e, top olo gy, and fr actal ge ometry , Springer, New Y ork, 2008. 4. M. Hata, O n the structur e of self-similar sets , Japan J. Appl. Math. 2 :2 (1985), 381–414. 5. M. Kwieci ´ nski, A lo c al ly co nne cte d c ontinuum which is not an IFS attr actor , Bull. Polish Acad. Sci. Math. 47 : 2 (1999), 127–132. 6. S. Nadler , Continuum the ory. A n intr o duction , Marcel Dekk er, Inc., New Y ork, 1992. Instytut M a tema tyki, Jan K ochanowski Univer sity, Kielce, Poland an d F acul ty of Mechanics and Ma thematic s, Iv an Franko Na tional University of L viv, Ukraine E-mail addr ess : t.o.b anakh@gmail.co m Instytut M a tema tyki, Jan K ochanowski Univer sity, Kielce, Poland an d Instytut Ma tema tyki, Jagiellonian University, Krak ow, Poland E-mail addr ess : magda lena.nowak805@ gmail.com