TOPOLOGICAL ENTROPY for SET-VALUED MAPS Dante

DISCRETE AND CONTINUOUS doi:10.3934/dcdsb.2015.20.3461 DYNAMICAL SYSTEMS SERIES B Volume 20, Number 10, December 2015 pp. 3461–3474

TOPOLOGICAL ENTROPY FOR SET-VALUED MAPS

Dante Carrasco-Olivera Departamento de Matemática Universidad del B´ıo,B´ıoAv. Collao # 1202 Casilla 5-C, VIII-Región,Concepción,Chile Roger Metzger Alvan Instituto de Matemáticay Ciencias Afines (IMCA) Universidad Nacional de Ingenier´ıa Calle Los Biólogos245, Urb. San CésarLa Molina Lima 12, Lima, Peru Carlos Arnoldo Morales Rojas Instituto de Matemática Universidade Federal do Rio de Janeiro P. O. Box 68530, 21945-970 Rio de Janeiro, Brazil

Abstract. In this paper we deﬁne and study the topological entropy of a set- valued dynamical system. Actually, we obtain two entropies based on separated and spanning sets. Some properties of these entropies resembling the single- valued case will be obtained.

1. Introduction. The concept of entropy was introduced in thermodynamics by Clausius ([13], [14]), in statistical and quantum mechanics by Boltzmann and von Neumann ([7], [34]) and in information theory by Shannon , Khinchin and McMillan ([31], [20], [23]). In 1958, Kolmogorov [21] defined metric entropy for K-systems and Sinai [29] extended his definition to all measure-preserving transformations. Adler, Konheim and McAndrews [1] defined topological entropy on compact topological spaces whereas Bowen [15] and Dinaburg [15] obtained an equivalent definition in the metrizable case. Katok [18] deduced the metric entropy of ergodic transformations from spanning sets. Basic references including extended historical notes are [16], [18], [30]. On the other hand, the study of the set-valued dynamical systems has been increasing along these years following the success of the single-value case. For in- stance, [22] investigated the existence of endpoints for set-valued dynamical systems through the notion of stable sets. The results are formulated by means of the inverse of a Pareto-minimal point of a vectorial Lyapunov function. In [5] it is introduced the notion of invariant measure for set-valued dynamical systems, generalizing the same concept in the single-valued case. Further notions of invariant or coincidence

2010 Mathematics Subject Classiﬁcation. Primary: 37B40; Secondary: 54C60. Key words and phrases. Set-valued map, topological entropy, metric space, spanning set, separated set. C-O was partially supported by FONDECYT project 11121598, CONICYT (Chile) and Pos- Doutorado Verano 2015, IMPA, Rio de Janeiro, Brazil. R.M. was partially supported by Pos- Doutorado Verano 2015, IMPA, Rio de Janeiro, Brazil.

3461 3462 D. CARRASCO-OLIVERA, R. METZGER ALVAN AND C. A. MORALES ROJAS measures are given in [24], [25] and [32]. In [36] it is studied the asymptotic behav- ior of the trajectories of a set-valued dynamical system. In particular, necessary and sufficient conditions for the existence of global attractors for dispersive systems as well as motivations coming from economic models are given. In [26] and [27] there were analized the notion of pseudo-orbit and inverse tracing properties for set-valued dynamical systems. Continuous time set-valued dynamical systems and its relationship with economic models were defined and studied in [11]. See also the monographs [2], [4], [3], [10] for further information about set-valued analysis. In this paper we will introduce and study the notion of topological entropy for set- valued maps on metric spaces. Indeed, we obtain two entropies given by separated and spanning sets respectively. We prove that they keep a number of well-known properties of the classical topological entropy of single-valued maps [1], [8], [15]. Our motivation comes not only from the aforementioned works but also from the hope that the notion of entropy would play an important role in the analysis of chaotic set-valued systems (as in the single-valued case). This paper is organized as follows. In Section2 we define the separated and spanning entropies for a discrete set-valued dynamical system. In Section3 we prove the main results dealing with properties of these entropies. In Section4 we present some related examples. In Section5 we present some conclusions.

2. Definition. Let X be a metric space. Given A, B ⊂ X we define d(A, B) = inf{d(a, b): a ∈ A, b ∈ B}. Denote by 2X the set formed by the subsets of X. By a set-valued map on X we mean a map f : X → 2X . We say that f is single-valued if card(f(x)) = 1 for all x ∈ X, where card denotes cardinality. There is an obvious correspondence between single-valued maps f : X → 2X and maps f : X → X. Hereafter we shall assume that every set-valued map f is strict, i.e., f(x) 6= ∅ for every x ∈ X. + Let f be a set-valued map of X. For all n ∈ N we define the map dn : X ×X → R+ by n n dn(x, y) = inf max d(xi, yi):(xi)i=0, (yi)i=0 are sequence satisfying (1) 0≤i≤n−1 x0 = x, y0 = y, xi+1 ∈ f(xi) and yi+1 ∈ f(yi) for all i with 0 ≤ i ≤ n − 1 .

f The notation dn indicates dependence on f. These maps are metrics in the single- valued case. In general they are only semimetrics [6], [17] (see Example 4.5). Deﬁne the -balls centered at x ∈ X with respect to dn,

Bn[x, ] = {y ∈ X : dn(x, y) ≤ }. f Again the notation Bn[x, ] indicates dependence on f. Given > 0 and F ⊂ X we say that F is (n, )-separated (for f) if

Bn[x, ] ∩ F = {x}, ∀x ∈ F. Deﬁne s(n, ) = sup{card(F ): F is (n, )-separated}. We write s(n, , f) to empha- size f. It is possible that s(n, ) = ