CONTINUITY OF THE MAPS
f
→
x∈Iω(x,
f
) AND
f
→ {
ω(x,
f
) :
x
∈
I
}
T. H. STEELE
Received 29 September 2004; Revised 11 December 2005; Accepted 12 February 2006
We study the behavior of two maps in an effort to better understand the stability ofω -limit setsω(x,f) as we perturb eitherx or f, or both. The first map is the set-valued functionΛtaking f in C(I,I) to its collection ofω-limit pointsΛ(f)=x∈Iω(x,f), and the second is the mapΩtaking f in C(I,I) to its collection ofω-limit setsΩ(f)= {ω(x,f) :x∈I}.We characterize those functions f in C(I,I) at which each of our maps ΛandΩis continuous, and then go on to show that bothΛandΩare continuous on a residual subset ofC(I,I). We then investigate the relationship between the continuity of ΛandΩat some functionf inC(I,I) with the chaotic nature of that function.
Copyright © 2006 Hindawi Publishing Corporation. All rights reserved.
1. Introduction
We begin with a brief historical overview in an effort to place this paper’s results in con-text. Let f be a continuous self-map of the unit intervalI=[0, 1], and take as our start-ing point the iterates of this map. In particular, we begin by considerstart-ing the trajectories generated by our function f ∈C(I,I) for particular initial conditions x in I, and let
τ(x,f)= {fk(x)}∞
k=0= {x,f(x),f(f(x)),. . .,fn(x),. . .}be the trajectory ofx∈I
gener-ated by f. Central to the study of dynamical and chaotic systems is the desire to under-stand how trajectoriesτ(x,f)= {fk(x)}∞
k=0 are affected by slight changes in the initial
conditionx. In order to somewhat simplify this study, one oftentimes considers theω -limit set generated byxrather than the trajectory. We takeω(x,f)—theω-limit set gen-erated byx—to be the collection of subsequential limit points of the trajectoryτ(x,f). Whenever f is a continuous self-map of a compact interval, theω-limit setω(x,f) enjoys several nice properties. In particular,ω(x,f) is always nonempty, closed, and strongly in-variant under the generating functionf so that f(ω(x,f))=ω(x,f). Nonetheless, as [1] shows,ω-limit sets and the trajectories that give rise to them can be wildly complicated (see Theorem2.2). One thread found in the literature, then, is to investigate when and how these complicatedω-limit sets can be approximated by simpler structures. If we let
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International Journal of Mathematics and Mathematical Sciences Volume 2006, Article ID 82623, Pages1–15
Λ(f)=x∈fω(x,f) represent the closed set ofω-limit points of f, one natural question to consider is what conditions on f insure thatP(f)=Λ(f), whereP(f) is the collection of the periodic points of f. Our first theorem is from [13].
Theorem 1.1. If f ∈C(I,I) is piecewise monotonic, thenP(f)=Λ(f).
Examples constructed in [10,18], however, show thatP(f)=Λ(f) is possible when we restrict our attention to either the class of Lipschitz or continuously differentiable functions.
We find another approach in [9] where Coven and D’Aniello study the relationship between the setsP(f) andΛ(f) and the chaotic nature of the generating function f ∈ C(I,I).
Theorem 1.2. LetA= {f ∈C(I,I) :P(f)=Λ(f)}. ThenAis dense in C(I,I) and any map f inAis Li-Yorke chaotic.
In [3] the relationship between periodic orbits andω-limit sets is studied directly, and a very interesting result dealing with wandering intervals and the prevalence of periodic orbits is established.
Theorem 1.3. If f ∈C(I,I) and each wandering interval of f converges to a periodic orbit, then the family of periodic orbits of f is dense with respect to the Hausdorffmetric in the collection ofω-limit sets of f.
We note thatP(f)=Λ(f) whenever the family of periodic orbits of f is dense with respect to the Hausdorffmetric in the collection ofω-limit sets.
Another recurring thread in the literature is to investigate the stability of certain struc-tures when the generating function is perturbed. Consider the following result from [14].
Theorem 1.4. Suppose f ∈C(I,I), f has zero topological entropy,P(f) is nowhere dense, and any simple system of f has nonempty interior. Then, for anyε >0, there existn(ε) a natural number andδ(ε)>0 so that the following condition holds: iff −g< δ(ε), then for anyω-limit setω0ofgthere exists a 2k-cyclepofgsuch thatk≤n(ε) and the Hausdorff
distance betweenωand pis less thanε.
What this theorem tells us is that everyω-limit set of a functiong can beε -approx-imated in the Hausdorffmetric space by one of its 2k-cycles wheneverg is sufficiently close to a particularly well-behaved function f. Recalling Theorem1.3, it is worth not-ing that if f ∈C(I,I), f has zero topological entropy, and any simple system of f has nonempty interior, then the 2k-cycles of f are dense with respect to the Hausdorffmetric in the collection ofω-limit sets of f.
such as those functions that are in some way nonchaotic, or those functions that satisfy a particular smoothness condition. We will see that in answering Bruckner’s queries we are able to develop insight into, or extend, the previously mentioned results. Here is how we proceed.
We work in four metric spaces. We use the regular, Euclidean metricdonI=[0,1], and make occasional use of neighborhoods of closed setsF of the formBε(F)= {x∈
I:d(x,y)< ε,y∈F}. WithinC(I,I) we use the supremum metric given by f −g =
sup{|f(x)−g(x)|:x∈I}. Our third metric space (K,H) is composed of the class of nonempty closed setsKinI endowed with the HausdorffmetricH given byH(E,F)= inf{δ >0 :E⊂Bδ(F),F⊂Bδ(E)}. This space is compact [6]. Our final metric space (K∗,H∗) consists of the nonempty closed subsets ofK. Thus,K∈K∗ifKis a nonempty family of nonempty closed sets inI such thatK is closed inK with respect toH. We endowK∗with the metricH∗, so thatK1 andK2 are close with respect toH∗if each
member ofK1is close to some member ofK2with respect toH, and vice versa.
