R E S E A R C H
Open Access
Separation properties for infinite iterated
function systems
Hui Liu and Xiangling Zhu
**Correspondence: [email protected]
Department of Mathematics, JiaYing University, Meizhou, GuangDong 514015, China
Abstract
In this paper, we study the infinite iterated function systems (IFSs) of contractive similitudes with overlaps. We extend the notions of the weak separation condition and the generalized finite type condition for finite IFSs to the infinite case. We show that for an infinite IFS of contractive similitudes the generalized finite type condition implies the weak separation condition.
MSC: Primary 46N99; secondary 28A80
Keywords: infinite iterated function systems; open set condition; weak separation condition; generalized finite type condition
1 Introduction
The separation properties are useful for studying the IFSs. At first, let’s recall the sep-aration property of finite IFSs. SupposeXis a nonempty compact subset ofRd. LetSi
(i= , , . . . ,m) be a contractive self-map onX⊂Rd. We call{S
i}mi= a finite similar IFS of
contractive similitudes onRdif there existsC∈(, ) such that
Si(x) –Si(y)=C|x–y| for every ≤i≤m.
There exists a nonempty subsetKofXsuch that
K=
m
i=
Si(K).
Kis called the invariant set or attractor of the system (see,e.g.[–]).
We say that {Si}mi= satisfies the open set condition (OSC) if there exists a nonempty
bounded open setU⊆Rdsuch that
Si(U)⊆U and Si(U)∩Sj(U) =∅ fori=j.
Such aUis called a basic open set for{Si}mi=. If, moreover,U∩K=∅, the{Si}mi=is said to
satisfy the strong open set condition (SOSC). It is a classical result (see [, , ]) that if a similar IFS satisfies the OSC, then it satisfies the SOSC. Fanet al.(see [–]) extended the result to finite conformal IFSs.
IFSs that do not satisfy the OSC are said to have overlaps. In this case, it is in gen-eral much harder to get acquainted with the structure of the corresponding invariant
set K. The weak separation condition (WSC) and the generalized finite type condition are weaker than the OSC but still strong enough to obtain good results (see [–]etc.).
However, the circumstances for infinite IFSs are distinct []. Szardk and Wedrychow-ica [] showed that for infinite IFSs, the OSC does not imply the SOSC. Moran [] also showed that self-similar set generated by a countable system of similitudes may not be s-set even if the open set condition is satisfies. So it is necessary to look for some sepa-ration conditions for infinite IFSs that are weaker than the OSC. Moran defined a weak separation property for infinite IFSs []. Suppose{Si}∞i= is an infinite conformal IFS on
an open setU. Moran [] defined the infinite IFS{Si}∞i=satisfies the finite open set
con-dition if for any integern, there is a nonempty open setUn⊂Usuch thatSi(Un)⊂Un
for any ≤i≤nandSi(Un)∩Sj(Un) =∅for anyi,j≤n,i=j. The finite strong open set
condition holds if furthermoreUn∩J=∅. It is easy to see that if the IFS{Si}∞i= satisfies
the OSC then it satisfies the finite strong open set condition. Moran uses this separation property to study the Hausdorff dimension of invariant set with respect to the infinite IFS [].
Our goal in this paper is to study the infinite iterated function systems (IFSs) of contrac-tive similitudes with overlaps. We define the WSC and the generalized finite type condi-tion for the infinite IFSs. Next, we study the relacondi-tionship of the two separacondi-tion condicondi-tions. We show that the generalized type condition implies the WSC. Our main result is Theo-rem ..
Theorem . Let{Si}∞i= be an infinite iterated function system of contractive similitude
onRd.If{S
i}∞i= is of generalized finite type condition,then it satisfies the weak separation
condition.
This paper is organized as follows. In Section , we define the weak separation condition for infinite IFSs and give some examples. In Section , we introduce the generalized finite type condition for infinite IFSs and provide examples of IFSs satisfying this condition. Finally, in Section , we prove that the generalized finite type condition implies the weak separation condition (i.e.Theorem .).
2 The weak separation condition
Let={, , . . . ,n, . . .},n={I= (i,i, . . . ,in) :in∈}and∗=n≥n. Let{Si}∞i=be an
IFS of contractions defined on a compact subsetX⊂RdwithX=∅. Letρ
ibe the
con-tractive ratio ofSi, andρI=ρiρi· · ·ρin, forI= (i,i, . . . ,in). We defineρmax=maxi≥ρi. Definition . We say that an IFS{Si}∞i=satisfies the weak separation condition (WSC)
if there existx∈Xandγ ∈Nsuch that, for anyI∈∗, the ball of radiusbcontains at
mostγ points of{S(SI(x)) :S∈Ab}for any <b≤. Here we let
Ab={SI:I∈τb} and τb=
I= (i,i, . . . ,in) :ρI≤b≤ρii···in–
.
