A NOTE ON STABLE ITERATED FUNCTION SYSTEMS
ROY A. MIMNA AND THOMAS SMOTZER
Received 30 November 2005; Revised 20 April 2006; Accepted 4 May 2006
The properties of continuous stable iterated function systems are investigated, and equi-continuity is studied in relation to stable iterated function systems.
Copyright © 2006 Hindawi Publishing Corporation. All rights reserved.
1. Introduction and preliminaries
LetXdenote a compact metric space with metricd, and let f :X→Xdenote a continu-ous self-map onX. For any subsetEofX, we letCl(E) denote the closure ofE. Following [1], we denote by{,X}an iterated function system, or IFS, onX. That is,is a finite fam-ily{f1,. . .,fm}of continuous self-maps onX. In this paper we do not consider the case in whichis an infinite family. Let∞ denote the family of infinite sequences of
com-positions of functions in. That is, a given sequenceF in∞ is composed of arbitrary
compositions of elements of, so that for each positive integern,Fn=f1◦f2◦ ··· ◦fn,
where each fi, 1≤i≤n, is an element of{,X}.
An IFS {,X} is hyperbolic if there is a real numberλ, where 0< λ <1, such that d(f(x),f(y))≤λ∗d(x,y) for all pointsxandyinXand for all f ∈ . An IFS{,X}is stable if for every infinite sequenceF, diam(Fn(X))→0 asn→ ∞. Equivalently,{,X}is stable if{Fn(X)}n≥1has a one-point set as its limit. If{,X}consists of a single function f, then{,X}is stable if diam(fk(X))→0 ask→ ∞, where fkis thek-fold composition of f. By the definition of stability, each sequenceFλin∞, whereλ∈Λfor some
index-ing setΛ, converges to a unique pointxλ. Let F denote the sequence of nested decreasing compact sets formed by the family of sequences{Fλ
n(X)}λ∈Λ. For example, in the well-known IFS{f1,f2, [0, 1]}, where f1(x)=(1/3)xand f2(x)=(1/3)x+ 2/3, F is the nested
sequence of closed subintervals of [0, 1] converging to the Cantor ternary set.
An IFS{,X}is conjugate to an IFS{,X}if there exists a homeomorphismh:X→
Xsuch that=h◦ ◦h−1= {h◦f◦h−1:f ∈ }. The conjugacy then assures that the
qualitative dynamic behavior of the systemsandare largely the same. Clearly, stable
systems exist which are not themselves hyperbolic, but which are conjugate to hyperbolic systems. In [1] it is shown that functions which are either continuous and monotone or
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International Journal of Mathematics and Mathematical Sciences Volume 2006, Article ID 81431, Pages1–5
belong to the Lip[0,1] class are conjugate to hyperbolic functions. Moreover, there exists a continuous stable IFS which is not conjugate to any hyperbolic IFS (see [1]). Although noncontinuous stable IFSs exist, we confine our investigation to the case where all fiin {,X}are continuous. We refer to such systems as “continuous stable IFSs.” We consider some of the properties of continuous stable systems, including systems which are neither hyperbolic nor conjugate to any hyperbolic system. As observed in [1], continuous stable IFSs which are not hyperbolic and are not conjugate to any hyperbolic system are residual in the set of all stable IFSs.
2. Properties of continuous stable iterated function systems
We now consider an IFS{,X}which is made up of a finite number of self-maps, each of which, taken individually, is stable. It turns out that such an iterated function system is not necessarily a stable IFS.
Proposition 2.1. An iterated function system{,X}, whereis a finite family{f1,. . .,fm} of self-maps, each of which, taken individually, is stable, need not be a stable IFS.
Proof. Let
f(x)= ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1
8 if 0≤x≤
1 2, 2x−7
8 if 1 2≤x≤
3 4, 5
8 if
3
4≤x≤1,
(2.1)
and let
g(x)= ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 5
8 if 0≤x≤
1 8, − 1 2
x+11 16 if
1 8≤x≤
5 8, 3
8 if
5
8≤x≤1.
(2.2)
Then
gf(x) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 5
8 if 0≤x≤ 1 2, 9
8−x if 1 2≤x≤
3 4, 3
8 if
3
4≤x≤1.