Our interest in, and the utility of, the metric spaces (K,H) and (K∗,H∗) stems at least in part from the following two theorems from [4,15], respectively.
Theorem 1.5. For any f inC(I,I), the setΛ(f) is closed inI.
Theorem 1.6. For any f inC(I,I), the setΩ(f) is closed in (K,H).
These theorems allow us to formulate earlier stability queries via the mapsΛ: (C(I,I),
· )→(K,H) given by f →Λ(f) andΩ: (C(I,I), · )→(K∗,H∗) given by f →Ω(f). Specifically, Bruckner asked how one could characterize those functions f at which each of the mapsΛ: (C(I,I), · )→(K,H) andΩ: (C(I,I), · )→(K∗,H∗) is continuous. In order to make these ideas explicit, three examples are developed in some detail. These examples will provide some insight into the behavior of the functionsΛandΩas well as focus efforts in the ensuing sections.
Example 1.7. Consider fn(x)=x(n−1)/n. Asngoes to infinity, we see that fngoes to the identity function f. Thus,Λ(f)=[0, 1]. SinceΛ(fn)= {0, 1}for alln, we see thatΛis not continuous at f, so thatΩmust necessarily be discontinuous there, too. While this does rule out the best possible result—thatΩ, and thereforeΛ, are continuous—our example does not rule out a natural generalization of the theorem found in [4].
Recall that our four authors in [4] show that if{ωn} ⊆Ω(f), andωn→ωinK, then
ω∈Ω(f).In Example1.7,{0} ∈Ω(fn) for everyn, and{0} ∈Ω(f). Perhaps, then, the following is true: ifωn∈Ω(fn) for eachn, fn→f andωn→L, thenL∈Ω(f). This con-jecture simplifies to the result of [4] if we letfn=f for alln.
For our next example, we need the following definition. LetM be a nowhere dense compact set inI, withA= {a0,a1,. . .,ak−1} = ∅a set of limit points ofM.Suppose there
is a system{Mi
n}∞n=0,i=0, 1,. . .,k−1, of nonempty pairwise disjoint compact subsets of M such thatM\i,nMni =Aand limn→∞Min=aifor each i. Let f :M→M be a con-tinuous map withAak-cycle of f such that f(ai)=ai−1fori >0 and f(a0)=ak−1. If
f(Mi
n)=Mni−1fori >0 and anyn, f(Mn0)=Mkn−−11forn >0, and f(M00)=ak−1, thenMis
Example 1.8. We will construct a sequence of homoclinicω-limit setsωnfor functions
fnin C(I,I) so thatωn→L, fn→ f, yetL is not contained inΛ(f). This negates our conjectured generalization of the result from [4].
We begin by constructing ourω-limit setsωn. For each portionMi
n, we take a scaled copy of the middle-third Cantor set with the indicated convex closure.
Forω1, leta0=1/2 and convMn0=[1/2 + 1/22+n, 1/2 + 1/22+n+ 1/23+n].SetA0=a0∪ {∞n=0Mn0}.Now, leta1=0 and convMn1=[1/22+n, 1/22+n+ 1/23+n].
Forω2, we begin with the setA0 described above, and takea1=1/4 and convMn1= [1/4 + 1/23+n, 1/4 + 1/23+n+ 1/24+n]; letA
1=a1∪ {
∞
n=0Mn1}.Now, leta2=0 and conv M2
n=[1/23+n, 1/23+n+ 1/24+n].
In general, forωm, we begin with the setsA0,A1,. . .,Am−2and takeam−1=1/2m and
convMm−1
n =[1/2m+1 + 1/2m+2+n, 1/2m+1+ 1/2m+2+n+ 1/2m+3+n]; let Am−1 =am−1∪ {∞n=0Mnm−1}. Now, letam=0 and convMnm=[1/2m+2+n, 1/2m+2+n+ 1/2m+3+n].
We see that each of our setsωnwill be homoclinic of ordern+ 1, and the sequence
{ωn}converges inKto the setL= {0} ∪ {
∞
n=0An}.How our functions fn:ωn→ωnare defined is clear from our definition of a homoclinic trajectory as well as the construction of the setsωn. Moreover, since each resulting fnis continuous, we can use [8] to extend
fn:ωn→ωnto a function we will also call fnthat is inC(I,I) and has the property that
ωn=ω(x,fn) for somex∈I. Since we can take fn|A1∪ ··· ∪Am= fk|A1∪ ··· ∪Am for alln and k greater thanm+ 2, and An→0 asn→ ∞, we can take our fn so that
f =limn→∞fnexists, and f(x)=0 forx∈[1/2, 1].Thus,Λ(f)∩[1/2, 1]= ∅as f(0)= limn→∞fn(0)=limn→∞1/2n.
It is worth pointing out that not only isLnot anω-limit set of f, but we lose a consider-able portion of ourω-limit points as well. For eachn,A0⊆ωn⊆Λ(fn) withA0⊆[1/2, 1],
yetΛ(f)∩[1/2, 1]= ∅.
Example 1.9 [5]. Let f(x)=xonI, and forε >0, choose 1/n < ε. An appropriate polyg-onal function fnthat possesses the orbit 0→1/n→2/n→ ··· →(n−1)/n→1→(n− 1/2)/n→(n−3/2)/n→ ··· →1/2/n→0 has a periodic orbit that spansI, and the prop-erty thatf −fn ≤1/n. SinceΩ(f)= {{x}:x∈I}, it follows thatH∗(Ω(fn),Ω(f))= 1/2 for alln.By choosing a subsequence if necessary, one may assume that limn→∞Ω(fn) exists, since (K∗,H∗) is compact. Then limn→∞fn= f, andH∗(limn→∞Ω(fn),Ω(f))= 1/2. Thus,Ωis discontinuous at the identity function, a function with zero topological entropy. Unlike Example1.8, however, in this example we did not lose anyω-limit points in going fromΛ(fn) toΛ(f), asΛ(f)=[0, 1], but we did lose all of our nontrivialω-limit sets in the limit.
