A class of continua that are not attractors of any IFS

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=1

that for everyx;y∈Xwe haved(f(x);f(y))≤λd(x;y). A mapf:X→Xis calledcontractiveif for everyx;y∈X,

x6 ,

ehavd( y) d ,se milyF :f t :

❆❜ai:_❛❝i:✿ This paper presents a sufficient condition for a continuum in R to be embeddable in R in such a way that its of any weak iterated function system is also given.

image is not an attractor of any iterated function system. An example of a continuum in R that is not an attractor

n n

2

▼❙❈✿ 28A80, 54F15, 37B25, 54H20

❑❡②✇♦_❞a✿ Fractal • Continuum • Iterated function system • Attractor

© Versita Sp. z o.o.

Cent. Eur. J. Math. • 10(6) • 2012 • 2073-2076 DOI: 10.2478/s11533-012- 0108-5

❈❡♥�L❛❧❊✉L♦♣❡❛♥❏♦✉L♥❛❧♦❢▼❛,1:❤❡♠❛,1:✐❝2

A class of continua that are not attractors ofanyIFS

MarcinKulczycki^1∗,MagdalenaNowak^1,2†

❘❡ ❡❛ ❝❤❆ ✐❝❧❡

1 InstituteofMathematics,FacultyofMathematicsandComputerScience,JagiellonianUniversity,Łojasiewicza6,30-348 Kraków,Poland

2 InstituteofMathematics,JanKochanowskiUniversity,Świętokrzyska15,25-406Kielce,Poland

❘❡❝❡✐✈❡❞ ✶✺ ❋❡❜ ✉❛ ② ✷✵✶✷❀ ❛❝❝❡♣ ❡❞ ✷✷ ▼❛② ✷✵✶✷

1. Introduction

Tfehretilneofitieolndooffarenseitaerrcahteadsfwuneclltiaosnasyvsetresmati(laebabnredv.usIFefSu)l,tionotrloidnulcoesdsybydaJtoahncoH muptrcehsisniosonn(eisnp1e9c8ia1ll[y4]w,hhaesrepirmovaegnetdoatbaeias

caonnacettrrnaecdto).rTofhiasnpIaFpSe.rWiseanostwudryecianllosnoemsepebcaisfiiccatesrpmeicntoolofgthye.theorythepossibilityofen codingaparticularsetas

Let (X ; d) be a complete metric space. A mapf:X→Xis called acontractionif there exists a constantλ∈[0;1) such

ca

=

lle

y

da

w

(weak

e

)ite

f

r

(

a

x

t

)

e

;

d

f(

fun

)c<

tion

(x

s

;

y

y

s

)

tem,

e

se

[2

e

].

[

1

A

].

fa

Givena

=

com

{f

p

1

a

;c:

t

:

B

;⊂n}

X

o

,

fd(c

e

o

fi

n

ne

ractivemaps)contractionsfiX→Xis

F ( B ) = [

in

fi(B):

∗

†

E

E

-

-

m

m

a

a

i

i

l

l

:

:

Mmaagrcdianl.eKnualc.nzoywck aik@80im5@.ujg.emdaui.lp.clom

✷✵✼✸

Barnsley–Hutchinsonoperator.AfixedpointofFiscalledtheattractorof(weak)IFS.

It eryIFh uq

t to s ctma b kF l

complex.OurresultsprovideasufficientconditiSonfordacontinuumtobeembeddableinR^nsothatitsimageisnotan ItiselementaRrytocheckthateverycontinuumdiinRisanattractorofsomeIFS.Moreover,anyembeddingofsuch

compactness ofXis assumed). M. Hata proved in [3] that if the attractor of some IFS is connected, then it is

alsolocallycot M. s owed n[] a y fn len i anattr

ε&0

IFS .

neighbourhoodofps

\

uchthatde(x;y;C\Uxy)<+∞.Thenthereexistsanembeddingh:C→Rⁿsuch

tha

x

t

y

h(C)isnot

length.Figure1illustratestheprocessforn=2.

Thendefinethe embedding h:C→ asthecompositionh2◦h¹.