(2.3)
point under further iteration:
g◦f◦g◦f(x)= ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1
2 if 0≤x≤ 1 2, x if1
2≤x≤ 5 8, 5
8 if 5
8≤x≤1.
(2.4)
The conclusion now follows.
Recall that for any compact metric spaceXwith metricd, we may define the Hausdorff metrichon the spaceH(X) of closed subsets ofXso that for any “points”CandEin the spaceH(X),h(C,E)=min{δ >0 :C⊂Bδ(E) andE⊂Bδ(C)}, whereBδ(C) andBδ(E) are openδ-balls aboutCandE, respectively. Equivalently,h(C,E)=max[max{d(x,E) : x∈C}; max{d(C,x) :x∈E}]. It is well known that if (X,d) is compact, then (H(X),h) is also compact. Let{,X} = {f1,. . .,fm,X}be an IFS and define the functionF:H(X)→
H(X), so thatF(C)=m
i=1fi(C) for everyC∈H(X). Then consider the sequence of
it-eratesFk(H(X)) in the metric space (H(X),h), so thatFk(H(X))=F(Fk−1(H(X))) for k=1, 2,. . .. If the IFS{F,H(X)}is hyperbolic, it is well known that there exists a unique nonempty compact invariant setA, where A=F(A)=m
i=1fi(A), so that{Fk(X)}∞k=1
converges toAin the Hausdorffmetric. This result has also been proved for systems con-jugate to hyperbolic systems (see [3, page 114]). InTheorem 2.3below we prove a similar result for iterated function systems which are continuous and stable. First, we prove a result on the continuity of the functionF:H(X)→H(X).
Proposition 2.2. Let{,X} = {f1,. . .,fm,X}be a continuous stable IFS on a compact
met-ric space (X,d). Let (H(X),h) denote the space of all closed subsets ofXwith the Hausdorff metrich. Then the functionF:H(X)→H(X), whereFis defined so thatF(C)=m
i=1fi(C)
for allCinH(X), is continuous in the Hausdorffmetric.
Proof. Let>0 be given. LetA∈H(X) be any point in the domain ofF. LetB(F(A),) be an-ball aboutF(A) in the Hausdorffmetric. We show that there existsδ >0 such thatF(B(A),δ)⊂B(F(A),). Observe that since each fiis uniformly continuous on the compact spaceX, there existsδi>0 for each fi, so thatd(x1,x2)< δiimplies thatd(fi(x1), fi(x2))<, for allx1,x2∈X. Letδ=min1≤i≤m{δi}. To show thatF(B(A),δ)⊂B(F(A),), letCbe any point inB(A,δ). IfA∩Cis not empty in the topology of the underlying metricd, then all points in (C∩A) are mapped intoB(F(A),). Now consider all points inCin the metricdwhich are not inA. Every such point isδ-close to some point inA in thed-metric, becauseh(A,C)=min{δ >0 :C⊂Bδ(A); A⊂Bδ(C)}. By the uniform continuity of the fi, it follows that every point inF(C) is-close to some point inF(A). By a similar argument, every point inF(A) is-close to some point inF(C). Then by the definition of the Hausdorffmetric,C∈B(F(A),). Hence,F(B(A),δ)⊂B(F(A),), and
Fis continuous on (H(X),h).
Theorem 2.3. Let{,X} = {f1,. . .,fm,X}be a continuous stable IFS on a compact metric
metrich. Then the sequence{Fn(H(X))}∞
n=1converges to a pointAin (H(X),h). That is,
the iterated function system{F,H(X)}is (continuous and) stable. Proof. First, observe that the iterates{Fn(H(X))}∞
n=1are nested. To show{Fn(H(X))}∞n=1
converges in the Hausdorffmetric to a pointA∈H(X), we show thatn∞=1Fn(H(X))= A. SinceH(X) is compact andFis continuous, thenFn(H(X))=A
nis compact for every n. For eachn,n=1, 2,. . ., letan∈Anbe a point in the underlying metric space (X,d). Then the sequence{an}∞n=1 is contained in the setA1. Since A1 is compact, there is a
convergent subsequence{ank}∞k=1 with limit ain A1. But that subsequence, except for
possibly its first element, is also contained inA2, and sinceA2is compact, the limit must
be contained inA2. By induction, the limit is contained in everyAn. Hence, the limitais
contained in the intersectionA=∞n=1Fn(H(X)), andAis not empty. Observe thatAis
compact and thus is a point inH(X). The above argument also shows that{Fn(H(X))}∞ n=1
converges toAin the Hausdorffmetric.
