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The persistence of ω-limit sets defined on compact spaces
J. Math. Anal. Appl. 413 (2014) 789–799
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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
The persistence of ω-limit sets defined on compact spaces ✩ Emma D’Aniello a , T.H. Steele b,∗ a Dipartimento di Matematica e Fisica, Seconda Università degli Studi di Napoli, Viale Lincoln n. 5, 81100 Caserta, Italy b Department of Mathematics, Weber State University, Ogden, UT 84408-1702, USA
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 14 June 2013 Available online 14 December 2013 Submitted by B. Cascales Keywords: ω-Limit set Transport property Hausdorff metric topology
Let K be the class of closed subsets of a compact metric space X, and K consist of the nonempty closed subsets of K. We study the maps f → L(f ) and f → L (f ) defined so that L(f ) is the collection of ω-limit sets of f , and L (f ) = {L ⊆ X: L is closed, f (L) = L, and F ∩ f (L \ F ) = ∅ whenever F is a nonempty and proper closed subset of L}. We show that L (f ) is always closed in K, hence L (f ) ∈ K , and that the map L : C(X, X) → K is upper semicontinuous. Using the notion of a periodic orbit stable under perturbation, we give a sufficient condition on f for L to be continuous there, and establish a residual subset of C(M, M ), when M is an n-manifold with n 1, where L is continuous. These results on L are fundamental to our study of the map L. We characterize those f in C(I, I) at which L is continuous, and show that L is continuous on a residual subset of C(I, I). Similarly, the map f → L(f ) is continuous on a residual subset of C(M, M ), and we characterize those functions in C(M, M ) at which f → L(f ) is upper semicontinuous. © 2013 Elsevier Inc. All rights reserved.
1. Introduction Let X be a compact metric space, with C(X, X) the collection of continuous self maps of X. For x ∈ X and f ∈ C(X, X), let γ(x, f ) := {x, f (x), f (f (x)), . . .} be the trajectory of x with the ω-limit set ω(x, f ) given by the collection of the subsequential limit points of γ(x, f ). Take Λ(f ) := x∈X ω(x, f ) as the set of ω-limit points of f , with L(f ) := {ω(x, f ): x ∈ X} being the collection of ω-limit sets of f . Finally, set L (f ) := L ⊆ X: L is closed, f (L) = L, and F ∩ f (L \ F ) = ∅
whenever F is a nonempty and proper closed subset of L .
We consider the maps Λ, L and L taking f → Λ(f ), f → L(f ) and f → L (f ), respectively. ✩ This research has been partially supported by “Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni dell’Istituto Nazionale di Alta Matematica F. Severi”. * Corresponding author. E-mail addresses: [email protected] (E. D’Aniello), [email protected] (T.H. Steele).
0022-247X/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jmaa.2013.12.026
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At the Twentieth Summer Symposium in Real Analysis, A.M. Bruckner posed several questions regarding the iterative stability of continuous functions as they undergo small perturbations, as well as why these questions are of general interest [3]. In particular: How is the collection of ω-limit sets affected by slight changes in the generating function? To make precise the meaning of stability, in addition to the compact space X with metric d(·,·), we work in three metric spaces. Within C(X, X) we use the supremum metric given by f − g = sup{d(f (x), g(x)): x ∈ X}. Our second metric space (K, H) is composed of the class of nonempty closed sets K in X endowed with the Hausdorff metric H given by H(E, F ) = inf{δ > 0: E ⊂ Bδ (F ), F ⊂ Bδ (E)}, where Bδ (A) := {x ∈ X: d(x, A) < δ} and d(x, A) := inf{d(x, a): a ∈ A}. This space is compact [4]. Our final metric space (K , H ) consists of the nonempty closed subsets of K. Thus, K ∈ K if K is a nonempty family of nonempty closed sets in X such that K is closed in K with respect to H. We endow K with the metric H so that K1 and K2 are close with respect to H if each member of K1 is close to some member of K2 with respect to H, and vice-versa. This metric space also is compact [3,13]. Hence, if f , g ∈ C(X, X) are close with respect to the supremum norm, then we investigate whether the corresponding sets φ(f ) and φ(g) are close with respect to a Hausdorff metric, where φ denotes Λ, L and L . As one sees from various examples found in [3] and [19], both the set Λ(f ) and the collection L(f ) can be affected dramatically by small perturbations. When one considers continuous self-maps of the unit interval I = [0, 1], considerable progress has been made towards resolving Bruckner’s queries. Theorem 1.1. (See [16].) The map Λ : C(I, I) → K is continuous at f if and only if the p-stable periodic orbits of f are dense in the set of chain recurrent points of f . Theorem 1.2. (See [16].) The map L : C(I, I) → K is continuous at f if and only if all of the following hold: (1) the periodic points of f are dense in CR(f ), (2) all the periodic points of f belong to p-stable periodic orbits, (3) if L ∈ K, f (L) = L and, for any proper closed subset F of L, F ∩ f (L \ F ) = ∅, then L ∈ L(f ). Let f ∈ C(I, I). A point x ∈ I is said to be recurrent if x ∈ ω(x, f ) and nonwandering if every open set containing x contains at least two points of some trajectory. Let R(f ) and N W (f ) denote respectively the set of recurrent and nonwandering points. The closure R(f ) of the set of recurrent points is known as the center of f . Let C(f ) denote the center of f [1]. Theorem 1.3. (See [16,9].) Let φ : C(I, I) → K be any of the maps f → Λ(f ), f → C(f ), f → N W (f ), f → CR(f ). Then the following conditions are equivalent: (1) φ is continuous at f ∈ C(I, I), (2) the set of points in p-stable periodic orbits of f is dense in CR(f ), (3) Λ(f ) = CR(f ) and the set of points in p-stable periodic orbits of f is dense in the set of periodic points of f . Suppose f is a continuous self-map of a compact metric space X, and ω ∈ L(f ). From the definition of an ω-limit set, one sees easily that ω is closed in X, f is strongly invariant on ω so that f (ω) = ω, and ω has the following transport property known as “weak incompressability” [15]: F ∩ f (ω \ F ) = ∅ whenever F is a proper, nonempty and closed subset of ω [1]. Theorem 1.3 shows that these three properties of ω-limit sets must, in fact, characterize the elements of L(f ) whenever L : C(I, I) → K is continuous there. The
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goal of this paper is to better understand the map f → L(f ) for the general compact metric space X. We do this by first considering the map L : C(X, X) → K . In Section 2 we recall some definitions, and results needed in the sequel. In Section 3 we develop the basic properties of the set L (f ) as well as those of the map L : C(X, X) → K . Interestingly, L (f ) frequently behaves as a set valued analogue of CR(f ), the chain recurrent set of f . In particular, CR(f ) is closed for any f in C(X, X), and so is L (f ). The map f → CR(f ) is upper semicontinuous, as is f → L (f ). Moreover, if x ∈ CR(f ) and X is an n-manifold, then for any > 0 there is a map g in B (f ) so that x is a periodic point of g. Similarly, if L ∈ L (f ) and > 0, then there exists g in B (f ) and a periodic orbit K of g so that H(L, K) < , where H is the Hausdorff metric. Proposition 3.2 shows that L is continuous at f whenever the p-stable periodic orbits of f are dense in L (f ), and Theorem 3.7 establishes the continuity of L on a residual subset of C(M, M ), whenever M is a compact n-manifold with n 1. Our principal results are found in Section 4. Theorem 4.1 develops another characterization of those maps f in C(I, I) at which L : C(I, I) → K is continuous, and Theorem 4.2 establishes the continuity of L on a residual subset of C(I, I). Theorem 4.4 establishes the continuity of f → L(f ) on a residual set of functions in the more general C(M, M ), and Theorem 4.3 characterizes those functions in C(M, M ) at which f → L(f ) is upper semicontinuous. We conclude with a condition on f ∈ C(X, X) sufficient to insure the continuity of L : C(X, X) → K at f . 