Remark For any starting pointy∈X, it is easy to see that{Si}∞i=will satisfy the WSC if
there existsn> such that, for anyJ,J∈τb, either
For anya> and any bounded subsetsD⊆XandU⊆Rd, we let
Aa,U,D=
S∈Aa|U|:S(D)∩U=∅, γa,D=sup U
#Aa,U,D.
Here|U|denotes the diameter ofU. We have two lemmas with respect to the definition of the weak separation condition which are needed to prove our main result.
Lemma . Let{Si}∞i=be an infinite IFS of contractive similitudes onRd,for any a> and
any nonempty subset D⊆X,γa,D<∞.Then{Si}∞i=satisfies the WSC.
Proof Letx∈X. LetD⊆Xbe a nonempty subset and letAbbe defined as above. Then
for anyI∈∗and any ballBbof radiusb,
#SSI(x)
∈Bb:S∈Ab
≤#S∈Ab:S
SI(x)
∈Bb
≤S∈A
|Bb|:S(D)∩Bb=∅
≤γ ,D
<∞,
which yields the statement.
Lemma . Let{Si}∞i= be an infinite IFS of contractive similitudes on a compact subset
X⊆Rd.If there exist a constantγ ∈Nand a subset D⊆X with D◦=∅,such that,for any
<b< and x∈X,
#S∈Ab:x∈S(D)
≤γ.
Then{Si}∞i=satisfies the WSC.
Proof We denote byBb(x) the closed ball with radiusband centerx. LetLdenote the
Lebesgue measure onRd. LetS∈Absuch thatS(D)∩Bb(x)=∅, andy∈Dsuch thatS(y)∈
Bb(x). Then for anyz∈D,
S(z) –x≤S(z) –S(y)+S(y) –x ≤c|z–y|+b
≤b +c|D|
.
So
S(D)⊆Bb(+c|D|)(x).
Letx∈D,I∈∗, <b< , we have
#SSI(x)
∈B:S∈Ab
≤#S∈Ab:S
SI(x)
∈B ≤#S∈Ab:S(D)∩Bb(x)=∅
Supposeρ:=min{ρi: ≤i≤n}. By assumption,
(bρ)dL(D)#S∈Ab:S(D)∩Bb(x)=∅
≤ LS(D):S∈Ab:S(D)∩Bb(x)=∅
≤γLBb(+c)(x)
.
That means
#S∈Ab:S(D)∩Bb(x)=∅
≤γ( +c|D|)
d
ρdL(D) :=N.
This completes the proof of the lemma.
Lemma . Suppose{Si}∞i=is an infinite IFS on a compact subset X⊆Rd,and it satisfies
the OSC.Then it satisfies the WSC.
Proof SupposeUis an open set guaranteed by the open set condition. For any <b≤, we writeAb={SI:I∈τb}and
τb=
I= (i,i, . . . ,in) :ρI≤b≤ρii···in–
.
For anyI,I∈Ab withI=I, the open set condition implies thatSI(U)∩SI(U) =∅.
Since
ρ·b|U| ≤SIi(U)≤b|U| (i= , ).
So|SI() –SI()| ≥ρb. Then the remark implies that{Si}∞i=satisfies the WSC.
Example . S(x) =x,S(x) = x+,Si(x) =(i–+i–x) + (i≥). This IFS satisfies
the WSC. It is easy to see that this infinite IFS does not satisfy the OSC. We know that S∩Si=∅(i≥),S∩Si=∅(i≥),Sj∩Sj=∅(j,j≥, andj=j). By Lemma . and the
example in []{Si}∞i=satisfies the WSC.
3 The generalized finite type condition
In this section we promote the generalized finite type condition to infinite IFSs. The gen-eralized finite type condition for infinite IFSs is slightly modified from that for finite IFSs []. The definition consists of two parts. The first part concerns the sequence of nested index sets. The second part entails the concept of neighborhood types.