We proceed through several sections. After presenting the definitions and previously known results needed in the sequel, we begin our analysis in Section3. There we study the mapΛ: (C(I,I), ◦ )→(K,H) given byf →Λ(f), and characterize those functions
f ∈C(I,I) at whichΛis continuous with Theorem3.1. We show thatΛis continuous on a residual subset ofC(I,I) with Proposition3.5and Theorem3.6. Section4is dedi-cated to the study of the mapΩ: (C(I,I), ◦ )→(K∗,H∗) given by f →Ω(f). Theo-rem4.8characterizes those functions f at whichΩis continuous when we restrict the domain to the setE= {f ∈C(I,I) : f has zero topological entropy}, and Theorem4.7
characterizes those functions at whichΩis continuous without any domain restrictions. Section4 concludes with Propositions4.11and4.12which establish the continuity of Ω: (C(I,I), ◦ )→(K∗,H∗) on a residual subset ofC(I,I). Section5addresses the re-lationship between the chaotic nature of a function and the behavior ofΛandΩthere.
2. Preliminaries
We make the following definitions in addition to those already presented in Section1. Let
P(f) represent those pointsx∈Ithat are periodic under f, and ifxis a periodic point of periodnfor whichfn(x)−xtakes on both positive and negative values in any deleted neighborhood ofx, thenxis called a stable periodic point; we letS(f) represent the stable periodic points of f. We letS(f)= {ω:ωis a stable periodic orbit of f inC(I,I)}be the collection of stable periodic orbits of f, and setΩ(f)= {L:L⊂[0, 1] is nonempty and closed, f(L)=L, and for any nonempty proper closedF⊂Lone hasF∩f(L−F)= ∅}. Now, let ε >0 be given, and takex and yto be any points in [0, 1]. An ε-chain from
xto ywith respect to a function f is a finite set of points{x0,x1,. . .,xn}in [0, 1] with
x=x0,y=xnand |f(xk−1)−xk|< ε fork=0, 1,. . .,n−1. We call x a chain recurrent point of f if there is anε-chain fromxto itself for anyε >0, and writex∈CR(f). We note that for every f in C(I,I),S(f)⊆P(f)⊆Λ(f)⊆CR(f). Central to the ensuing study of the mapsΛ: (C(I,I), ◦ )→(K,H) andΩ: (C(I,I), ◦ )→(K∗,H∗) is the notion of semicontinuity. Consider the set-valued functionF: (C(I,I), ◦ )→(K,H) with f ∈C(I,I). We say that Fis upper semicontinuous at f if for anyε >0 there exists
δ >0 so thatF(g)⊂Bε(F(f)) wheneverf −g< δ. Similarly, Fis lower semicontinuous at f if for anyε >0 there existsδ >0 so thatF(f)⊂Bε(F(g)) wheneverf−g< δ.
In part of the sequel we will restrict our attention to a closed subsetEofC(I,I) com-posed of those functions f having zero topological entropy, denoted by h(f)=0. The reader is referred to [11, Theorem A] for an extensive list of equivalent formulations of topological entropy zero. For our purposes, it suffices to note that every periodic orbit of a continuous function with zero topological entropy has cardinality of a power of two [12]. The following theorem, due to Sm´ıtal [16], sheds considerable light on the structure of infiniteω-limit sets for functions with zero topological entropy.
Theorem 2.1. Ifωis an infiniteω-limit set off ∈C(I,I) possessing zero topological entropy, then there exists a sequence of closed intervals{Jk}∞k=1in [0, 1] such that
(1) for eachk,{fi(J k)}2
k
i=1are pairwise disjoint, andJk=f2k(Jk); (2) for eachk,Jk+1∪f2
k
(3) for eachk,ω⊂2k
i=1fi(Jk); (4) for eachkandi,ω∩fi(J
k)= ∅.
LetJbe an interval inIso thatJ,f(J),. . .,fn−1(J) are pairwise disjoint, and fn(J)=J. We callJa periodic interval, and refer to{fi(J)}n−1
i=1 =orbJas a cycle of intervals. Now, take J0⊃J1⊃J2⊃ ··· to be periodic intervals with periods m0,m1,m2,. . ., so thatmi must dividemi+1for anyi. Ifmi→ ∞, then the intervals{Ji}∞1=0 generate a solenoidal system
of f, and any invariant closed setS⊂L=i≥0orbJiis called a solenoidal set of f. From Sm´ıtal’s theorem one sees that every map f with zero topological entropy is solenoidal on each of its infiniteω-limit sets, withJkhaving period 2kfor everyk. When the period ofJkis 2kfor every intervalJk, we refer to the solenoidal system as a simple system.
Now, let us suppose that f has a cycle of intervalsM=J∪f(J)∪ ··· ∪fn−1(J), and
consider the set{x∈M: for any relative neighborhoodUofxinMwe have orbU=M}. This closed setM is invariant, and we refer toMas a basic set of f, provided that it is infinite. A fundamental structure associated with both positive topological entropy and basic sets is a horseshoe. If f ∈C(I,I) has positive topological entropy, then there exist intervalsK andLinIhaving at most one point in common such thatK∪L⊂fm(K)∩
fm(L), for some natural numberm. This horseshoe structure gives rise to a setF⊂I such that f(F)=F and fm|F is semiconjugate to the shift operator on two symbols. Speaking loosely, the horseshoe structure shows us that there is a considerable amount of expansion that takes place within basic sets that is not present in solenoidal systems.
The following theorem characterizesω-limit sets for continuous functions [1].
Theorem 2.2. Let F⊆I be a nonempty closed set. ThenF is an ω-limit set for some
f ∈C(I,I) if and only ifFis either nowhere dense, orFis the union of finitely many nonde-generate closed intervals.
3. Continuity ofΛ: (C(I,I), ◦ )→(K,H)
The main result of this section, Theorem3.1, characterizes those functions f ∈C(I,I) at which the mapΛ: (C(I,I), ◦ )→(K,H) is continuous.
Theorem 3.1. Λis continuous at f if and only ifS(f)=CR(f).