A class of continua that are not attractors of any IFS

This transformation, acting on the space of nonempty compact subsets ofXwith the Hausdorff metric, is called the

is shown in [4]t h a t v e S as a ni ue at rac r (ananalogou f a y not e true for a wea I

S,unesspronvneenctehda.tifaJ.Sisanadneerndsphointofisom7etharctAev⊂erRnarwchoichfihiatesthegpthropserties:ac torforsomeIFS.Additionally,hehas

(1) forallx;y∈A\{a}thelengthofthesubarcofAwithendpointsxandyisfinite, (2) foreveryx∈A\{a}thelengthofthesubarcofAwithendpointsxandaisinfinite,

othfeMn.AKwisiencoitńsakniafrtotmrac[5to]rmoafyaanlysoIFbSeaecatsinilgyomnoRⁿfi.edOntoeseaxtaimsfpylethoefsesua cshsuamnpatriocniss.theharmonicspiral[6].Theexample

continuum in still is an attractor of some IF . In imension two and higher, however, the situationbecomesmoreattractor of anyIFS.

2. Mainresults

Definition 2.1.

x

Let

=

(X

y

;

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∈

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(

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x;y;A;ε).Thislimitmaybeinfinit

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ItiselementarythatifA⊂Bthende(x;y;A)≥de(x;y;B).

Theorem2.2.

i=1 i i+1 sequences.

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=

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{

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a

tan attractor ofany Proof. Byapplyinganaffinetrann−s1formationwemayassunmewnithoutlossofgeneralitythatB={0}×R^n−1,

p=(0;:::;0),andC⊂[0;1]×[−1;1] . Next defineh1; h2: R→Ra s

h1(x1;:::;xn)=Ix1;1x 0

1

0x²;:::;1x 0

1

0xⁿ

\

; h2(x1;:::;xn)=x

1;√

x1sinx1−1+

x2;x3;:::;xn

1:

Rⁿ Rthoeugfuhnlcytisopnea√

kxinsgi,nhx1⁻¹t.raAnssfoarmressuCltionftothaessheacropndneterdanles,fowrhmialetiohn2tbheen dnseethdaletbneeceodmleesto,sfipteianktiongaitmhipcrkeecniseedl-yu,pofgrinafipnhitoef

neighbourhoodofh(p)suchthatde(x;y;h(C)\Ux2 y)<+∞.

Themaph1doesnotincreasedistance,andthereforeforeveryx;y∈h1(C\{p})thereexistsUx1

ywhichisaneighbour-

✷✵✼✹

h

co

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(p

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ity

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thatf(y)=6 h(p).Thende(x;y;h(C))wouldbefiniteandde(f(x);f(y);h(C))wouldbeinfinite,whichcontradictsthe If,ontheotherhand,f(h(p))=h(p)thenthereexistx∈h (C)suchthatf(x)=h(p)andy∈h (C)\{h(p)}such

λde(x;f(x);h(C)).Butthiswouldimplythatden(x−;1h1(p);h(C))isalsofinite,whileitisnot,sinceitcanbeseenfromthe

forthecontinuathatsatisfythemdirectly,butalsoforthecontinuathatarehomeomorphictosubsetsof^nsatisfying

t e

m os ia t d a f eho r su . e p

toourbestknowledge,open.WeshallnowgiveanexampleofasubcontinuumofR²thatisnotanakattractorofany

interse0ctions,cons−instingn−onffinit−enl−y2ma−nnyin−tenr−v2als,thatstartsatp⁰,eSn_∞dsatpn

r , o

hasthetotallengthnof2ⁿ,andi s

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Figure 1.The maphforn=2

Cthoennsifd(ehr(Cno))w=a{chon(ptr)a}c.tTioontfhi:she(Cnd),→conhj(eCct)uwreiththaatLifpsischniotztccoonnssttaannttλan<d1h.

(Wp)e∈wfo(uhl(dCl)ik).etoprovethatifh(p)∈f(h(C)) c

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))

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ote

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∈

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h

(x

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;f

a

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(C

6=

))

h

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(p

fi

).

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th

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h

)

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;

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::

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)≤

is

definitionofh2thatdex;h(p);h([0;1]×[−1;1] )isinfinite.