Let {Si}∞i= be an infinite IFS of contractive similitudes on a compact subset X ⊆
Rd, ={, , . . . ,n, . . .}, n={I= (i,i, . . . ,in) :in∈} and∗ =
n≥n. For any I=
(i,i, . . . ,im)∈m,J= (j,j, . . . ,jn)∈n, we letIJ= (i,i, . . . ,im,j,j, . . . ,jn). ForI,J∈∗,
ifIis an initial segment ofJorI=J, we writeIJ. We denote byIJifIJdoes not hold. Consider a sequence of index sets{Mk}∞k=, where for allk≥,Mkis a finite subset
of∗. Let
mk=mk(Mk) :=min
and
mk=mk(Mk) :=max
|I|:I∈Mk
.
Definition . We say that{Mk}∞k=is a sequence of nested index sets if it satisfies the
following conditions:
(i) Both{mk}and{mk}are nondecreasing, andlimk→∞mk=limk→∞mk=∞. (ii) For eachk≥,Mkis an antichain in∗.
(iii) For eachJ∈∗with|J|>mk, there existsI∈Mksuch thatIJ. (iv) For eachJ∈∗with|J|<mk, there existsI∈Mksuch thatJI.
(v) There exists a positive integern, independent ofk, such that, for allI∈Mkwith
IJ, we have|J|–|I| ≤n.
By lettingMk=kfor allk≥, we obtain an example of sequence of nested index sets.
To define neighborhood types, we fix a sequence of nested index sets{Mk}∞k=. For each
integerk≥, letVkbe the set ofkth level vertices (with respect to{Mk}) defined as
V:=
(I, ) and Vk:=
(SI,k) :I∈Mk
for allk≥.
We call (I, ) the root vertex and denote it byvroot. LetV=
k≥Vkbe the set of all vertices.
Forv= (SI,k)∈Vk, we use the convenient notationSv:=SI.
Fix any nonempty bounded open set⊆Xwhich is invariant under{Si}∞i=. Twokth
levelv,v∈Vk(allowingv=v) are said to be neighbors (with respect toand{Mk}) if
Sv()∩Sv()=∅. The set of vertices
N(v) :=v:v∈Vkis a neighbor ofv
is called the neighborhood of v(with respect to and{Mk}). Note thatv∈N(v) by
definition.
Next, we define an equivalence relation onv.
Definition . Under the above assumptions, two verticesv∈Vkandv∈Vkare
equiv-alent, denoted byv∼vif, forτ:=SvS–v :Rd→Rd, the following conditions hold:
(i) {Su:u∈N(v)}={τSu:u∈N(v)}.
(ii) Foru∈N(v)andu∈N(v)such thatSu=τSu, and for any positive integerl≥,
an indexI∈∗satisfies(SuSI,k+l)∈Vk+lif and only if it satisfies
(SuSI,k+l)∈Vk+l.
It is easy to see that∼is an equivalence relation. We denote the equivalence class con-tainingvby [v] and call it the neighborhood type ofv(with respect toand{Mk}). We
define two important infinite directed graphsGandGR. The graphGhas vertex setVand
directed edges defined as follows. Letv∈Vk andu∈Vk+. Suppose there existI∈Mk,
J∈Mk+, andL∈such that
Then we connect a directed edgeLfromvtou and denote this byv→L u. We callva parent ofuinGanduan offspring ofvinG. We writeG= (V,E), whereEis the set of all directed edges defined above.
The reduced graphGRis obtained fromGby first removing all but the smallest (in the
lexicographical order) directed edge going to a vertex. More precisely, let vk
Lk
→u, k= , , . . . ,n, . . . , be all the directed edges going to the vertexu∈Vk+, wherevk∈Vk are
distinct and thusLkare distinct also. SupposeL<L<· · ·<Ln<· · ·in the lexicographical
order. Then we keep onlyLin the reduced graph and remove all the edgesLk(k≥).
Next, we denote the resulting graph byGR. It is possible that a vertex inVdoes not have any offspringGR (see the example in []). We remove all vertices that do not have any
offspring inGR, together with all vertices and edges leading only to them. The resulting graph is the reduced graph, denoted byGR= (VR,ER), whereVR is the set of vertices and
ERis the set of all edges.
It follows from the invariance ofunder{Si}∞i=that only vertices in(v) can be parents
of any offspring ofvinG. In fact, ifu= (SvSl,k+ )∈Vk+is an offspring ofvinGand if
w∈Vk\N(v), then for any indexI∈∗,
SwSI()∩Su()⊆Sw()∩Sv() =∅.
Hencewcannot be a parent ofu.