2. Preliminaries We shall be concerned with C(X, X) and the iterative properties this class of functions possesses. For f in C(X, X) and any integer n 1, f n denotes the n-th iterate of f . We also take f 0 to be the identity map, defined by f 0 (x) = x for every x in X. A point x ∈ X is said to be a periodic point of f with period m if f m (x) = x, f k (x) = x for 1 k < m. Let P (f ) represent those points x ∈ X that are periodic under f , and let P (f ) be the set of periodic orbits of f . If K is a periodic orbit of f of period n 1, and for any > 0 there exists δ > 0 so that any g ∈ Bδ (f ) = {h ∈ C(X, X): h − f < δ} has a periodic orbit L of period n satisfying H(L, K) < , then we call K a perturbation-stable, or p-stable, periodic orbit of f . Let S(f ) be the collection of the p-stable periodic orbits of f with S(f ) = {x ∈ P (f ): ω(x, f ) ∈ S(f )} the set of p-stable periodic points. Should x be a periodic point of period n for f in C(I, I), a sufficient condition for x to be p-stable is that f n (x) − x not be unisigned on any neighborhood of x. The set CR(f ) of all chain recurrent points of f is defined by x ∈ / CR(f ) if there exists an open set U with f (U ) ⊆ U such that x ∈ / U , f (x) ∈ U . Let > 0 and let x and y be points of X. An -chain, or pseudo-orbit, from x to y is a finite sequence {x0 , x1 , . . . , xn } of points of X with x0 = x and xn = y and d(f (xk−1 ), xk ) < for k = 1, . . . , n. A point x ∈ X is chain recurrent if and only if, for every > 0, there is an - chain from x to itself [1, p. 112]. For A ⊆ X, we denote by diam(A) the diameter of A, that is diam(A) := sup{d(x, y): x, y ∈ A} and by (A)0 the interior of A, that is the largest open set contained in A. In Section 3, we will occasionally restrict our attention to elements of C(M, M ), where M is an n-manifold with or without boundary. Unless otherwise stated M will denote an n-manifold with or without boundary. All of our manifolds are compact metric spaces [11]. The main property of manifolds used is that every point of a manifold has an arbitrarily small neighborhood whose closure is homeomorphic to I n , the unit cube of Rn . It is well known that every continuous map f : I n → I n has a fixed point [14]. Therefore, if M is homeomorphic to I n and g : M → M is continuous, then g has a fixed point. If φ is a function from X into the class of nonempty subsets of a metric space Y , then we say that φ is upper semicontinuous (lower semicontinuous) if for each open (resp. closed) subset V of Y the set {x ∈ X: φ(x) ⊆ V } is open (resp. closed) in X. Hence, in the case of F : (C(X, X), · ) → (K, H) with
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f ∈ C(X, X), F is upper semicontinuous at f if for any > 0 there exists δ > 0 so that F (g) ⊆ B (F (f )) whenever f − g < δ. That is, for g sufficiently close to f in C(X, X), if y ∈ F (g) closed in X, then d(y, F (f )) = inf{d(y, z): z ∈ F (f )} < . In the case of F : (C(X, X), · ) → (K , H ), F (g) and F (f ) are collections of closed subsets of X, each closed with respect to K. The map F is upper semicontinuous at f if for any > 0 there exists δ > 0 so that F (g) ⊂ B (F (f )) whenever f − g < δ. That is, for g sufficiently close to f in C(X, X), if K ∈ K is in F (g) then d(K, F (f )) = inf{H(K, L): L ∈ F (f )} < . In our study of the map L : C(X, X) → K given by f → L (f ) we make use of the notion of oscillation. Define the oscillation of L on B (f ) such that OL B (f ) = sup H L (f1 ), L (f2 ) : f1 , f2 ∈ B (f ) . We take the oscillation of L at f to be OL (f ) = inf OL B (f ) = lim+ OL B (f ) . >0
→0
It follows that {f ∈ C(X, X): OL (f ) < } is open for every > 0, and L is continuous at f if and only if OL (f ) = 0 [4]. We now turn our attention to the Baire category theorem. Let (X, d) be a metric space. A set is of the first category in (X, d) if it can be written as a countable union of nowhere dense sets; otherwise, the set is of the second category. A set is residual if it is the complement of a first category set; an element of a residual subset of (X, d) is called a typical element of X. With these definitions in mind, we recall Baire’s theorem on category. Theorem 2.1. Let (X, d) be a complete metric space with S a first category subset of X. Then X \ S is dense in X. We make reference to the following three results. Theorem 2.2. (See [8].) If f ∈ C(I, I) and P (f ) is dense in the set of chain recurrent points of f , then L(f ) is contained in the Hausdorff closure of the periodic orbits of f . Theorem 2.3. (See [11].) If M is homeomorphic to I n , and f is a continuous function from a closed subset F of M into M , then f can be extended so that the domain of f is all of M , and f (M ) ⊆ M . For the sake of convenience, we say that a subset J of an n-manifold M is an n-cube if J is homeomorphic to I n and J has nonempty interior relative to M . Lemma 2.4. (See [6].) Suppose x ∈ M , f ∈ C(M, M ), and x = x0 , x1 , . . . , xm are distinct points such that f (xi ) = xi+1 for all 0 i m − 1 and f (xm ) = xl for some 0 l m. Then, for every > 0, there is g ∈ B (f ) and n-cubes J0 , J1 , . . . , Jm with diameters less than such that (1) xi ∈ (Ji )0 and diam(Ji ) < for all i, (2) g(xi ) = f (xi ) for 0 i m, (3) g(Ji ) ⊆ Ji+1 for 0 i m − 1, and g(Jm ) ⊆ Jl . 3. The map L : C(X, X) → K This section is dedicated to developing the basic properties of the set L (f ) and the map L : C(X, X) → K . From [2], we know that {ω(x, f ): x ∈ I} is closed in (K, H) whenever f ∈ C(I, I). While the collection
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of ω-limit sets need not be closed when f ∈ C(X, X) for the arbitrary compact metric space X [10], the set L (f ) is always closed, and the map f → L (f ) is upper semicontinuous. These results follow readily from the definition of L (f ) as well as the observations that (K, H) is compact and C(X, X) is complete with respect to the supremum norm. Since they follow the analysis for continuous self-maps of the interval found in [18, Propositions 4.9 and 4.10], we record them in the first proposition. Proposition 3.1. Let X be a compact metric space. If f ∈ C(X, X), then L (f ) is closed in (K, H), and the map L : (C(X, X), · ) → (K , H ) is upper semicontinuous. Proposition 4.11 in [18] shows that L is continuous at some f whenever S(f ) = L (f ). Since L(f ) is closed whenever f ∈ C(I, I), this is equivalent to requiring S(f ) = L(f ) = L (f ). Our next proposition extends this result to the continuity of L at f in C(X, X) whenever S(f ) = L (f ). Proposition 3.2. If f ∈ C(X, X) and S(f ) = L (f ), then L : C(X, X) → K is continuous at f . Proof. Here, the closure S(f ) is taken with respect to the topology generated by the metric space (K, H). Notice that for f in C(X, X), S(f ) ⊆ L (f ) since a periodic orbit of f belongs necessarily to the closed set L (f ). Let us fix f and > 0. Since L : C(X, X) → K is upper semicontinuous at f , there exists δ1 > 0 so that L (g) ⊂ B (L (f )) whenever f − g < δ1 . Since S(f ) is dense in L (f ), there exists δ2 > 0 so that L (f ) ⊂ B (P (g)) whenever f − g < δ2 . In particular, since L (f ) is closed in (K, H) compact, there exist n L1 , . . . , Ln in L (f ) such that L (f ) ⊆ i=1 B (Li )). Since S(f ) is dense in L (f ), for every 1 i n, there exists Ki ∈ S(f ) such that H(Li , Ki ) < 2 . As each Ki is a p-stable periodic orbit, there exists δ2 > 0 such that any g ∈ Bδ2 (f ), for every 1 i n, has a periodic orbit Ti satisfying H(Ti , Ki ) < 2 , and hence H(Ti , Li ) H(Ti , Ki ) + H(Ki , Li ) <
+ = . 2 2
Therefore, L (f ) ⊂
n
B Li ⊆ B P (g)
i=1
whenever f − g < δ2 . If f − g < min{δ1 , δ2 }, then L (g) ⊂ B (L (f )) and L (f ) ⊂ B (P (g)) ⊂ B (L (g)). It follows that H(L (g), L (f )) < , and L : C(X, X) → K is continuous at f . Our goal is to describe a residual subset of C(M, M ) on which L : C(M, M ) → K is continuous. We continue our analysis on C(M, M ) since we can apply the fixed point properties on n-cubes to elements of C(M, M ); this fixed point property need not hold for continuous self-maps of an arbitrary compact metric space X. The following lemma gives some insight into how the elements of L (f ) are “almost” ω-limit sets. It generalizes the analogous result in the case of M = I contained in Proposition 4.5 in [18]. We point out that the proof, in the more general setting of M an n-manifold, requires a more thoughtful application of the Tietze extension theorem as well as Lemma 2.7 from [7]. Lemma 3.3. Let f ∈ C(M, M ) with L ∈ L (f ) and > 0. There exist g in B (f ) and K a periodic orbit of g so that H(L, K) < . Proof. As M is a compact n-manifold, as remarked in [7], there exists ˜ > 0 such that, for each x ∈ M , B˜(x) is an n-cube. Therefore, for every x ∈ M , B (x) is an n-cube for every 0 < < ˜. Fix > 0 with
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< ˜. Since f ∈ C(M, M ) is uniformly continuous on M , there exists δ1 > 0 so that d(f (x), f (y)) < 4 whenever d(x, y) < δ1 . Choose δ > 0 so that δ < min{δ1 , 4 }. As L is a compact metric space, it is totally n bounded. Hence, there exists a δ-net {x1 , . . . , xn } for L; that is, L ⊆ i=1 Bδ (xi ) with {x1 , . . . , xn } ⊆ L. We perturb f to get a function g in C(M, M ) possessing a periodic ω-limit set ω so that f − g < and n {x1 , . . . , xn } ⊆ ω ⊆ i=1 Bδ (xi ), as this implies H(ω, L) < . Suppose n = 1, so that f (Bδ (x1 )) ⊆ B (x1 ). Using the Tietze extension theorem [11, Corollary 1, p. 82], we find g ∈ C(M, M ) so that (1) f − g < , (2) g(x) = f (x) whenever x ∈ X \ Bδ (x1 ), and (3) g(x1 ) = x1 . Take h such that h(x1 ) = x1 and h = f on M \ Bδ (x1 ). As Bδ (x1 ) is an n-cube, by Tietze’s extension theorem, h can be extended to a map g on all of M such that g is continuous, and g(Bδ (x1 )) ⊆ B (x1 ). This is the desired function g. Let us now prove the existence of such a g in the case n = 2. Say L ⊆ Bδ (x1 )∪Bδ (x2 ). From our transport property, we know that F ∩ f (L \ F ) = ∅ for any nonempty proper closed set F ⊆ L. Let F = L ∩ Bδ (x1 ). It follows that there exists x ∈ Bδ (x2 ) such that f (x) ∈ Bδ (x1 ). Now d f (x2 ), x1 d f (x2 ), f (x) + d f (x), x1 < + = . 4 4 2
()
Similarly, d f (x1 ), x2 < . 2 Hence, f Bδ (x1 ) ⊆ B (x2 )
and f Bδ (x2 ) ⊆ B (x1 ).