This result follows from Lemmas3.3and3.4, as Lemma3.3characterizes those contin-uous functions at whichΛis upper semicontinuous, and Lemma3.4characterizes those continuous functions at whichΛis lower semicontinuous. In the proof of Lemma3.3, it is helpful to recall the following result from [2].
Lemma 3.2. Ifx∈CR(f), then any open neighborhood of f inC(I,I) contains a function
gfor whichx∈P(g).
Lemma 3.3. Let f ∈C(I,I). Then for anyε >0 there existsδ >0 so thatΛ(g)⊂Bε(Λ(f)) wheneverf −g< δif and only ifΛ(f)=CR(f).
f −g< δ [2]. By hypothesis, we have thatΛ(f)=CR(f), so that Λ(g)⊂CR(g)⊂ Bε(Λ(f)), and our conclusion follows.
Now, let us suppose thatx∈CR(f)−Λ(f). Then there exists{fn} ⊂C(I,I) so that
fn→ f andx∈P(fn) for eachn, so thatx∈limΛ(fn)−Λ(f).
Lemma 3.4. Let f ∈C(I,I). Then for anyε >0 there existsδ >0 so thatΛ(f)⊂Bε(Λ(g)) wheneverf −g< δif and only ifS(f)=Λ(f).
Proof. The sufficiency of our lemma follows immediately from the definition of a stable periodic orbit, and the compactness ofΛ(f). As for the necessity, let us supposeS(f) is a proper subset ofΛ(f), and letJbe an open interval in [0, 1] for whichΛ(f)∩J= ∅, butS(f)∩J= ∅. IfP(f)∩J= ∅, then there existsKan open interval contained inJfor whichP(f)∩K⊆Pn(f), for some natural numbern. IfP(f)∩J= ∅, setK=J. In either case, then, there exists{fn} ⊂C(I,I) and an open intervalK for whichK∩Λ(f)= ∅, butP(fn)∩K= ∅for all natural numbersn, and fn→ f. Since we may take fnto be piecewise monotonic onI,Λ(fn)=P(fn) for eachn[2], so that by taking a subsequence of{fn}if necessary, we haveΛ(fn)=P(fn)→Fin (K,H), withF∩K= ∅. Thus,Λ(f) is
not contained in limΛ(fn).
Theorem3.6shows that the mapΛ: (C(I,I), ◦ )→(K,H) is continuous on a resid-ual subset ofC(I,I). This will follow immediately from Proposition3.5, where we show thatS(g)=CR(g) for the typicalf inC(I,I).
Proposition 3.5. The setS= {f ∈C(I,I) :S(f)=CR(f)}is residual in (C(I,I), ◦ ).
Proof. SinceS(f)⊆CR(f) andCR(f) is closed inIit follows thatS(f)⊆CR(f). To show thatH(S(f),CR(f))< ε, then, it suffices to show that for anyx∈CR(f) there exists y∈ S(f) so that|x−y|< ε. SetSn= {f ∈C(I,I) :H(S(f),CR(f))<1/n}. SinceS=∞n=1Sn, we need to show thatSnis both dense and open inC(I,I).
We first verify thatSn is a dense subset of C(I,I). Let f ∈C(I,I)−Sn withε >0. SinceCR:C(I,I)→Kis upper semicontinuous, there existsδ >0 so thatf −g< δ im-pliesCR(g)⊂Bε(CR(f)). Takeδ >0 so thatCR(g)⊂B1/2n(CR(f)) wheneverf −g<
δ, and let{x1,x2,. . .,xm} ⊆CR(f) be a 1/2n-net ofCR(f). Now, chooseg∈C(I,I) so that xi∈S(g) for 1≤i≤mandf −g<min{δ,ε}. Then CR(f)⊂B1/2n(S(g)) since
{x1,x2,. . .,xm} ⊆S(g) andCR(g)⊂B1/2n(CR(f)), so thatCR(g)⊂B1/n(S(g)). We con-clude thatH(S(g),CR(g))<1/n.
We now show thatSnis an open subset ofC(I,I). Let f ∈Snwithn≥4. SayH(S(f),
CR(f))=α <1/n, and setγ=1/n−α. Letδ1>0 so thatf −g< δ1impliesCR(g)⊂ Bγ/n(CR(f)). Take{x1,x2,. . .,xm} ⊆S(f) to be an (α+γ/n)-net ofCR(f). Now, there ex-istsδ2>0 so thatf −g< δ2impliesS(g)∩Bγ/n(xi)= ∅fori=1, 2,. . .,m. Ifg∈C(I,I) for whichf−g<min{δ1,δ2}, thenmi=1xi⊂Bγ/n(S(g)),CR(f)⊂Bα+γ/n(mi=1xi) and
CR(g)⊂Bγ/n(CR(f)). It follows that CR(g)⊂B1/n(S(g)) so that H(S(g),CR(g))<1/n,
andg∈Sn.
Theorem1.2 shows that the setA= {f ∈C(I,I) :P(f)=Λ(f)}is dense inC(I,I); Proposition3.5shows thatAcannot be too large, however, since it is contained in a set of the first category. Moreover, not only isP(f)=Λ(f) for the class of “nice” piecewise monotonic maps (Theorem1.1), butP(f)=Λ(f) also is true for the typical f ∈C(I,I). We note that the typical element ofC(I,I) has positive topological entropy, so while Li-Yorke chaos is necessary forP(f)=Λ(f), it is not sufficient.
4. Continuity ofΩ: (C(I,I), ◦ )→(K∗,H∗)
As withΛ: (C(I,I), ◦ )→(K,H), semicontinuity is central to our analysis ofΩ: (C(I,
I), ◦ )→(K∗,H∗). Since the upper semicontinuity of Λ: (C(I,I), ◦ )→(K,H) is a necessary condition for the upper semicontinuity of Ω: (C(I,I), ◦ )→(K∗,H∗), Lemma3.3provides a necessary condition for the upper semicontinuity ofΩ.
Proposition 4.1. A necessary condition for the function f to be a point of upper semicon-tinuity of the mapΩ: (C(I,I), ◦ )→(K∗,H∗) is thatΛ(f)=CR(f).