Proposition . Letv,v∈Vk andu,ube their offspring.Ifvandvare not neighbors,
then neither areuandu.
Proof LetSu=SvSwandSu=SvSwfor somew,w∈∗. SinceSw()⊆andSw()⊆,
we have
Su()∩Su()⊆Sv()∩Sv() =∅.
This leads to the conclusion.
Proposition . Letbe a bounded invariant open set for the IFS{Si}∞i=and letGRbe the
corresponding reduced graph.Then there exists a unique path inGR from the root vertex vrootto any given vertex.
Proof The existence of a path is obvious. Next, we prove the uniqueness. Supposev∈V,
if there are two different paths inGRfromvroottov, then the vertex at which the two paths
cross will have two parents inGR, it is a contradiction.
Proposition . Suppose v∈Vk andv∈Vk be two vertices with offspringui and uj
(i,j≥)inGR.Suppose[v] = [v]and let
N(v) ={v,v,v, . . .}, N
Suppose that in the graphGwe have edges k,ksuch that
vi k
→u, vj k
→u,
vi→k u, vj→k u.
Thenu=uif and only ifu=uandu,uare neighbors if and only ifu,uare.
Proof Observe that
Su=SviSk=τSviSk=τSu. ()
Similarly we haveSu=τSu. HenceSu=Su if and only ifSu=Su, and sou=uif and
only ifu=u. Furthermore,
Su()∩Su() =∅ if and only if τSu()∩τSu() =∅.
This proves the second part of the proposition.
Proposition . Suppose v∈Vk andv∈Vk be two vertices with offspring ui and uj
(i,j≥)inGR.Suppose that[v] = [v],and let
N(v) ={v,v,v, . . .}, N
v=v,v,v, . . .. Then
[ui]|i≥
=uj|j≥. ()
Proof The proposition says roughly that two vertices of the same neighborhood type have equivalent offspring. Let W andWbe the set of offspring of the vertices inN(v) and N(v). We define a mapχ:W→Was follows: Suppose thatuis an offspring ofviinG
by an edgek. We letχ(u) be the offspring ofviby an edgek. Propositions . and . show thatχis a one-to-one correspondence. Furthermore, by () we have
Sχ(u) =τSu.
By Proposition . only vertices inN(v) can be parents of any offspring ofvinG. Again by Propositions . and .,uis an offspring ofvinGRif and only ifχ(u) is an offspring
ofvinGR. This yields ().
Definition . Let{Si}∞i= be an self-map IFS on a subsetX⊂Rd. We say that{Si}∞i= is
of generalized finite type if there exist a sequence of nested index sets {Mk}∞k=and a
nonempty invariant open set ⊆X such that, with respect to and{Mk}∞k=,V/∼=
{[v]|v∈V}is a finite set. In this case, we say thatis a generalized finite type set.
Proposition . If{Si}∞i=is of OSC,then it is of the generalized finite type condition.
Proof Letbe the open set of OSC. SupposeMk=k. For eachv∈V={(Si, ) :i≥},
the OSC implies thatN(v) ={v}. Letτ =IS–
v ,i.e.τSv=I, we havev∼(I, ) =vroot. By
Proposition .,V/∼={[vroot]}. So{Si}∞i= satisfies the generalized finite type condition.
Example . S(x) =x,S(x) =x+,S(x) =x+,Si(x) = –ix– –i++(i= , , . . .),
this IFS satisfies the generalized finite type condition.
Proof Let= (, ). For eachk≥ letMk=k. Upon iterating the IFS once, the root
vertex generates the following vertices:
v= (S, ), v= (S, ), . . . , vn= (Sn, ), . . . .
Obviously,N(vk) ={vk}(k≥). Sovk∼vroot(k≥) withτ =ISv–k. Upon one more
it-eration, there is no new neighborhood type generated (see the example . in []). So V/∼={[vroot], [v], [v]}and the result follows.
4 Proof of the main theorem
Proof of Theorem. Assume that{Si}∞i=is a finite type similar IFS onXand let{Mk}∞k=
andbe defined as above. We will show that there existsγ ∈Nsuch that, for all <b≤ andx∈X,
#S∈Ab:x∈S()
≤γ.
Let={S∈Ab:x∈S()}. List all elements ofasSI,SI, . . . . ForIjthere exists a unique
Ij∈Mkjsuch thatI
jIj. The choice of the particularIjdoes not affect the following proof.
We assume thatIjis chosen such thatkjis maximum,i.e., ifIj IjandIj∈Mlfor somel,
thenl≤kjandIjIj. We assume without loss of generality that
k=min{kj:SIj∈}=k and I
∈Mk.