Choose δ˜ so that Bδ˜(x1 ) ∩ Bδ˜(x2 ) = ∅. Now, using the Tietze extension theorem, we find g ∈ C(M, M ) so that (1) f − g < , (2) g(x) = f (x) whenever x ∈ X \ Bδ˜(x1 ) ∪ Bδ˜(x2 ), and (3) g(x1 ) = x2 and g(x2 ) = x1 . Let us see this in detail: Take h such that h(x1 ) = x2 , h(x2 ) = x1 and h = f on M \ Bδ˜(x1 ) ∪ Bδ˜(x2 ), where δ˜ is chosen so that Bδ˜(x1 ) ∩ Bδ˜(x2 ) = ∅. By Tietze’s extension theorem h can be extended to a map g on all of M such that g is continuous, g(Bδ (x1 )) ⊆ B (x2 ) and g(Bδ (x2 )) ⊆ B (x1 ). This is the desired function g. The proof for n 3 follows by induction and from the fact that F ∩ f (L \ F ) = ∅ for any nonempty proper closed set F ⊆ L: (1) Let F = L ∩ ( j=i Bδ (xj )) to see that there exists x in Bδ (xi ) such that f (x) ∈ Bδ (xj ) for some j = i, for any i = 1, 2, . . . , n. (2) Let F = L ∩ Bδ (xi ) to see that there exists x ∈ Bδ (xj ) for some j = i so that f (x) ∈ Bδ (xi ), for any i = 1, 2, . . . , n. (3) Let T ⊆ {1, 2, . . . , n} with F = L ∩ ( j ∈T / Bδ (xj )) to see that there exists x ∈ i∈T Bδ (xi ) so that f (x) ∈ Bδ (xj ) for some j ∈ {1, 2, . . . , n} \ T .
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Lemma 3.4. Let f ∈ C(M, M ) with K a periodic orbit of f . Then for any > 0 there exists g in B (f ) so that K ∈ S(g). Proof. Let K = {x0 , x1 , . . . , xm } so that xi = xj whenever i = j, f (xi ) = xi+1 for 0 i m − 1, and f (xm ) = x0 . Fix > 0. From Lemma 2.4 we may assume that there are disjoint n-cubes V0 , V1 , . . . , Vm 0 with diameters less than so that xi ∈ (Vi ) whenever 0 i m, f (Vi ) ⊆ Vi+1 , for 0 i m − 1 and 0 0 f (Vm ) ⊆ V0 . Let Ui be an n-cube such that Ui ⊆ (Vi ) and xi ⊆ (Ui ) . Using the Tietze extension theorem, we obtain g in B (f ) such that m 0 (1) g = f on M \ i=0 (Vi ) , (2) g(Vi ) ⊆ Vi+1 for 0 i m − 1, and g(Vm ) ⊆ V0 , (3) g(Ui ) = f (xi ) = xi+1 for all 0 i m − 1, and g(Um ) = f (xm ) = x0 . We show that K ∈ S(g), that is, that for any α > 0 there exists τ > 0 so that any h ∈ Bτ (g) has a periodic orbit L of period n satisfying H(L, K) < α. Let α > 0, and take 0 < σ < α so that Bσ (xi ) ⊆ (Ui )0 , for all 0 i m. From our choice of g, if d(g, h) < τ = σ2 , then h(Bσ (xi )) ⊆ Bσ (xi+1 ) whenever 0 i m − 1, and h(Bσ (xm )) ⊆ Bσ (x0 ). In particular, hm (Bσ (xi )) ⊆ Bσ (xi ), so that Bσ (xi ) contains a fixed point yi of hm by the Brower fixed point theorem [12], for all i. It follows that H(K, ω(yi , h)) σ < α. Lemma 3.5. The set In = {f ∈ C(M, M ): H (S(f ), L (f )) <
1 n}
is dense in C(M, M ).
Proof. Let f ∈ C(M, M ) so that H (S(f ), L (f )) n1 . Since L : C(X, X) → K is upper semicontinuous 1 (L (f )) whenever g ∈ Bδ (f ). Since L (f ) is closed in K at f , there exists δ > 0 so that L (g) ⊂ B 4n 1 m and (K, H) is compact, there exist {Li }m i=1 ⊂ L (f ) so that {Li }i=1 is a 4n -net of L (f ). We construct 1 for g ∈ C(M, M ) so that g ∈ Bδ (f ) and there is a p-stable periodic orbit Ki ∈ S(g) so that H(Ki , Li ) < 4n i = 1, 2, . . . , m. We do this by applying Lemmas 3.3 and 3.4 to each Li in turn and insuring that Kj ∩Kl = ∅ for l = 1, 2, . . . , j − 1 as j goes from 1 to m. This allows us to develop an appropriate function g even if the sets Li are not pairwise disjoint. It follows that 1 1 1 L (g) ⊂ B 4n S(g) , L (f ) ⊂ B 2n {Li }m i=1 ⊂ B 2n so that 1 . H S(g), L (g) < 2n Lemma 3.6. If f ∈ In , then OL (f ) <
3 n.
Proof. Let f ∈ In . We recall that S(f ) ⊆ L (f ) for all f in C(M, M ), and that L : C(M, M ) → K is upper semicontinuous. Let > 0. It follows, then, that there exists δ > 0 so that, whenever gi ∈ Bδ (f ), L (gi ) ⊆ B L (f ) ⊆ B+ n1 S(f ) and S(f ) ⊆ B P (gi ) ⊆ B L (gi ) . Thus, 1 H S(f ), L (gi ) < + , n
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so that 2 H L (g1 ), L (g2 ) < 2 + n for any {g1 , g2 } ⊆ Bδ (f ), and hence 2 OL Bδ (f ) 2 + . n We conclude that 3 2 OL (f ) = lim OL Bδ (f ) < . δ→0 n n We are now in a position to establish the section’s main result. Theorem 3.7. The map L : C(M, M ) → K is continuous on a residual subset of C(M, M ). Proof. The Baire category theorem is applicable to C(M, M ) since this space is complete with the supremum norm [5,4]. Recall from [4] that En = {f ∈ C(M, M ): OL (f ) < n1 } is open so that, from Lemmas 3.5 and 3.6, En is both dense and open. Now, E :=
∞
En = f ∈ C(M, M ): OL (f ) = 0 = f ∈ C(M, M ): L is continuous at f ,
n=1
is a dense Gδ subset of C(M, M ). The set L(f ) need not be closed for an arbitrary element of C(M, M ) [10]. As the final result of the section shows, however, L(f ) is typically dense in L (f ). Theorem 3.8. The set T = {f ∈ C(M, M ): L(f ) = L (f )} is residual in C(M, M ). Proof. It suffices to show that In = {f ∈ C(M, M ): H (L(f ), L (f )) < n1 } contains a dense and open subset of C(M, M ). Using the notation from Lemma 3.5, since In ⊆ In , and In is dense, it follows immediately that In is dense, too. Now, let f ∈ In . We show that there exists some δ > 0 so that Bδ (f ) ⊆ In . Say H (L(f ), L (f )) = σ ˜ < n1 , choose σ > 0 with σ ˜ < σ < n1 and set γ = n1 − σ. Since the map L is upper semicontinuous, there exists δ1 > 0 so that L (g) ⊆ B γ4 (L (f )) whenever f − g < δ1 . By the definition of a p-stable periodic orbit, there exists δ2 > 0 so that S(f ) ⊆ B γ4 (P (g)) whenever f − g < δ2 . Now, take δ < min{δ1 , δ2 } with f − g < δ. Then L (g) ⊆ B γ4 L (f ) ⊆ Bσ+ γ2 S(f ) ⊆ Bσ+ 34 γ P (g) ⊆ Bσ+ 34 γ L(g) , so we conclude that H (L(f ), L (f )) <
1 n.