Ccoonntsreaqctuievenntleys,sifoFffi,scothmeplBeatirnngsltehye–

Hpruotocfhtinhsaotnifofpetaraketosrvfaolruesohm(ep)IFoSnaatndleaFs(tho(Cne))a⊂rghu(mCe)n,tththeennFi(thh(aCs))tombaeyccoonm

sptarinste.

ohf(C{h)(ips)n}oatnadnpaotstrsaibcltyoraolsfoFfi.nitelymanyotherclosedsetsnotcontainingh(p).ButthenF(h(C)) h(C),provingthat

Theassumptionso fTheorem2. 2 aret echnicalandmayseemv eryrestrictive.Itsassertion,h owever,ist

Rruen oto nly inhe

ts hea

cs os

nu tinp

ut ui

mn A.⊂Th

Ris

ns

ci ag

nnbfi eccn

only new

cti ee

dn is

nt Ahe

bycl as

ps na+o

t1hs oe

fts fint

ih tet

lene gte

hm ,this

enui tef

cal nfo

br eF

e a

o s

r ily

xa s

meepnle t

, h

i a

fta a

n n

y y

tw on

o e-p

o o

in in

ts t

unionofAand[0;1]ishomeomorphictoasubsetofR for which the assumptions of Theorem2.2aresatisfied.

oAfftseormtheeIFreSs.uTlthoefeHxaatmap[l3e]iotfhKawsibeeceińnskain[5o]pepnropvirdoebdleamnwehgeatthiveereavnesr

wyelro,cbaulltythcoensnaemcteedquceosnttiionnuufomriwneR^nIiFsSa’nsarettmraacitnosr,weak IFS.

I Pn

utth pisd

=efi (n

0i

;ti 0o

)n aw

ndes pwit

=ch (t

2o−nt

;h 2e−ns

)ta fn

od ra

nrd

≥pol 1a

.rc Fo

oo rrd

ai nn

yate ns≥yst

1em ch(

or o;θ

se)o an

bR² k,

et nha

lt ini

e s

s( ex

g;y m)

en= tl(rc

wo is

thθ o;

ur tsi

sn eθ

lf) -. contained in

[0;

2 )×(2

−2

;2 +2 )∪ {pn}. DefineP= _i=1l_i. Remark2.3.

Definition2.4.

Suppose thatf:P→Pis contractive. We shall examine how many of the pointspican belong tof(P). If

notincreaseone-dimensionalmeasure).Consequently,onlyfinitelymanyofthepointspibelongtof(P).

pointspiand almost all of the setsf(li).Note that only finitely many of the setsf(li)may reach the outside ofU.

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Theorem 2.5.

The space P as a subset ofR²with the Euclidean metric is not an attractor of any weak IFS.

f(p0)6=p0thenthereisaneighbourhoodUoff(p0)suchthatthedistanced(p0;U)>0andUcontainsfinitelymanyeAllesmoeonbtsaerryv eprtohpaetretyacohffc(olni)trcaocvteiorsnsatcamnosbtefipnriotevleynmeaitnhyerpobiyntussipnigbδec-

achuaseintsh,eorl,eansgtihns[o5f],lbiyaruesinnogttinhcerefaacstedthbaytffd(toheiss Proof

.

ofF.

[5]1K2w7i–

e1c3iń2skiM.,AlocallyconnectedcontinuumwhichisnotanIFSattractor,Bull.Pol.Acad.Sci.Math.,1999,47(2), [3]HataM.,Onthestructureofself-similarsets,JapanJ.Appl.Math.,1985,2(2),381–414

of these sets are too small to traverse the wholeln. But no other point inPcan be mapped ontopnbyf,becausef

A class of continua that are not attractors of any IFS

Figure 2.The space P

If,ontheotherhand,f(p0)=p0,then,givenn≥1,notethatpnmaynotbelongtof(li)fori<n,becausethelengthsidfeFcreiassaeswtheae kdiIsFtSan,ctehebnetowneleynfipn0itaenlydmanaynyotohferthpeoipnoti.nTtshepriefcoarne,btheeloonnglytopoFin(Pt)p,iap nrdestehnetreinfofr(ePP)isispn0.otInacnonacttlurasciotonr,

Acknowledgements

TThheesaeuctohnodrsawutohuolrdwliaksestuoppthoarntekdtbhyetrheefeEreSeFsfHorumthaenirCtahpoirtoaulgOhpwero artkioannadlPsurogggreasmtimngegseravnerta6l/1im/8p.o2r.1ta/PntOcKlaLr/i2fi0c0a9t.ions.

References

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[4] Hutchinson J.E., Fractals and self similarity, Indiana Univ. Math. J., 1981, 30(5), 713–747

[[67]]SSaannddeerrssMM..JJ..,,NAnonn-a-

ctterlalcitnorsRⁿo⁺f¹ittehraatteidsfnuocttiothnesyastttreamcsto,rTeoxfaasnPyroIFjeSctoNneRxTⁿ⁺J¹o,uMrniasls,o2u0r0 i3J,.1M,a1t–h9.S ci . , 2009,21(1),

13–20