For eachj, hereSIj∈, letIjbe the initial segment ofIjsuch thatIj∈Mk. In particular,
I=I. Sincex∈S() for allS∈, it follows that
v= (SI,k), . . . , vm= (SIm,k), . . . ,
are neighbors ofv= (SI,k). The finite type condition implies that the number of vertices
in each neighborhood is uniformly bounded by some constantMindependent ofx,b, and the choice ofIj. That is,
#{v,v, . . . ,vm, . . .} ≤M,
i.e.
#SIj:SIj∈,I j Ij
Since eachSIjbelongs toAb, we have
ρb||<SIj()≤b||, SIj∈.
Also, by the definition of{Mk}∞k=, there exists a constantn, independent ofxandb, such
that|Ij|–|Ij| ≤nfor allIj, hereSIj∈. Hence,
ρb||<SIj()≤bρ–n||, SIj∈,I jIj.
It yields
ρ≤|SI
()|
|SIj()|
≤ρ–(n+). ()
It also implies that there exists a constantc> , independent ofxandb, such that
c– ≤| SIj()|
|SI ()|
≤c, SIj∈. ()
Combining () and () yields
c– ≤| SI
j()|
|SIj()|
≤c, SIj∈. ()
Herec:=ρ–(n+)c.
We writeIj= (Ij,Lj), for anyj≥. For eachSIj∈, () implies that
ρI
j ≤ρIjc=ρIjρLjc.
Hence
ρ|L
j|
max≥ρIj≥c– .
If we letl:= [–log(c)/logρmax] + , then|Lj| ≤l. The finite type condition implies that
#SIj:I
I
j
≤Ml.
Thus,#{S∈Ab:x∈S()} ≤γ follows by takingγ =Ml+. Lemma . implies that the
{Si}∞i=satisfies the WSC.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
References
1. Falconer, K: Fractal Geometry: Mathematical Foundations and Applications. Wiley, New York (2007) 2. Falconer, K: Techniques in Fractal Geometry. Wiley, Chichester (1997)
3. Hutchinson, J: Fractals and self-similarity. Indiana Univ. Math. J.30, 713-747 (1981)
4. Bandt, C, Graf, S: Self-similar sets. VII. A characterization of self-similar fractals with positive Hausdorff measure. Proc. Am. Math. Soc.114, 995-1001 (1992)
5. Schief, A: Separation properties for self-similar sets. Proc. Am. Math. Soc.122, 111-115 (1994) 6. Fan, A, Lau, K: Iterated function system and Ruelle operator. J. Math. Anal. Appl.231, 319-344 (1999)
7. Peres, Y, Rams, M, Simon, K, Solomyak, B: Equivalence of positive Hausdorff measure and the open set condition for self-conformal sets. Proc. Am. Math. Soc.129, 2689-2699 (2001)
8. Ye, Y: Separation properties for self-conformal sets. Stud. Math.152, 33-44 (2002)
9. Lau, K, Ngai, S: Multifractal measures and a weak separation condition. Adv. Math.141, 45-96 (1999)
10. Lau, K, Ngai, S: A generalized finite type condition for iterated function systems. Adv. Math.208, 647-671 (2007) 11. Zerner, M: Weak separation properties for self-similar sets. Proc. Am. Math. Soc.124, 3529-3539 (1996) 12. Nguyen, N: Iterated function systems of finite type and the weak separation property. Proc. Am. Math. Soc.130,
483-487 (2002)
13. Ngai, S, Wang, Y: Hausdorff dimension of self-similar sets with overlaps. J. Lond. Math. Soc.63, 655-672 (2001) 14. Lau, K, Ngai, S, Wang, X: Separation conditions for conformal iterated function systems. Monatshefte Math.156,
325-355 (2009)
15. Mauldin, R: Infinite iterated function systems: theory and applications. In: Fractal Geometry and Stochastics, vol. 37, pp. 91-110 (1995)
16. Szarek, T, Wedrychowicz, S: The OSC does not imply the SOSC for infinite iterated function systems. Proc. Am. Math. Soc.133, 437-440 (2005)
17. Moran, M: Hausdorff measure of infinitely generated self-similar sets. Monatshefte Math.122, 387-399 (1996) 18. Wang, X: Iterated function systems with a weak separation condition. Stud. Math.161, 249-268 (2004)
doi:10.1186/1029-242X-2014-118