We say that the map f :X→X has equicontinuous iterates if for anyx∈X and for any>0 there exists aδ >0 such thatd(fn(x),fn(y))< wheneverd(x,y)< δfor all y∈X and for all positive integers n. Since X is compact, every transformation on X with equicontinuous iterates has equiuniformly continuous iterates. That is, for any>0, there existsδ >0 such that d(x,y)< δ implies thatd(fn(x),fn(y))< for all positive integersnand for allxand yinX. Clearly, all hyperbolic maps of the form f :X→X have equiuniformly continuous iterates. Also, observe that since a stable system{f,X} converges to a one-point set, we may think of this convergence as convergence of the functions f1,f2,. . .,fk,. . ., to a constant function f. We letS(X,X) denote the set of all
continuous stable self-maps with the supremum metricρ; that is, each element inS(X,X) is a self-map of the form f :X→X, and for any two distinct elements f andginS(X,X), ρ(f,g)=supx∈X{d(f(x),g(x)) :x∈X}. We now show that a necessary condition for a map f to be continuous and stable is that f has equiuniformly continuous iterates.
Proposition 2.4. Let f be an element ofS(X,X). Then the iterates of f are equiuniformly continuous.
Proof. Fix>0. Let{f,X}be a continuous stable IFS comprised of a single function. Since {fn}∞
n=1 converges to a constant functiong=z, for somez∈X, then there is a
positive integerNsuch thatd(fn(x),z)</2 andd(fn(y),z)</2, and henced(fn(x), fn(y))<, for alln≥Nand for allx,yinX. Sincefnis uniformly continuous for everyn, then there existsδn>0 such thatd(x,y)< δnimplies thatd(fn(x),fn(y))<for eachn, 1≤n < N.Then forδ=min{δn: 1≤n < N},d(x,y)< δimplies thatd(fn(x),fn(y))< for alln, 1≤n < N, and for allx,yinX. Hence, f has equiuniformly continuous iterates.
The reader will easily find examples to show that the converse ofProposition 2.4is not true.
Corollary 2.5. Each sequence{Fn(X)}∞
Corollary 2.6. Let f ∈S(X,X). Then f is not surjective.
Proof. Since f ∈S(X,X), then byProposition 2.4, f has equiuniformly continuous it-erates. By [2, Corollary 8], any surjective self-map f on a compact metric space whose iterates are equicontinuous is a homeomorphism. If f is a homeomorphism, then so are all of the iterates of f. But it then follows that the iterates of f(X) do not converge to a
single point, a contradiction.
Remark 2.7. The proof of Proposition 2.4shows that the sequence{fn}∞
n=1 converges
uniformly to the constant mapg. That is, for any>0, there exists a positive integerN, so that for alln≥N, and for allx∈X,ρ(fn,g)<.
Proposition 2.8. Let{,X} = {f1,. . .,fm,X}be a continuous stable IFS on a compact
met-ric space (X,d). Let (H(X),h) denote the space of all closed subsets ofXwith the Hausdorff metrich. Then the functionFonH(X), as defined inTheorem 2.3above, has equiuniformly continuous iterates.
Proof. By Theorem 2.3 above, the system {F,H(X)} is stable. Now by mimicking the
proof ofProposition 2.4above, the result follows.
References
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iterated function systems, Real Analysis Exchange 25 (1999/2000), no. 1, 449–461.
[2] A. M. Bruckner and T. Hu, Equicontinuity of iterates of an interval map, Tamkang Journal of Mathematics 21 (1990), no. 3, 287–294.
[3] P. F. Duvall Jr., J W. Emert, and L. S. Husch, Iterated function systems, compact semigroups, and
topological contractions, Continuum Theory and Dynamical Systems, Lecture Notes in Pure and
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Roy A. Mimna: Department of Mathematics, Youngstown State University, Youngstown, OH 44555, USA
E-mail address:[email protected]
Thomas Smotzer: Department of Mathematics, Youngstown State University, Youngstown, OH 44555, USA