4. The map f → L(f ) Let C = {f ∈ C(I, I): P (f ) = CR(f ), S(f ) = P (f ) and L(f ) = L (f )}, and D = {f ∈ C(I, I): S(f ) = L (f )}. From [16], the map L : C(I, I) → K is continuous at f in C(I, I) if and only if f ∈ C. In [17], it is proved that the map L : C(I, I) → K is upper semicontinuous at f if and only if L(f ) ⊇ L (f ), and, from
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Proposition 3.3 in [17], it follows that if S(f ) = L(f ) then the map L : C(I, I) → K is lower semicontinuous. In this section we endeavor to extend these results and to better understand the map f → L(f ) on more general metric spaces by taking advantage of the recent results concerning L . Theorem 4.1. The map L : C(I, I) → K is continuous at f if and only if S(f ) = L(f ) = L (f ). Proof. As, from [16], the map L : C(I, I) → K is continuous at f in C(I, I) if and only if f ∈ C, it suffices to show that C = D. Let f ∈ C; since P (f ) = CR(f ), from [8] it follows that every ω-limit set of f is contained in the Hausdorff closure of periodic orbits. Now, S(f ) = P (f ), so, in fact, S(f ) = L(f ) = L (f ). Thus, f ∈ D, and one concludes that C ⊆ D. Now, let f ∈ D. As we saw in the proof of Proposition 3.2, since S(f ) = L(f ), L is lower semicontinuous at f , and since L(f ) = L (f ), L is upper semicontinuous at f . It follows that L : C(I, I) → K is continuous at f , so that f ∈ C, too, and D ⊆ C. By applying Lemmas 3.5 and 3.6 to the specific case of continuous self-maps of I, we establish the continuity of L : C(I, I) → K on a residual subset of C(I, I). This result was announced in Theorem 4.13 in [18]. The proof there, however, is irreparably flawed. The proof presented below necessarily rests on a fundamentally different analysis. Theorem 4.2. The map L : C(I, I) → K is continuous on a residual subset of C(I, I). Proof. From [4] we know that Tn = {f ∈ C(M, M ): OL(f ) < closed in K for all f in C(I, I) [2], it follows that
3 n}
is open, and since {ω(x, f ): x ∈ I} is
S(f ) ⊆ L(f ) ⊆ L (f ). In particular, Tn is both dense and open. One now proceeds as in the proof of Theorem 3.7. When working in C(I, I), one has the benefit of knowing that L(f ) is closed in K for every f in C(I, I). As one sees from [10], L(f ) need not be closed when f ∈ C(M, M ), for an arbitrary n-manifold M . Because of this, we consider the map L : C(M, M ) → K given by f → L(f ) as we turn our attention to analogues of Theorems 4.1 and 4.2 on the more general space C(M, M ). The next result was proved for L in [18] in the case of continuous self-maps of the interval, for which L(f ) = L(f ). We generalize this result to the map L defined on the space of continuous self-maps of an n-manifold. Theorem 4.3. The map L : C(M, M ) → K is upper semicontinuous at f if and only if L(f ) = L (f ). Proof. Suppose L : C(M, M ) → K is upper semicontinuous at f , and let L ∈ L (f ). We show that L ∈ L(f ). By Lemma 3.3 there exists {fn } ⊆ C(M, M ) with Kn ∈ L(fn ) ⊆ L(fn ) = L(fn ) for any n so that limn fn = f and limn Kn = L. Since L : C(M, M ) → K is upper semicontinuous at f , L ∈ L(f ). Now, suppose that L(f ) = L (f ); we show that L : C(M, M ) → K is upper semicontinuous at f . Let {fn } ⊆ C(M, M ) with Ln ∈ L(fn ) for any n so that fn → f and Ln → L. Since L ∈ L (f ), and L(f ) = L (f ), it follows that L ∈ L(f ) and L : C(M, M ) → K is upper semicontinuous at f . Theorem 4.4. The map L : C(M, M ) → K is continuous on a residual subset of C(M, M ). Proof. Recall from [4] that Sn = {f ∈ C(M, M ): OL(f ) < n3 } is open. It suffices to show that In = {f ∈ C(M, M ): H (S(f ), L (f )) < n1 } ⊆ Sn , as In is dense by Lemma 3.5. Let f ∈ In and > 0. Arguing as in Lemma 3.6 we find δ > 0 so that, whenever gi ∈ Bδ (f ),
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L (gi )) ⊆ B+ n1 S(f ) , so that, as L(gi ) ⊆ L (gi ), L(gi )) ⊆ B+ n1 S(f ) . Moreover, S(f ) ⊆ B P (gi ) , so that, as P (gi ) ⊆ L(gi ), S(f ) ⊆ B L(gi ) . Thus, 1 H S(f ), L(gi ) < + , n so that 2 H L(g1 ), L(g2 ) < 2 + n for any {g1 , g2 } ⊆ Bδ (f ), and hence 2 OL Bδ (f ) 2 + . n We conclude that 3 2 OL(f ) = lim OL Bδ (f ) < . δ→0 n n Now, S :=
∞
Sn = f ∈ C(M, M ): OL(f ) = 0 = f ∈ C(M, M ): L is continuous at f ,
n=1
is a dense Gδ subset of C(M, M ). One now needs only recall that the Baire category theorem is applicable to C(M, M ) since this space is complete with respect to the supremum norm [5,4]. We conclude by showing that the characterization of those f in C(I, I) at which L is continuous, is sufficient on any compact space X. It would be interesting to know if the converse is true, so that S(f ) = L (f ) must necessarily hold whenever L : C(X, X) → K is continuous at f . Theorem 4.5. If f ∈ C(X, X) and S(f ) = L (f ), then L : C(X, X) → K is continuous at f . Proof. Fix f in C(X, X) and > 0. We find δ > 0 so that H (L(g), L(f )) < whenever f − g < δ. Since L : C(X, X) → K is upper semicontinuous, there exists δ1 > 0 so that L (g) ⊂ B (L (f )), and consequently L(g) ⊂ B (L (f )), whenever f − g < δ1 . Since S(f ) is dense in L (f ), there exists δ2 > 0 so that L (f ) ⊆ B (P (g)) ⊆ B (L(g)) whenever f − g < δ2 . If f − g < δ = min{δ1 , δ2 }, then
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L(g) ⊂ B (L (f )) and L (f ) ⊆ B (L(g)), so that H (L(g), L (f )) < . Since S(g) ⊆ L(g) ⊆ L (g) for any g in C(X, X), we have in the case of f that S(f ) = L(f ) = L (f ), and H L(g), L (f ) = H L(g), L(f ) < . References [1] L.S. Block, W.A. Coppel, Dynamics in One Dimension, Lecture Notes in Math., vol. 1513, Springer-Verlag, Berlin, 1992. [2] A. Blokh, A.M. Bruckner, P.D. Humke, J. Smítal, The space of ω-limit sets of a continuous map of the interval, Trans. Amer. Math. Soc. 348 (1996) 1357–1372. [3] A.M. Bruckner, Stability in the family of ω-limit sets of continuous self maps of the interval, in: Summer Symposium in Real Analysis, XX, Windsor, ON, 1996, Real Anal. Exchange 22 (1996/1997) 52–57. [4] A.M. Bruckner, J.B. Bruckner, B.S. Thomson, Real Analysis, Prentice-Hall, New Jersey, 1997. [5] D.L. Cohn, Measure Theory, Birkhäuser, Boston, 1980. [6] E. D’Aniello, U.B. Darji, T.H. Steele, Ubiquity of odometers in topological dynamical systems, Topology Appl. 156 (2008) 240–245. [7] E. D’Aniello, P.D. Humke, T.H. Steele, The space of adding machines generated by continuous self maps of manifolds, Topology Appl. 157 (2010) 954–960. [8] E. D’Aniello, T.H. Steele, Approximating ω-limit sets with periodic orbits, Aequationes Math. 75 (2008) 93–102. [9] J. Dvorakova, Stability of chain recurrent points of continuous maps on interval, J. Difference Equ. Appl. 18 (2012) 1027–1031. [10] J.L. García Guirao, Hausdorff compactness on a space of ω-limit sets, Topology Appl. 153 (2005) 833–843. [11] W. Hurewicz, H. Wallman, Dimension Theory, Princeton University Press, 1948. [12] S. Karamadian, Fixed Points, Algorithms and Applications, Acad. Press, 1977. [13] K. Kuratowski, Topology, PWN and Academic Press Inc., 1968. [14] J.R. Munkres, Topology, Prentice-Hall, New Jersey, 2000. [15] A.N. Sharkovsky, et al., Dynamics of One-Dimensional Maps, Kluwer Academic Publishers, 1997. [16] J. Smítal, T.H. Steele, Stability of dynamical structures under perturbation of the generating function, J. Difference Equ. Appl. 15 (2009) 77–86. [17] T.H. Steele, The persistence of ω-limit set under perturbation of the generating function, Real Anal. Exchange 26 (2000/2001) 963–974. [18] T.H. Steele, Continuity of the maps f → x∈I ω(x, f ) and f → {ω(x, f ): x ∈ I}, Int. J. Math. Math. Sci. (2006), Art. ID 82623, 15 pp. [19] T.H. Steele, Continuity and chaos in discrete dynamical systems, Aequationes Math. 71 (2006) 300–310.
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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
The persistence of ω-limit sets defined on compact spaces ✩ Emma D’Aniello a , T.H. Steele b,∗ a Dipartimento di Matematica e Fisica, Seconda Università degli Studi di Napoli, Viale Lincoln n. 5, 81100 Caserta, Italy b Department of Mathematics, Weber State University, Ogden, UT 84408-1702, USA
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 14 June 2013 Available online 14 December 2013 Submitted by B. Cascales Keywords: ω-Limit set Transport property Hausdorff metric topology
Let K be the class of closed subsets of a compact metric space X, and K consist of the nonempty closed subsets of K. We study the maps f → L(f ) and f → L (f ) defined so that L(f ) is the collection of ω-limit sets of f , and L (f ) = {L ⊆ X: L is closed, f (L) = L, and F ∩ f (L \ F ) = ∅ whenever F is a nonempty and proper closed subset of L}. We show that L (f ) is always closed in K, hence L (f ) ∈ K , and that the map L : C(X, X) → K is upper semicontinuous. Using the notion of a periodic orbit stable under perturbation, we give a sufficient condition on f for L to be continuous there, and establish a residual subset of C(M, M ), when M is an n-manifold with n 1, where L is continuous. These results on L are fundamental to our study of the map L. We characterize those f in C(I, I) at which L is continuous, and show that L is continuous on a residual subset of C(I, I). Similarly, the map f → L(f ) is continuous on a residual subset of C(M, M ), and we characterize those functions in C(M, M ) at which f → L(f ) is upper semicontinuous. © 2013 Elsevier Inc. All rights reserved.
1. Introduction Let X be a compact metric space, with C(X, X) the collection of continuous self maps of X. For x ∈ X and f ∈ C(X, X), let γ(x, f ) := {x, f (x), f (f (x)), . . .} be the trajectory of x with the ω-limit set ω(x, f ) given by the collection of the subsequential limit points of γ(x, f ). Take Λ(f ) := x∈X ω(x, f ) as the set of ω-limit points of f , with L(f ) := {ω(x, f ): x ∈ X} being the collection of ω-limit sets of f . Finally, set L (f ) := L ⊆ X: L is closed, f (L) = L, and F ∩ f (L \ F ) = ∅
whenever F is a nonempty and proper closed subset of L .
We consider the maps Λ, L and L taking f → Λ(f ), f → L(f ) and f → L (f ), respectively. ✩ This research has been partially supported by “Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni dell’Istituto Nazionale di Alta Matematica F. Severi”. * Corresponding author. E-mail addresses: [email protected] (E. D’Aniello), [email protected] (T.H. Steele).
0022-247X/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jmaa.2013.12.026
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At the Twentieth Summer Symposium in Real Analysis, A.M. Bruckner posed several questions regarding the iterative stability of continuous functions as they undergo small perturbations, as well as why these questions are of general interest [3]. In particular: How is the collection of ω-limit sets affected by slight changes in the generating function? To make precise the meaning of stability, in addition to the compact space X with metric d(·,·), we work in three metric spaces. Within C(X, X) we use the supremum metric given by f − g = sup{d(f (x), g(x)): x ∈ X}. Our second metric space (K, H) is composed of the class of nonempty closed sets K in X endowed with the Hausdorff metric H given by H(E, F ) = inf{δ > 0: E ⊂ Bδ (F ), F ⊂ Bδ (E)}, where Bδ (A) := {x ∈ X: d(x, A) < δ} and d(x, A) := inf{d(x, a): a ∈ A}. This space is compact [4]. Our final metric space (K , H ) consists of the nonempty closed subsets of K. Thus, K ∈ K if K is a nonempty family of nonempty closed sets in X such that K is closed in K with respect to H. We endow K with the metric H so that K1 and K2 are close with respect to H if each member of K1 is close to some member of K2 with respect to H, and vice-versa. This metric space also is compact [3,13]. Hence, if f , g ∈ C(X, X) are close with respect to the supremum norm, then we investigate whether the corresponding sets φ(f ) and φ(g) are close with respect to a Hausdorff metric, where φ denotes Λ, L and L . As one sees from various examples found in [3] and [19], both the set Λ(f ) and the collection L(f ) can be affected dramatically by small perturbations. When one considers continuous self-maps of the unit interval I = [0, 1], considerable progress has been made towards resolving Bruckner’s queries. Theorem 1.1. (See [16].) The map Λ : C(I, I) → K is continuous at f if and only if the p-stable periodic orbits of f are dense in the set of chain recurrent points of f . Theorem 1.2. (See [16].) The map L : C(I, I) → K is continuous at f if and only if all of the following hold: (1) the periodic points of f are dense in CR(f ), (2) all the periodic points of f belong to p-stable periodic orbits, (3) if L ∈ K, f (L) = L and, for any proper closed subset F of L, F ∩ f (L \ F ) = ∅, then L ∈ L(f ). Let f ∈ C(I, I). A point x ∈ I is said to be recurrent if x ∈ ω(x, f ) and nonwandering if every open set containing x contains at least two points of some trajectory. Let R(f ) and N W (f ) denote respectively the set of recurrent and nonwandering points. The closure R(f ) of the set of recurrent points is known as the center of f . Let C(f ) denote the center of f [1]. Theorem 1.3. (See [16,9].) Let φ : C(I, I) → K be any of the maps f → Λ(f ), f → C(f ), f → N W (f ), f → CR(f ). Then the following conditions are equivalent: (1) φ is continuous at f ∈ C(I, I), (2) the set of points in p-stable periodic orbits of f is dense in CR(f ), (3) Λ(f ) = CR(f ) and the set of points in p-stable periodic orbits of f is dense in the set of periodic points of f . Suppose f is a continuous self-map of a compact metric space X, and ω ∈ L(f ). From the definition of an ω-limit set, one sees easily that ω is closed in X, f is strongly invariant on ω so that f (ω) = ω, and ω has the following transport property known as “weak incompressability” [15]: F ∩ f (ω \ F ) = ∅ whenever F is a proper, nonempty and closed subset of ω [1]. Theorem 1.3 shows that these three properties of ω-limit sets must, in fact, characterize the elements of L(f ) whenever L : C(I, I) → K is continuous there. The
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goal of this paper is to better understand the map f → L(f ) for the general compact metric space X. We do this by first considering the map L : C(X, X) → K . In Section 2 we recall some definitions, and results needed in the sequel. In Section 3 we develop the basic properties of the set L (f ) as well as those of the map L : C(X, X) → K . Interestingly, L (f ) frequently behaves as a set valued analogue of CR(f ), the chain recurrent set of f . In particular, CR(f ) is closed for any f in C(X, X), and so is L (f ). The map f → CR(f ) is upper semicontinuous, as is f → L (f ). Moreover, if x ∈ CR(f ) and X is an n-manifold, then for any > 0 there is a map g in B (f ) so that x is a periodic point of g. Similarly, if L ∈ L (f ) and > 0, then there exists g in B (f ) and a periodic orbit K of g so that H(L, K) < , where H is the Hausdorff metric. Proposition 3.2 shows that L is continuous at f whenever the p-stable periodic orbits of f are dense in L (f ), and Theorem 3.7 establishes the continuity of L on a residual subset of C(M, M ), whenever M is a compact n-manifold with n 1. Our principal results are found in Section 4. Theorem 4.1 develops another characterization of those maps f in C(I, I) at which L : C(I, I) → K is continuous, and Theorem 4.2 establishes the continuity of L on a residual subset of C(I, I). Theorem 4.4 establishes the continuity of f → L(f ) on a residual set of functions in the more general C(M, M ), and Theorem 4.3 characterizes those functions in C(M, M ) at which f → L(f ) is upper semicontinuous. We conclude with a condition on f ∈ C(X, X) sufficient to insure the continuity of L : C(X, X) → K at f . 2. Preliminaries We shall be concerned with C(X, X) and the iterative properties this class of functions possesses. For f in C(X, X) and any integer n 1, f n denotes the n-th iterate of f . We also take f 0 to be the identity map, defined by f 0 (x) = x for every x in X. A point x ∈ X is said to be a periodic point of f with period m if f m (x) = x, f k (x) = x for 1 k < m. Let P (f ) represent those points x ∈ X that are periodic under f , and let P (f ) be the set of periodic orbits of f . If K is a periodic orbit of f of period n 1, and for any > 0 there exists δ > 0 so that any g ∈ Bδ (f ) = {h ∈ C(X, X): h − f < δ} has a periodic orbit L of period n satisfying H(L, K) < , then we call K a perturbation-stable, or p-stable, periodic orbit of f . Let S(f ) be the collection of the p-stable periodic orbits of f with S(f ) = {x ∈ P (f ): ω(x, f ) ∈ S(f )} the set of p-stable periodic points. Should x be a periodic point of period n for f in C(I, I), a sufficient condition for x to be p-stable is that f n (x) − x not be unisigned on any neighborhood of x. The set CR(f ) of all chain recurrent points of f is defined by x ∈ / CR(f ) if there exists an open set U with f (U ) ⊆ U such that x ∈ / U , f (x) ∈ U . Let > 0 and let x and y be points of X. An -chain, or pseudo-orbit, from x to y is a finite sequence {x0 , x1 , . . . , xn } of points of X with x0 = x and xn = y and d(f (xk−1 ), xk ) < for k = 1, . . . , n. A point x ∈ X is chain recurrent if and only if, for every > 0, there is an - chain from x to itself [1, p. 112]. For A ⊆ X, we denote by diam(A) the diameter of A, that is diam(A) := sup{d(x, y): x, y ∈ A} and by (A)0 the interior of A, that is the largest open set contained in A. In Section 3, we will occasionally restrict our attention to elements of C(M, M ), where M is an n-manifold with or without boundary. Unless otherwise stated M will denote an n-manifold with or without boundary. All of our manifolds are compact metric spaces [11]. The main property of manifolds used is that every point of a manifold has an arbitrarily small neighborhood whose closure is homeomorphic to I n , the unit cube of Rn . It is well known that every continuous map f : I n → I n has a fixed point [14]. Therefore, if M is homeomorphic to I n and g : M → M is continuous, then g has a fixed point. If φ is a function from X into the class of nonempty subsets of a metric space Y , then we say that φ is upper semicontinuous (lower semicontinuous) if for each open (resp. closed) subset V of Y the set {x ∈ X: φ(x) ⊆ V } is open (resp. closed) in X. Hence, in the case of F : (C(X, X), · ) → (K, H) with
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f ∈ C(X, X), F is upper semicontinuous at f if for any > 0 there exists δ > 0 so that F (g) ⊆ B (F (f )) whenever f − g < δ. That is, for g sufficiently close to f in C(X, X), if y ∈ F (g) closed in X, then d(y, F (f )) = inf{d(y, z): z ∈ F (f )} < . In the case of F : (C(X, X), · ) → (K , H ), F (g) and F (f ) are collections of closed subsets of X, each closed with respect to K. The map F is upper semicontinuous at f if for any > 0 there exists δ > 0 so that F (g) ⊂ B (F (f )) whenever f − g < δ. That is, for g sufficiently close to f in C(X, X), if K ∈ K is in F (g) then d(K, F (f )) = inf{H(K, L): L ∈ F (f )} < . In our study of the map L : C(X, X) → K given by f → L (f ) we make use of the notion of oscillation. Define the oscillation of L on B (f ) such that OL B (f ) = sup H L (f1 ), L (f2 ) : f1 , f2 ∈ B (f ) . We take the oscillation of L at f to be OL (f ) = inf OL B (f ) = lim+ OL B (f ) . >0
→0
It follows that {f ∈ C(X, X): OL (f ) < } is open for every > 0, and L is continuous at f if and only if OL (f ) = 0 [4]. We now turn our attention to the Baire category theorem. Let (X, d) be a metric space. A set is of the first category in (X, d) if it can be written as a countable union of nowhere dense sets; otherwise, the set is of the second category. A set is residual if it is the complement of a first category set; an element of a residual subset of (X, d) is called a typical element of X. With these definitions in mind, we recall Baire’s theorem on category. Theorem 2.1. Let (X, d) be a complete metric space with S a first category subset of X. Then X \ S is dense in X. We make reference to the following three results. Theorem 2.2. (See [8].) If f ∈ C(I, I) and P (f ) is dense in the set of chain recurrent points of f , then L(f ) is contained in the Hausdorff closure of the periodic orbits of f . Theorem 2.3. (See [11].) If M is homeomorphic to I n , and f is a continuous function from a closed subset F of M into M , then f can be extended so that the domain of f is all of M , and f (M ) ⊆ M . For the sake of convenience, we say that a subset J of an n-manifold M is an n-cube if J is homeomorphic to I n and J has nonempty interior relative to M . Lemma 2.4. (See [6].) Suppose x ∈ M , f ∈ C(M, M ), and x = x0 , x1 , . . . , xm are distinct points such that f (xi ) = xi+1 for all 0 i m − 1 and f (xm ) = xl for some 0 l m. Then, for every > 0, there is g ∈ B (f ) and n-cubes J0 , J1 , . . . , Jm with diameters less than such that (1) xi ∈ (Ji )0 and diam(Ji ) < for all i, (2) g(xi ) = f (xi ) for 0 i m, (3) g(Ji ) ⊆ Ji+1 for 0 i m − 1, and g(Jm ) ⊆ Jl . 3. The map L : C(X, X) → K This section is dedicated to developing the basic properties of the set L (f ) and the map L : C(X, X) → K . From [2], we know that {ω(x, f ): x ∈ I} is closed in (K, H) whenever f ∈ C(I, I). While the collection
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of ω-limit sets need not be closed when f ∈ C(X, X) for the arbitrary compact metric space X [10], the set L (f ) is always closed, and the map f → L (f ) is upper semicontinuous. These results follow readily from the definition of L (f ) as well as the observations that (K, H) is compact and C(X, X) is complete with respect to the supremum norm. Since they follow the analysis for continuous self-maps of the interval found in [18, Propositions 4.9 and 4.10], we record them in the first proposition. Proposition 3.1. Let X be a compact metric space. If f ∈ C(X, X), then L (f ) is closed in (K, H), and the map L : (C(X, X), · ) → (K , H ) is upper semicontinuous. Proposition 4.11 in [18] shows that L is continuous at some f whenever S(f ) = L (f ). Since L(f ) is closed whenever f ∈ C(I, I), this is equivalent to requiring S(f ) = L(f ) = L (f ). Our next proposition extends this result to the continuity of L at f in C(X, X) whenever S(f ) = L (f ). Proposition 3.2. If f ∈ C(X, X) and S(f ) = L (f ), then L : C(X, X) → K is continuous at f . Proof. Here, the closure S(f ) is taken with respect to the topology generated by the metric space (K, H). Notice that for f in C(X, X), S(f ) ⊆ L (f ) since a periodic orbit of f belongs necessarily to the closed set L (f ). Let us fix f and > 0. Since L : C(X, X) → K is upper semicontinuous at f , there exists δ1 > 0 so that L (g) ⊂ B (L (f )) whenever f − g < δ1 . Since S(f ) is dense in L (f ), there exists δ2 > 0 so that L (f ) ⊂ B (P (g)) whenever f − g < δ2 . In particular, since L (f ) is closed in (K, H) compact, there exist n L1 , . . . , Ln in L (f ) such that L (f ) ⊆ i=1 B (Li )). Since S(f ) is dense in L (f ), for every 1 i n, there exists Ki ∈ S(f ) such that H(Li , Ki ) < 2 . As each Ki is a p-stable periodic orbit, there exists δ2 > 0 such that any g ∈ Bδ2 (f ), for every 1 i n, has a periodic orbit Ti satisfying H(Ti , Ki ) < 2 , and hence H(Ti , Li ) H(Ti , Ki ) + H(Ki , Li ) <
+ = . 2 2
Therefore, L (f ) ⊂
n
B Li ⊆ B P (g)
i=1
whenever f − g < δ2 . If f − g < min{δ1 , δ2 }, then L (g) ⊂ B (L (f )) and L (f ) ⊂ B (P (g)) ⊂ B (L (g)). It follows that H(L (g), L (f )) < , and L : C(X, X) → K is continuous at f . Our goal is to describe a residual subset of C(M, M ) on which L : C(M, M ) → K is continuous. We continue our analysis on C(M, M ) since we can apply the fixed point properties on n-cubes to elements of C(M, M ); this fixed point property need not hold for continuous self-maps of an arbitrary compact metric space X. The following lemma gives some insight into how the elements of L (f ) are “almost” ω-limit sets. It generalizes the analogous result in the case of M = I contained in Proposition 4.5 in [18]. We point out that the proof, in the more general setting of M an n-manifold, requires a more thoughtful application of the Tietze extension theorem as well as Lemma 2.7 from [7]. Lemma 3.3. Let f ∈ C(M, M ) with L ∈ L (f ) and > 0. There exist g in B (f ) and K a periodic orbit of g so that H(L, K) < . Proof. As M is a compact n-manifold, as remarked in [7], there exists ˜ > 0 such that, for each x ∈ M , B˜(x) is an n-cube. Therefore, for every x ∈ M , B (x) is an n-cube for every 0 < < ˜. Fix > 0 with
794
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< ˜. Since f ∈ C(M, M ) is uniformly continuous on M , there exists δ1 > 0 so that d(f (x), f (y)) < 4 whenever d(x, y) < δ1 . Choose δ > 0 so that δ < min{δ1 , 4 }. As L is a compact metric space, it is totally n bounded. Hence, there exists a δ-net {x1 , . . . , xn } for L; that is, L ⊆ i=1 Bδ (xi ) with {x1 , . . . , xn } ⊆ L. We perturb f to get a function g in C(M, M ) possessing a periodic ω-limit set ω so that f − g < and n {x1 , . . . , xn } ⊆ ω ⊆ i=1 Bδ (xi ), as this implies H(ω, L) < . Suppose n = 1, so that f (Bδ (x1 )) ⊆ B (x1 ). Using the Tietze extension theorem [11, Corollary 1, p. 82], we find g ∈ C(M, M ) so that (1) f − g < , (2) g(x) = f (x) whenever x ∈ X \ Bδ (x1 ), and (3) g(x1 ) = x1 . Take h such that h(x1 ) = x1 and h = f on M \ Bδ (x1 ). As Bδ (x1 ) is an n-cube, by Tietze’s extension theorem, h can be extended to a map g on all of M such that g is continuous, and g(Bδ (x1 )) ⊆ B (x1 ). This is the desired function g. Let us now prove the existence of such a g in the case n = 2. Say L ⊆ Bδ (x1 )∪Bδ (x2 ). From our transport property, we know that F ∩ f (L \ F ) = ∅ for any nonempty proper closed set F ⊆ L. Let F = L ∩ Bδ (x1 ). It follows that there exists x ∈ Bδ (x2 ) such that f (x) ∈ Bδ (x1 ). Now d f (x2 ), x1 d f (x2 ), f (x) + d f (x), x1 < + = . 4 4 2
()
Similarly, d f (x1 ), x2 < . 2 Hence, f Bδ (x1 ) ⊆ B (x2 )
and f Bδ (x2 ) ⊆ B (x1 ).
Choose δ˜ so that Bδ˜(x1 ) ∩ Bδ˜(x2 ) = ∅. Now, using the Tietze extension theorem, we find g ∈ C(M, M ) so that (1) f − g < , (2) g(x) = f (x) whenever x ∈ X \ Bδ˜(x1 ) ∪ Bδ˜(x2 ), and (3) g(x1 ) = x2 and g(x2 ) = x1 . Let us see this in detail: Take h such that h(x1 ) = x2 , h(x2 ) = x1 and h = f on M \ Bδ˜(x1 ) ∪ Bδ˜(x2 ), where δ˜ is chosen so that Bδ˜(x1 ) ∩ Bδ˜(x2 ) = ∅. By Tietze’s extension theorem h can be extended to a map g on all of M such that g is continuous, g(Bδ (x1 )) ⊆ B (x2 ) and g(Bδ (x2 )) ⊆ B (x1 ). This is the desired function g. The proof for n 3 follows by induction and from the fact that F ∩ f (L \ F ) = ∅ for any nonempty proper closed set F ⊆ L: (1) Let F = L ∩ ( j=i Bδ (xj )) to see that there exists x in Bδ (xi ) such that f (x) ∈ Bδ (xj ) for some j = i, for any i = 1, 2, . . . , n. (2) Let F = L ∩ Bδ (xi ) to see that there exists x ∈ Bδ (xj ) for some j = i so that f (x) ∈ Bδ (xi ), for any i = 1, 2, . . . , n. (3) Let T ⊆ {1, 2, . . . , n} with F = L ∩ ( j ∈T / Bδ (xj )) to see that there exists x ∈ i∈T Bδ (xi ) so that f (x) ∈ Bδ (xj ) for some j ∈ {1, 2, . . . , n} \ T .
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Lemma 3.4. Let f ∈ C(M, M ) with K a periodic orbit of f . Then for any > 0 there exists g in B (f ) so that K ∈ S(g). Proof. Let K = {x0 , x1 , . . . , xm } so that xi = xj whenever i = j, f (xi ) = xi+1 for 0 i m − 1, and f (xm ) = x0 . Fix > 0. From Lemma 2.4 we may assume that there are disjoint n-cubes V0 , V1 , . . . , Vm 0 with diameters less than so that xi ∈ (Vi ) whenever 0 i m, f (Vi ) ⊆ Vi+1 , for 0 i m − 1 and 0 0 f (Vm ) ⊆ V0 . Let Ui be an n-cube such that Ui ⊆ (Vi ) and xi ⊆ (Ui ) . Using the Tietze extension theorem, we obtain g in B (f ) such that m 0 (1) g = f on M \ i=0 (Vi ) , (2) g(Vi ) ⊆ Vi+1 for 0 i m − 1, and g(Vm ) ⊆ V0 , (3) g(Ui ) = f (xi ) = xi+1 for all 0 i m − 1, and g(Um ) = f (xm ) = x0 . We show that K ∈ S(g), that is, that for any α > 0 there exists τ > 0 so that any h ∈ Bτ (g) has a periodic orbit L of period n satisfying H(L, K) < α. Let α > 0, and take 0 < σ < α so that Bσ (xi ) ⊆ (Ui )0 , for all 0 i m. From our choice of g, if d(g, h) < τ = σ2 , then h(Bσ (xi )) ⊆ Bσ (xi+1 ) whenever 0 i m − 1, and h(Bσ (xm )) ⊆ Bσ (x0 ). In particular, hm (Bσ (xi )) ⊆ Bσ (xi ), so that Bσ (xi ) contains a fixed point yi of hm by the Brower fixed point theorem [12], for all i. It follows that H(K, ω(yi , h)) σ < α. Lemma 3.5. The set In = {f ∈ C(M, M ): H (S(f ), L (f )) <
1 n}
is dense in C(M, M ).
Proof. Let f ∈ C(M, M ) so that H (S(f ), L (f )) n1 . Since L : C(X, X) → K is upper semicontinuous 1 (L (f )) whenever g ∈ Bδ (f ). Since L (f ) is closed in K at f , there exists δ > 0 so that L (g) ⊂ B 4n 1 m and (K, H) is compact, there exist {Li }m i=1 ⊂ L (f ) so that {Li }i=1 is a 4n -net of L (f ). We construct 1 for g ∈ C(M, M ) so that g ∈ Bδ (f ) and there is a p-stable periodic orbit Ki ∈ S(g) so that H(Ki , Li ) < 4n i = 1, 2, . . . , m. We do this by applying Lemmas 3.3 and 3.4 to each Li in turn and insuring that Kj ∩Kl = ∅ for l = 1, 2, . . . , j − 1 as j goes from 1 to m. This allows us to develop an appropriate function g even if the sets Li are not pairwise disjoint. It follows that 1 1 1 L (g) ⊂ B 4n S(g) , L (f ) ⊂ B 2n {Li }m i=1 ⊂ B 2n so that 1 . H S(g), L (g) < 2n Lemma 3.6. If f ∈ In , then OL (f ) <
3 n.
Proof. Let f ∈ In . We recall that S(f ) ⊆ L (f ) for all f in C(M, M ), and that L : C(M, M ) → K is upper semicontinuous. Let > 0. It follows, then, that there exists δ > 0 so that, whenever gi ∈ Bδ (f ), L (gi ) ⊆ B L (f ) ⊆ B+ n1 S(f ) and S(f ) ⊆ B P (gi ) ⊆ B L (gi ) . Thus, 1 H S(f ), L (gi ) < + , n
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so that 2 H L (g1 ), L (g2 ) < 2 + n for any {g1 , g2 } ⊆ Bδ (f ), and hence 2 OL Bδ (f ) 2 + . n We conclude that 3 2 OL (f ) = lim OL Bδ (f ) < . δ→0 n n We are now in a position to establish the section’s main result. Theorem 3.7. The map L : C(M, M ) → K is continuous on a residual subset of C(M, M ). Proof. The Baire category theorem is applicable to C(M, M ) since this space is complete with the supremum norm [5,4]. Recall from [4] that En = {f ∈ C(M, M ): OL (f ) < n1 } is open so that, from Lemmas 3.5 and 3.6, En is both dense and open. Now, E :=
∞
En = f ∈ C(M, M ): OL (f ) = 0 = f ∈ C(M, M ): L is continuous at f ,
n=1
is a dense Gδ subset of C(M, M ). The set L(f ) need not be closed for an arbitrary element of C(M, M ) [10]. As the final result of the section shows, however, L(f ) is typically dense in L (f ). Theorem 3.8. The set T = {f ∈ C(M, M ): L(f ) = L (f )} is residual in C(M, M ). Proof. It suffices to show that In = {f ∈ C(M, M ): H (L(f ), L (f )) < n1 } contains a dense and open subset of C(M, M ). Using the notation from Lemma 3.5, since In ⊆ In , and In is dense, it follows immediately that In is dense, too. Now, let f ∈ In . We show that there exists some δ > 0 so that Bδ (f ) ⊆ In . Say H (L(f ), L (f )) = σ ˜ < n1 , choose σ > 0 with σ ˜ < σ < n1 and set γ = n1 − σ. Since the map L is upper semicontinuous, there exists δ1 > 0 so that L (g) ⊆ B γ4 (L (f )) whenever f − g < δ1 . By the definition of a p-stable periodic orbit, there exists δ2 > 0 so that S(f ) ⊆ B γ4 (P (g)) whenever f − g < δ2 . Now, take δ < min{δ1 , δ2 } with f − g < δ. Then L (g) ⊆ B γ4 L (f ) ⊆ Bσ+ γ2 S(f ) ⊆ Bσ+ 34 γ P (g) ⊆ Bσ+ 34 γ L(g) , so we conclude that H (L(f ), L (f )) <
1 n.