That the condition of Proposition4.1is not sufficient to insure that f ∈C(I,I) is a point of upper semicontinuity ofΩ: (C(I,I), ◦ )→(K∗,H∗) follows from considera-tion of the hat maphwhereh(x)=2xforx∈[0, 1/2] andh(x)=2(1−x) forx∈(1/2, 1]. Since S(h)=CR(h)=[0, 1],Λis continuous at h. Now, let us consider g∈C(I,I) so thatg(0)=ε/2,g(1/2)=1,g(1)=0, andg is linear on both (0, 1/2) and (1/2, 1). Then
g−h< ε, yetω(x,g)∩[1/2, 1]= ∅for allxinI. Since{0} =ω(0,h)∈Ω(h), we see thatH∗(Ω(g),Ω(f))≥1/2, so thatΩis discontinuous ath.
Turning now to our next results, Proposition4.2recalls a couple of basic properties of
ω-limit sets [2], and Proposition4.3shows that these properties are shared by closed sets
Lwheneverfn→f,ωn→Landωn∈Ω(fn) for eachn.
Proposition 4.2. Suppose f ∈C(I,I) withω∈Ω(f).Then (1) f(ω)=ω,
(2) ifFis any nonempty proper closed subset ofω, thenF∩f(ω\F)= ∅.
Proposition 4.3. Suppose fn→f,ωn→Landωn∈Ω(fn) for eachn.Then (1) f(L)=L,
(2) ifFis any nonempty proper closed subset ofL, thenF∩f(L\F)= ∅.
Proof. We first show that f(L)=L.
f(L)⊆L: let y∈L, and take {yn} so that yn∈ωn for each n, and yn→y. Then
fn(yn)→f(y), and since fn(yn)∈ωn, it follows that f(y)∈L.
L⊆ f(L): let y∈L, and take{yn}so that yn∈ωnfor each n, andyn→y.Suppose
xn∈fn−1(yn)∩ωn, with{xnk} ⊆ {xn}a convergent subsequence; sayxnk→x.Thenx∈L,
and f(x)=y, as|f(x)−y| ≤ |f(x)−f(xnk)|+|f(xnk)−fnk(xnk)|+|fnk(xnk)−y|.
In order to prove the second part of our proposition, suppose, to the contrary, thatF
and f(L\F) are disjoint. Then there exist open setsG1,G2such thatL\F⊆G1,F⊆G2
andG2is disjoint from f(G1). Sayσ=min{|x−y|:x∈G2,y∈ f(G1)}.Sinceωn→L, there existsM a natural number such thatωn⊆G1∪G2andωn∩G1= ∅,ωn∩G2= ∅
for allm≥Nandx∈I. Let us taken, then, so thatn >max{M,N}, and setFn=ωn∩G2.
ThenFnis a closed, nonempty, proper subset ofωn, andG2is disjoint from fn(G1). Let xn∈I so thatωn=ω(xn,fn). For all largek, fnk(xn) belongs to either G1 orG2, and it
belongs to each of them infinitely often. Thus there is an infinite sequencek1< k2< k3< ··· so that fki
n (xn)∈G1, and fnki+1(xn)∈G2.Ifyis a limit point of the sequence fnki(xn),
theny∈G1, and f(y)∈G2, which is a contradiction.
Proposition4.3holds the key to characterizing those functions f ∈C(I,I) at which Ω: (C(I,I), ◦ )→(K∗,H∗) is upper semicontinuous. Anyω-limit set of a continuous functionf satisfies both parts of the conclusion of Proposition4.3; however, a functionf
will be a point of upper semicontinuity ofΩ: (C(I,I), ◦ )→(K∗,H∗) only when these conditions characterize itsω-limit sets. This is the content of Theorem4.4.
Theorem 4.4. The mapΩ: (C(I,I), ◦ )→(K∗,H∗) is upper semicontinuous at the function f if and only ifL∈Ω(f) wheneverL∈K for which f(L)=LandF∩f(L\F)= ∅for any nonempty proper closed subsetFofL.
Significant progress in proving Theorem4.4is made with the development of the fol-lowing proposition.
Proposition 4.5. Let f ∈C(I,I) withL∈Kfor which f(L)=LandF∩f(L\F)= ∅for any nonempty proper closed subsetFofL. For anyε >0 there existg∈C(I,I) andω∈Ω(g) so thatf−g< εandH(ω,L)< ε.
Proof. Letε >0. Since f ∈C(I,I), f is uniformly continuous onI, so that there exists
δ1>0 with the property that|f(x)−f(y)|< εwhenever|x−y|< δ1. Chooseδso that
0< δ <min{δ1,ε}, and take{x1,x2,. . .,xn} ⊆Lto be aδ-net forL. It suffices to perturb
f to get a functiong∈C(I,I) possessing a periodic attractorωso thatf −g< εand
{x1,x2,. . .,xn} ⊆ω⊆ni=1Bδ(xi), as this impliesH(ω,L)< ε. That this is possible follows from our hypothesis that F∩f(L\F)= ∅for any nonempty proper closed subset F
contained inL. In particular,
(1) letF=L\Bδ(xi) to see that there existsx∈Bδ(xi) such that f(x)∈Bδ(xj) for some j=i, for anyi=1, 2,. . .,n;
(2) letF=L\j=iBδ(xj) to see that there existsx∈Bδ(xj) for some j=iso that
f(x)∈Bδ(xi), for anyi=1, 2,. . .,n;
(3) letS⊆ {1, 2,. . .,n}withF=L\i∈SBδ(xi) to see that there existsx∈
i∈SBδ(xi) so that f(x)∈Bδ(xj) for some j∈ {1, 2,. . .,n}\S.
With Proposition4.5, a proof of Theorem4.4follows easily.
