TOPOLOGICAL STABILITY IN SET-VALUED DYNAMICS

. We propose a deﬁnition of topological stability for set-valued maps. We prove that a single-valued map which is topologically stable in the set- valued sense is topologically stable in the classical sense [14]. Next, we prove that every upper semicontinuous closed-valued map which is positively expan- sive [15] and satisﬁes the positive pseudo-orbit tracing property [9] is topologically stable. Finally, we prove that every topologically stable set-valued map of a compact metric space has the positive pseudo-orbit tracing property and the periodic points are dense in the nonwandering set. These results extend the classical single-valued ones in [1] and [14].

1. Introduction. The topological dynamics of set-valued maps has been studied recently in the literature. For instance, [4], [5] and [8] introduced the metric and topological entropies for set-valued maps. In [11] it is defined the specification and topologically mixing properties for set-valued maps. In [6] it is considered the continuum-wise expansivity for set-valued maps.
In this paper we will propose a definition of topological stability for set-valued maps. We prove that a single-valued map which is topologically stable in the setvalued sense is topologically stable in the classical sense [14]. Next, we prove that every upper semicontinuous closed-valued map which is positively expansive [15] and satisfies the positive pseudo-orbit tracing property [9] is topologically stable. Finally, we prove that every topologically stable set-valued map of a compact metric space has the positive pseudo-orbit tracing property and the periodic points are dense in the nonwandering set. These results extend the classical single-valued ones in [1] and [14].
2. Definitions and results. We start this section by introducing the concept of topologically stable set-valued map. This will require some basic notions of setvalued analysis [2]. Afterwards, we state our results.
Let X denote a metric space. Denote by 2 X the set formed by the subsets of X. By a set-valued map of X we mean a map f : X → 2 X . We say that f is single-valued if card(f (x)) = 1 for every x ∈ X, where card(·) denotes cardinality. There is an obvious correspondence between single-valued maps f : X → 2 X and maps f : X → X. In what follows all set-valued maps will be assumed to be strict, We say that f is upper semicontinuous if for every x ∈ X and every neighborhood U of f (x) there is η > 0 such that f (x ) ⊂ U for every x ∈ X satisfying d(x, x ) < η. This definition reduces to the usual continuity in the single-valued case.
The distance between single-valued maps f and g of X is defined by ).
Next we present the classical definition of topologically stable single-valued map by Walters [14].
Definition 2.1. A continuous single-valued map f : X → X is topologically stable, in the class of continuous maps (or topologically stable for short), if for every > 0 there is δ > 0 such that for every continuous map g : X → X with d(f, g) < δ there is a continuous mapĥ : X → X such that d(ĥ, Id X ) < and f •ĥ =ĥ • g, where Id X : X → X is the identity.
To extend this definition to the set-valued context we require further notations. Given A, B ⊂ X we define the distance The distance between the set-valued maps f and g of X is defined by Notice that d H (f, g) reduces to the distance d(f, g) when the involved set-valued maps f and g are single-valued.
Denote by X N = n∈N X the infinite product of copies of X, equipped with the distance Another distance to be considered in X N is We say that (x n ) n∈N ∈ X N is an orbit of a set-valued map f (or an f -orbit for short) if x n+1 ∈ f (x n ), ∀n ∈ N. The set lim ← f formed by the f -orbits is often called the inverse limit space induced by f (cf. [8]). The name inverse limit system is also used (cf. [1]). Precisely, It turns out that f induces a map, to be called left shift Let π : X N → X the projection in the first variable, i.e., π((x n ) n∈N ) = x 0 . Define the map π f : lim ← f → X as the restriction of π to lim ← f . Now we present our definition of topologically stable set-valued map.
Definition 2.2. An upper semicontinuous closed-valued map f of X is topologically stable, in the class of upper semicontinuous closed-valued maps (or topologically stable for short), if for every > 0 there is δ > 0 such that for every upper semicontinuous closed-valued map g with d H (f, g) < δ there is a continuous map The following remark holds.
1. An important difference between definitions 2.1 and 2.2 is that the domain of the semiconjugacy h in the latter definition depends on the perturbation g.
Since every continuous single-valued map is upper semicontinuous and closed valued as a set-valued map, it is natural to compare the definitions 2.1 and 2.2 in the single-valued context. This motivates the following result.
Theorem 2.1. Every continuous single-valued map of a metric space which is topologically stable as a set-valued map (Definition 2.2) is topologically stable in the classical sense (Definition 2.1).
Unfortunately we do not know if the converse of Theorem 2.1 holds, namely, if a single-valued map which is topologically stable in the classical sense (Definition 2.1) is also topologically stable when regarded as a set-valued map (Definition 2.2). The next theorem (and Example 2.1 below) give some light to this question.
Theorem 2.2. Every topologically stable single-valued map f of a compact metric space X satisfies the following property: • For every > 0 there is δ > 0 such that for every continuous single-valued map g : In [13] Walters proved that every positively expansive map with the positive pseudo-orbit tracing property of a compact metric space is topologically stable. Now we extend this result to the set-valued context. Previously we recall the concepts of positive expansivity and pseudo-orbit tracing property in the set-valued context.
such that x = y whenever x, y ∈ X satisfy that there there are f -orbits (x n ) n∈N and (y n ) n∈N such that x 0 = x, y 0 = y and d(x n , y n ) ≤ for every n ∈ N. Sometimes we will say that f is positively expansive with respect to d to emphasize the metric d of X.

Definition 2.4 ([9]
). We say that a set-valued map f of a metric space X has the positive pseudo-orbit tracing property (abbrev. POTP + ) if for every > 0 there is there is an f -orbit (q n ) n∈N satisfying d(p n , q n ) ≤ , ∀n ∈ N.
These definitions extend the classical single-valued ones by Utz [12], Eisenberg [7] and Bowen [3]. Using them we obtain the following set-valued version of Walters stability theorem [13].  Next we present a property of the topologically stable set-valued maps. Given a set-valued map f of X, we say that x ∈ X is a periodic point if there are an f -orbit (x n ) n∈N and m ∈ N + such that x 0 = x and x n+m = x n for every n ∈ N. The set of periodic points is denoted by P er(f ). The nonwandering set of f is the set Ω(f ) of those points x ∈ X such that for every neighborhood U of x there is m ∈ N + satisfying U ∩ f m (U ) = ∅. With these definitions we obtain the following result.
Theorem 2.4. Every topologically stable upper semicontinuous closed-valued map of a compact metric space has the POTP + . Moreover, P er(f ) is dense in Ω(f ).
A short application of this theorem in the single-valued context is as follows. Recall that, on every compact manifold, every single-valued map f which is topologically stable in the classical sense has the POTP + and P er(f ) is dense in Ω(f ). See for instance Theorem 2.4.8 in [1] or [13].
In the following corollary of Theorem 2.4 and Theorem 2.1 we obtain that, on every metric space, every single-valued map f which is topologically stable as a set-valued map (Definition 1.4) has the POTP + and P er(f ) is dense in Ω(f ). In other words we have the following result.
Corollary 2.5. Every continuous single-valued map f of a metric space which is topologically stable as a set-valued map (Definition 2.2) has the POTP + . Moreover, P er(f ) is dense in Ω(f ).
3. Proof of the theorems. In this section we will prove the theorems stated in the previous section. We start with a lemma about the left shift map for single-valued maps. Proof. Since f is single-valued, one has π f ((x) n ) n∈N ) = x if and only if x n = f n (x) for every n ∈ N. Then, π f is bijective with inverse π −1 f (x) = (f n (x)) n∈N . Also, for proving that π f is continuous.
On the other hand, for fixed γ > 0 we let diam(X) denote the diameter of X and we let n 0 ∈ N be such that Since f is continuous, there is ρ > 0 such that Then, , π * f (y)) = d * ((f n (x)) n∈N , (f n (y)) n∈N ) =  ) is a homeomorphism and the proof follows.
With this lemma we can prove Theorem 2.1.
Proof of Theorem 2.1. Let f be a continuous map of a metric space X which is topologically stable as a set-valued map (Definition 2.2).
Fix > 0 and let δ be given by that property. Take g : X → X continuous such that d(f, g) < δ. Since f and g are single-valued, d H (f, g) = d(f, g) and By Lemma 3.1, since both f and g are single-valued, we have that the maps for every x ∈ X, one has d(ĥ, Id X ) ≤ .
Next we prove Theorem 2.2.
Proof of Theorem 2.2. Fix > 0 and let δ be given by the topological stability of f . Take g : X → X continuous such that d H (f, g) < δ. Then, d(f, g) < δ and so there isĥ : X → X continuous such that d(ĥ, Id X ) ≤ and f •ĥ =ĥ • g.
On the other hand, by Lemma 3.1 we have that π f : (lim ← , d * ) → (X, d) and π g : (lim ← g, d * ) → (X, d) are homeomorphisms. Then, sinceĥ is continuous, the Since g is single-valued, x n = g n (x 0 ) for all (x n ) n∈N ∈ lim ← g and n ∈ N. Then, Moreover, Since is arbitrary, f satisfies the required property and the proof follows.
To prove the remainder theorems we need some short preliminars. The first one is a basic property of the upper semicontinuous closed valued maps (see Proposition 1.4.8 in [2]).
Since lim f (X) we obtain the following lemma. For the next lemma we will use an auxiliary definition.
Definition 3.1. We say that a set-valued map f of a metric space X has the finite shadowing property if for every > 0 there is δ > 0 such that for every finite set {p 0 , · · · , p m } satisfying d(p n+1 , f (p n )) < δ for every 0 ≤ n ≤ m − 1 there is a finite set {q 0 , · · · , q m } such that q n+1 ∈ f (q n ) and d(p n , q n ) < for every 0 ≤ n ≤ m − 1.
With this definition we obtain the following result. Proof. We only need to prove the sufficiency. Let f be an upper semicontinuous closed-valued map with the finite shadowing property of a compact metric space X. Let > 0 be given. Find a corresponding δ > 0 given by the finite shadowing property. Let (p n ) n∈N be a sequence satisfying d(p n+1 , f (p n )) ≤ δ for every n ∈ N. Then, by finite shadowing, for every m ∈ N there is a sequence {q m 0 , · · · , q m m } such that q m n+1 ∈ f (q m n ) and d(p n , q m n ) ≤ for every 0 ≤ n ≤ m. Since X is compact, we can assume by passing to subsequences if necessary that there is a sequence (q n ) n∈N such that q m n → q n as m → ∞ for every n ∈ N. Since f is upper semicountinuous, closed-valed and X is compact, Lemma 3.2 implies (q n ) n∈N ∈ lim ← f . By fixing n in d(p n , q m n ) ≤ and letting m → ∞ we obtain d(p n , q m ) ≤ for every n ∈ N. Then, f has the POTP + proving the result.
The next lemma is about the expansivity of the shift map for positively expansive set-valued maps. Proof. Let be a positive expansivity constant of f . Take ( ∀k ∈ N. The following result is the positively expansive version of Lemma 2 in [13] (with similar proof).
Lemma 3.6. Let r : Y → Y be a positively expansive continuous map of a compact metric space Y . Then, for every positive expansivity constantê and every ∆ > 0 there is N ≥ 1 such that d(x, y) ≤ ∆ whenever x, y ∈ Y satisfy d(r k (x), r k (y)) ≤ê for every 0 ≤ k ≤ N .
Next we prove the continuity of the left shift. Fix > 0 and let δ be given from POTP + for the constant 0 = min{ ,e,ê} 8 , where e is the positive expansivity constant of the set-valued map f . Fix a set-valued map g such that d H (f, g) ≤ δ 8 . Let (x n ) n∈N be a g-orbit. Since d H (g(x 0 ), f (x 0 )) ≤ δ 8 (by hypothesis) and x 1 ∈ g(x 0 ), we have d(x 1 , f (x 0 )) < δ.
Then, by the POTP + and the choice of δ, there is an f -orbit (y n ) n∈N such that d(x n , y n ) ≤ 0 , ∀n ∈ N.