4. The map f → L(f ) Let C = {f ∈ C(I, I): P (f ) = CR(f ), S(f ) = P (f ) and L(f ) = L (f )}, and D = {f ∈ C(I, I): S(f ) = L (f )}. From [16], the map L : C(I, I) → K is continuous at f in C(I, I) if and only if f ∈ C. In [17], it is proved that the map L : C(I, I) → K is upper semicontinuous at f if and only if L(f ) ⊇ L (f ), and, from
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Proposition 3.3 in [17], it follows that if S(f ) = L(f ) then the map L : C(I, I) → K is lower semicontinuous. In this section we endeavor to extend these results and to better understand the map f → L(f ) on more general metric spaces by taking advantage of the recent results concerning L . Theorem 4.1. The map L : C(I, I) → K is continuous at f if and only if S(f ) = L(f ) = L (f ). Proof. As, from [16], the map L : C(I, I) → K is continuous at f in C(I, I) if and only if f ∈ C, it suffices to show that C = D. Let f ∈ C; since P (f ) = CR(f ), from [8] it follows that every ω-limit set of f is contained in the Hausdorff closure of periodic orbits. Now, S(f ) = P (f ), so, in fact, S(f ) = L(f ) = L (f ). Thus, f ∈ D, and one concludes that C ⊆ D. Now, let f ∈ D. As we saw in the proof of Proposition 3.2, since S(f ) = L(f ), L is lower semicontinuous at f , and since L(f ) = L (f ), L is upper semicontinuous at f . It follows that L : C(I, I) → K is continuous at f , so that f ∈ C, too, and D ⊆ C. By applying Lemmas 3.5 and 3.6 to the specific case of continuous self-maps of I, we establish the continuity of L : C(I, I) → K on a residual subset of C(I, I). This result was announced in Theorem 4.13 in [18]. The proof there, however, is irreparably flawed. The proof presented below necessarily rests on a fundamentally different analysis. Theorem 4.2. The map L : C(I, I) → K is continuous on a residual subset of C(I, I). Proof. From [4] we know that Tn = {f ∈ C(M, M ): OL(f ) < closed in K for all f in C(I, I) [2], it follows that
3 n}
is open, and since {ω(x, f ): x ∈ I} is
S(f ) ⊆ L(f ) ⊆ L (f ). In particular, Tn is both dense and open. One now proceeds as in the proof of Theorem 3.7. When working in C(I, I), one has the benefit of knowing that L(f ) is closed in K for every f in C(I, I). As one sees from [10], L(f ) need not be closed when f ∈ C(M, M ), for an arbitrary n-manifold M . Because of this, we consider the map L : C(M, M ) → K given by f → L(f ) as we turn our attention to analogues of Theorems 4.1 and 4.2 on the more general space C(M, M ). The next result was proved for L in [18] in the case of continuous self-maps of the interval, for which L(f ) = L(f ). We generalize this result to the map L defined on the space of continuous self-maps of an n-manifold. Theorem 4.3. The map L : C(M, M ) → K is upper semicontinuous at f if and only if L(f ) = L (f ). Proof. Suppose L : C(M, M ) → K is upper semicontinuous at f , and let L ∈ L (f ). We show that L ∈ L(f ). By Lemma 3.3 there exists {fn } ⊆ C(M, M ) with Kn ∈ L(fn ) ⊆ L(fn ) = L(fn ) for any n so that limn fn = f and limn Kn = L. Since L : C(M, M ) → K is upper semicontinuous at f , L ∈ L(f ). Now, suppose that L(f ) = L (f ); we show that L : C(M, M ) → K is upper semicontinuous at f . Let {fn } ⊆ C(M, M ) with Ln ∈ L(fn ) for any n so that fn → f and Ln → L. Since L ∈ L (f ), and L(f ) = L (f ), it follows that L ∈ L(f ) and L : C(M, M ) → K is upper semicontinuous at f . Theorem 4.4. The map L : C(M, M ) → K is continuous on a residual subset of C(M, M ). Proof. Recall from [4] that Sn = {f ∈ C(M, M ): OL(f ) < n3 } is open. It suffices to show that In = {f ∈ C(M, M ): H (S(f ), L (f )) < n1 } ⊆ Sn , as In is dense by Lemma 3.5. Let f ∈ In and > 0. Arguing as in Lemma 3.6 we find δ > 0 so that, whenever gi ∈ Bδ (f ),
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L (gi )) ⊆ B+ n1 S(f ) , so that, as L(gi ) ⊆ L (gi ), L(gi )) ⊆ B+ n1 S(f ) . Moreover, S(f ) ⊆ B P (gi ) , so that, as P (gi ) ⊆ L(gi ), S(f ) ⊆ B L(gi ) . Thus, 1 H S(f ), L(gi ) < + , n so that 2 H L(g1 ), L(g2 ) < 2 + n for any {g1 , g2 } ⊆ Bδ (f ), and hence 2 OL Bδ (f ) 2 + . n We conclude that 3 2 OL(f ) = lim OL Bδ (f ) < . δ→0 n n Now, S :=
∞
Sn = f ∈ C(M, M ): OL(f ) = 0 = f ∈ C(M, M ): L is continuous at f ,
n=1
is a dense Gδ subset of C(M, M ). One now needs only recall that the Baire category theorem is applicable to C(M, M ) since this space is complete with respect to the supremum norm [5,4]. We conclude by showing that the characterization of those f in C(I, I) at which L is continuous, is sufficient on any compact space X. It would be interesting to know if the converse is true, so that S(f ) = L (f ) must necessarily hold whenever L : C(X, X) → K is continuous at f . Theorem 4.5. If f ∈ C(X, X) and S(f ) = L (f ), then L : C(X, X) → K is continuous at f . Proof. Fix f in C(X, X) and > 0. We find δ > 0 so that H (L(g), L(f )) < whenever f − g < δ. Since L : C(X, X) → K is upper semicontinuous, there exists δ1 > 0 so that L (g) ⊂ B (L (f )), and consequently L(g) ⊂ B (L (f )), whenever f − g < δ1 . Since S(f ) is dense in L (f ), there exists δ2 > 0 so that L (f ) ⊆ B (P (g)) ⊆ B (L(g)) whenever f − g < δ2 . If f − g < δ = min{δ1 , δ2 }, then
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L(g) ⊂ B (L (f )) and L (f ) ⊆ B (L(g)), so that H (L(g), L (f )) < . Since S(g) ⊆ L(g) ⊆ L (g) for any g in C(X, X), we have in the case of f that S(f ) = L(f ) = L (f ), and H L(g), L (f ) = H L(g), L(f ) < . References [1] L.S. Block, W.A. Coppel, Dynamics in One Dimension, Lecture Notes in Math., vol. 1513, Springer-Verlag, Berlin, 1992. [2] A. Blokh, A.M. Bruckner, P.D. Humke, J. Smítal, The space of ω-limit sets of a continuous map of the interval, Trans. Amer. Math. Soc. 348 (1996) 1357–1372. [3] A.M. Bruckner, Stability in the family of ω-limit sets of continuous self maps of the interval, in: Summer Symposium in Real Analysis, XX, Windsor, ON, 1996, Real Anal. Exchange 22 (1996/1997) 52–57. [4] A.M. Bruckner, J.B. Bruckner, B.S. Thomson, Real Analysis, Prentice-Hall, New Jersey, 1997. [5] D.L. Cohn, Measure Theory, Birkhäuser, Boston, 1980. [6] E. D’Aniello, U.B. Darji, T.H. Steele, Ubiquity of odometers in topological dynamical systems, Topology Appl. 156 (2008) 240–245. [7] E. D’Aniello, P.D. Humke, T.H. Steele, The space of adding machines generated by continuous self maps of manifolds, Topology Appl. 157 (2010) 954–960. [8] E. D’Aniello, T.H. Steele, Approximating ω-limit sets with periodic orbits, Aequationes Math. 75 (2008) 93–102. [9] J. Dvorakova, Stability of chain recurrent points of continuous maps on interval, J. Difference Equ. Appl. 18 (2012) 1027–1031. [10] J.L. García Guirao, Hausdorff compactness on a space of ω-limit sets, Topology Appl. 153 (2005) 833–843. [11] W. Hurewicz, H. Wallman, Dimension Theory, Princeton University Press, 1948. [12] S. Karamadian, Fixed Points, Algorithms and Applications, Acad. Press, 1977. [13] K. Kuratowski, Topology, PWN and Academic Press Inc., 1968. [14] J.R. Munkres, Topology, Prentice-Hall, New Jersey, 2000. [15] A.N. Sharkovsky, et al., Dynamics of One-Dimensional Maps, Kluwer Academic Publishers, 1997. [16] J. Smítal, T.H. Steele, Stability of dynamical structures under perturbation of the generating function, J. Difference Equ. Appl. 15 (2009) 77–86. [17] T.H. Steele, The persistence of ω-limit set under perturbation of the generating function, Real Anal. Exchange 26 (2000/2001) 963–974. [18] T.H. Steele, Continuity of the maps f → x∈I ω(x, f ) and f → {ω(x, f ): x ∈ I}, Int. J. Math. Math. Sci. (2006), Art. ID 82623, 15 pp. [19] T.H. Steele, Continuity and chaos in discrete dynamical systems, Aequationes Math. 71 (2006) 300–310.