Proof of Theorem 4.4. SupposeΩ: (C(I,I), ◦ )→(K∗,H∗) is upper semicontinuous at
f, andL∈Kfor which f(L)=LandF∩f(L\F)= ∅for any nonempty proper closed subsetFofL. By Proposition4.5, there exists{fn} ⊆C(I,I) withωn∈Ω(fn) for anynso that limn→∞fn= f and limn→∞ωn=L. SinceΩis upper semicontinuous at f, it follows thatL∈Ω(f).
anynso that limn→∞fn=f and limn→∞ωn=L∗. Sincef(L∗)=L∗andF∩f(L∗\F)=∅ for any nonempty proper closed subsetFofL∗by Proposition4.3, it follows thatL∗∈
Ω(f), so thatΩ: (C(I,I), ◦ )→(K∗,H∗) is upper semicontinuous at f. The next task is to develop some insight into those functions f in C(I,I) at which Ω: (C(I,I), ◦ )→(K∗,H∗) is lower semicontinuous.
Theorem 4.6. If f ∈C(I,I) for whichP(f)−S(f)= ∅, thenΩ: (C(I,I), ◦ )→(K∗,
H∗) is not lower semicontinuous at f.
Proof. Letx0∈P(f)−S(f), say of periodn. Sincex0∈P(f)−S(f), for each 0≤i≤ n−1 there exists a neighborhoodNicontainingxi=fi(x) so thatfn(xi)−xiis unisigned onNi− {xi}. By taking sufficiently small neighborhoods we may assume thatNi∩Nj=
∅ wheneveri= j.Now, letε >0 so thatBε(xi)⊆Ni for alliand f(Bε(xi))Ni+1 for
0≤i≤n−2 with f(Bε(xn−1))N0. We now takeδ >0 so thatg(Bε(xi))Ni+1, too,
for 0≤i≤n−2 withg(Bε(xn−1))N0 wheneverf −g< δ. Fixg ∈C(I,I) so that Pn(g)∩Ni= ∅for 0≤i≤n−1 andf −g< δ. Should fn(y)≥yfor ally∈N0− {x0},
it suffices to takeg(x)=f(x) +δ/2 forx∈Bε(xn−1),g(x)= f(x) forx /∈Nn−1, and extend g appropriately to the remainder ofNn−1. We show thatH(ω(x0,f),ω(y,g))> εfor all y in I. Suppose, to the contrary, that there exists y∗ inI so that H(ω(x0,f),ω∗)< ε,
whereω∗=ω(y∗,g). Thenω∗⊆Bε(ω(x0,f))⊆ n−1
i=0Ni, and by choosing g as we did we know thatg(ω∗∩Ni)=ω∗∩Ni+1for 0≤i≤n−2 withg(ω∗∩Nn−1)=ω∗∩N0.
Thus,gn(ω∗∩Ni)=ω∗∩Nifor alli, so that the convex closure conv(ω∗∩Ni) contains
a periodic point of periodn. This, however, contradicts our earlier choice ofg.
We are now in a position to state and prove a main result of the section. We note that condition (1) implies the continuity of the mapΛ: (C(I,I), ◦ )→(K,H) atf and condition (3) insures the upper semicontinuity of the mapΩ: (C(I,I), ◦ )→(K∗,H∗) there. Both of these are clearly necessary to the continuity ofΩat f.
Theorem 4.7. Let f ∈C(I,I). The mapΩ: (C(I,I), ◦ )→(K∗,H∗) is continuous at f
if and only if
(1)S(f)=CR(f),
(2) all the periodic points of f are stable,
(3)L∈Ω(f) wheneverL∈K for which f(L)=L and F∩ f(L\F)= ∅ for every nonempty proper closed subsetFofL.
Proof. Suppose the mapΩ: (C(I,I), ◦ )→(K∗,H∗) is continuous at f. It follows im-mediately from the definition of the mapΛ: (C(I,I), ◦ )→(K,H) that it, too, must be continuous there. Moreover, ifP(f)−S(f)= ∅, thenΩwould not be lower semicontin-uous at f, so thatΩcould not be continuous there. Finally, asΩis continuous at f,Ω must be upper semicontinuous there, so that (3) holds as it characterizes those functions at whichΩis upper semicontinuous.
We begin by showing that any infiniteω-limit set of f is contained in the Hausdorff closure of its periodic orbits. Letω∗be a basic set of f. From Theorem1.3, anyω-limit set contained inω∗ is in the closure of the periodic orbits of f whenever there is no wandering intervalJ converging to an infiniteω-limit set in ω∗. To see that this is not possible, assume thatJis such an interval and setA=∞n=0f−n(J). Thenω∗⊂Aso that A∩P(f)= ∅ and A∩CR(f)= ∅. But this contradicts (1). It remains to show that any simple setWis contained in the closure of the periodic orbits. Let{Jk}be a nested family of compact intervals such thatJk has period 2k andW⊂
∞
k=1 2k−1
i=0 fi(Jk)=L. Since every intervalU⊂Lconsists of nonperiodic chain recurrent points, condition (1) implies thatLhas empty interior so thatWcan be approximated by periodic orbits, too. We now show thatΩis lower semicontinuous at f. Ifω∈Ω(f) is finite, thenω is stable by (2), so that for anyε >0 there existsδ >0 such that anyg∈C(I,I) for which
f −g< δpossesses a periodic orbitαwith the property thatH(α,ω)< ε. Ifω∈Ω(f) is infinite, thenωcan be Hausdorffapproximated by stable periodic orbits; this provides
the lower semicontinuity ofΩat f.
Now, suppose that f ∈EandS(f)=CR(f). ThenintL= ∅for any simple system of
f, andP(f) has nonempty interior, so that such a function f satisfies the hypotheses of Price and Sm´ıtal’s Theorem1.4.
Theorem 4.8. If f ∈E, thenΩ:C(I,I)→K∗ is continuous at f if and only ifS(f)= CR(f).
This theorem from [19] sharpens the result of [14] and indicates that wheneverf ∈E, the mapΩis continuous at f if and only if the mapΛis continuous there. From the definition of a solenoidal system, f must possess a wandering interval that converges to an infiniteω-limit set of f whenever a solenoidal system has nonempty interior. As [3] indicates, this wandering interval may generate an infiniteω-limit set that is not found in the Hausdorffclosure of the periodic orbits so thatΩ:C(I,I)→K∗may not be lower semicontinuous there. Theorem4.4shows thatΩ:C(I,I)→K∗is never upper semicon-tinuous at such an f.
Our next objective is to establish the continuity ofΩ:C(I,I)→K∗on a residual subset ofC(I,I). Proposition4.9shows thatΩ(f) is closed in (K,H). This allowsΩto play a role analogous to that of the chain recurrent set in the analysis ofΛ:C(I,I)→Kin Section3.
Proposition 4.9. If f ∈C(I,I), thenΩ(f) is closed in (K,H).
Proof. Let{Lk}∞k=1⊂Kwith f ∈C(I,I) so thatLk= f(Lk) for anyk, andLk→LinK. Since f is continuous, it follows thatf(Lk)→ f(L) inK, too. We conclude thatL= f(L). Now, suppose that for eachk, the following holds forLk: IfF= ∅is closed such thatF
Lk, thenF∩f(Lk−F)= ∅. We show that for anyF= ∅closed,FL, it follows thatF∩
f(L−F)= ∅. Suppose, to the contrary, that there exists such anFso thatF∩f(L−F)= ∅; sayH(F,f(L−F))=σ. Letδ >0 so that|x−y|< δimplies|f(x)−f(y)|< σ/4 and choosensufficiently large so thatH(Ln,L)< γ,Ln∩Bγ(F)= ∅, andLn∩Bγ(L−F)= ∅, whereγ <min(δ,σ/8). Set F=Ln∩Bγ(F). ThenF∩f(Ln−F)= ∅sinceF⊂Bσ/4(F)
Proposition 4.10. The mapΩ : (C(I,I), ◦ )→(K∗,H∗) given by f →Ω(f) is upper semicontinuous.
Proof. Let fn→f in (C(I,I), ◦ ) withLn∈Ω(fn) for eachn, andLn→Lin (K,H). We show thatL∈Ω(f).
We first show thatL=f(L). Sincef ∈C(I,I), fn→ f uniformly andLn→Lin (K,H), we have H(L,f(L)) =H(L,Ln) +H(Ln,fn(Ln)) +H(fn(Ln),f(Ln)) +H(f(Ln),f(L)) where each of the terms on the right-hand side goes to zero asn→ ∞. It follows that
L= f(L).
Now, let us suppose to the contrary that there exists an appropriateFfor which our transport property does not hold for F,L−F and f. In particular, F= ∅ is closed,
FLandF∩f(L−F)= ∅. SayH(F,f(L−F))=σ. Since fn→ f uniformly, there is
N1 a natural number so thatn > N1 implies |f(x)− fn(x)|< σ/8 for allx∈I. Since
f is uniformly continuous onI there is aδ >0 so that|f(x)−f(y)|< σ/8 whenever
|x−y|< δ. SinceLn→Lin (K,H) there isN2a natural number so thatn > N2implies H(Lk,L)< γ,Lk∩Bγ(F)= ∅, andLk∩Bγ(L−F)= ∅, whereγ <min{δ,σ/8}. Now, set
F=Lk∩Bγ(F), so thatF⊂Bσ/4(F). Then fk(Lk−F)⊂Bσ/8(f(Lk−F)) and f(Lk−F)⊂
Bσ/8(f(L−F)) so that fk(Lk−F)⊂Bσ/4(f(L−F)) wheneverk > max{N1,N2}. This
im-pliesH(F,fk(Lk−F))> σ/2, a contradiction.
The next result ties the behavior of Ω: (C(I,I), ◦ )→(K∗,H∗) to the upper semi-continuity of the mapΩ : (C(I,I), ◦ )→(K∗,H∗).
Proposition 4.11. If f ∈C(I,I) for whichS(f)=Ω(f) in (K∗,H∗), thenΩ: (C(I,I),
◦ )→(K∗,H∗) is continuous at f.
Proof. Recall that for any f ∈C(I,I),S(f)⊂Ω(f)⊂Ω(f). Let us fix f andε >0. Since
Ω: (C(I,I), ◦ )→(K∗,H∗) is upper semicontinuous at f, there existsδ1>0 so that
Ω(g)⊂Bε/4(Ω(f)) wheneverf −g< δ1. SinceS(f) is dense inΩ(f), there existsδ2>0
so thatΩ(f)⊂Bε/4(S(g)) wheneverf−g< δ2. Iff −g<min{δ1,δ2}, thenΩ(g)⊂
Ω(g)⊂Bε/4(Ω(f))⊂Bε/2(S(g)) ⊂Bε/2(Ω(g)) so that Ω(g)⊂Bε/2(Ω(f)) and Ω(f)⊂
Ω(f)⊂Bε/4(Ω(g)). It follows that H(Ω(g),Ω(f))< ε/2, and Ω: (C(I,I), ◦ )→(K∗,
H∗) is continuous atf.
It remains to show thatS(f)=Ω(f) for the typical f inC(I,I).
Proposition 4.12. The setG= { f ∈C(I,I) :S(f)=Ω(f)}is residual in (C(I,I), ◦ ).
Proof. Let Bn= { f ∈C(I,I) :H∗(S(f),Ω(f))>1/n}. It suffices to show that Bn is nowhere dense for anyn.
We first show thatC(I,I)−Bnis dense. Let f ∈Bn. SinceΩ: (C(I,I), ◦ )→(K∗,H∗) is upper semicontinuous atf, there existsδ >0 so thatΩ(g)⊂B4n(Ω(f)) wheneverf−
ofΩ(f). Chooseg∈C(I,I) so that f −g< δand there is a stable periodic orbitKi∈
S(g) so that H(Ki,Li)<1/4nfor i=1, 2, 3,. . .,m. It follows thatΩ(g)⊂B1/4n(Ω(f))⊂
B1/2n({Li}im=1)⊂B1/2n(S(g)), so thatH∗(S(g),Ω(g))<1/2n.
We now show thatC(I,I)−Bnis open. Letf ∈C(I,I) such thatH∗(S(f),Ω(f))=σ < 1/n; say 1/n−σ=ε. Chooseδ1>0 so thatΩ(g)⊂Bε/4(Ω(f)) wheneverf−g< δ1, and
take{Li}mi=1⊂S(f) with the property thatH∗({Li}im=1,Ω(f))< σ+ε/4. Since{Li}mi=1⊂
S(f), there existsδ2>0 so thatf−g< δ2implies the existence, for anyi=1, 2,. . .,m, of Ki∈S(g) so thatH(Ki,Li)< ε/4. Letg∈C(I,I) withf −g<min{δ1,δ2}. ThenΩ(g)⊂ Bε/4(Ω(f))⊂Bσ+ε/2({Li}mi=1)⊂Bσ+3ε/4({Ki}mi=1)⊂Bσ+3ε/4(S(g)), so thatH∗(S(g),Ω(g))<
σ+ 3ε/4<1/n.
From Propositions4.11and4.12it now follows immediately thatΩ: (C(I,I), ◦ )→
(K∗,H∗) is continuous on a residual subset ofC(I,I).
Theorem 4.13. The mapΩ: (C(I,I), ◦ )→(K∗,H∗) given by f →Ω(f) is continuous at a residual set of functions f inC(I,I).
5. The relationship between stability and chaos
The goal of this section is to determine the relationship between the chaotic nature of a function f in C(I,I) and the behavior of the maps Λ: (C(I,I), ◦ )→(K,H) and Ω: (C(I,I), ◦ )→(K∗,H∗) at that function. We begin by considering functions f that are not chaotic in the sense of Li-Yorke, and then consider the evolving behavior ofΛ: (C(I,I), ◦ )→(K,H) and Ω: (C(I,I), ◦ )→(K∗,H∗) as we make the function f
progressively more chaotic.
Lemma 5.1. Suppose f ∈C(I,I) is not chaotic in the sense of Li-Yorke. Then one of the following possibilities must hold:
(1) the mapsΛ: (C(I,I), ◦ )→(K,H) andΩ: (C(I,I), ◦ )→(K∗,H∗) are both continuous at f;
(2) the mapsΛ: (C(I,I), ◦ )→(K,H) andΩ: (C(I,I), ◦ )→(K∗,H∗) are both discontinuous at f.
Proof. If f is not chaotic in the sense of Li-Yorke, then f has zero topological entropy, so thatΛandΩare either both continuous or discontinuous together at f. This follows
from Theorems3.1and4.8.
As the next pair of examples shows, each of the situations described in Lemma5.1
is possible. Suppose f(x)=0 for allx∈I. Then f is not Li-Yorke chaotic and both Λ andΩare continuous there. This follows from the observation thatS(f)=CR(f)= {0}. Now, let f(x)=x for allx∈I. Then f is not Li-Yorke chaotic and bothΛandΩare discontinuous there. We note thatS(f)= ∅whereasCR(f)=[0, 1].
Proposition 5.2. LetE= {f ∈C(I,I) :f has zero topological entropy}. If f is an element ofEchaotic in the sense of Li-Yorke, then the mapsΛ: (C(I,I), ◦ )→(K,H) andΩ: (C(I,I), ◦ )→(K∗,H∗) are both discontinuous atf.
Proof. Let f ∈Ebe chaotic in the sense of Li-Yorke. Since f ∈E,Λand Ωwill either be continuous or discontinuous together at f. From [7] we know that f must possess a simple systemLwith nonempty interior. Sinceint(L)∩S(f)= ∅andint(L)⊂CR(f), we see thatS(f)CR(f), and our conclusion follows from Theorem3.10.
We now apply Proposition5.2to functions f for which the mapωf :I→K is not in the first class of Baire but do still possess zero topological entropy.
Corollary 5.3. Suppose f is an element ofEand the mapωf :I→K is not in the first class of Baire. ThenΛ: (C(I,I), ◦ )→(K,H) andΩ: (C(I,I), ◦ )→(K∗,H∗) are both discontinuous at f.
With Proposition5.4we consider the behavior ofΛandΩat a function f possessing positive topological entropy.
Proposition 5.4. LetT = {f ∈C(I,I) :f has positive topological entropy}, with f ∈T. Then one of the following possibilities must hold:
(1)ΛandΩare both continuous at f;
(2)Λis continuous at f, butΩis discontinuous there; (3)ΛandΩare both discontinuous at f.
Proof. This proposition follows readily from Theorems3.1and4.7.
We provide examples illustrating each of the three possibilities found in Proposi-tion5.4. From Theorem4.13we know thatΩis continuous on a residual subset ofC(I,I). SinceT is also residual inC(I,I), it follows that the set{f ∈T:Ωis continuous at f}is also residual inC(I,I). Thus, our first possibility holds on a residual subset ofC(I,I).
As for the second possibility, consider the hat maph(x) given byx→2xforx∈[0, 1/2] and x→2(1−x) forx∈(1/2, 1]. Then S(h)=CR(h)=[0, 1], so that Λ: (C(I,I), ◦ )→(K,H) is continuous ath. Since{0} ∈P(h)−S(h), by Theorem4.7 we see that Ω: (C(I,I), ◦ )→(K∗,H∗) is discontinuous at f.
We turn our attention to the third possibility. Consider a function f ∈T possessing a wandering interval Lsuch that the closure of the orbit ofL contains a basic set ω0.
SinceS(f)∩int(orb(L))= ∅andint(orb(L))∩CR(f)= ∅, we see thatS(f)CR(f), so thatΛ: (C(I,I), ◦ )→(K,H), and henceΩ: (C(I,I), ◦ )→(K∗,H∗), must be discontinuous there.
Acknowledgments
The author would like to thank Professor Jaroslav Sm´ıtal and the Silesian University for their support and hospitality; much of the work found in Section 4 was done in conjunction with Professor Sm´ıtal while visiting the Institute of Mathematics at the Sile-sian University. This paper endeavors to draw together many of the results from [17,19–
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T. H. Steele: Department of Mathematics, Weber State University, Ogden, UT 84408-1702, USA
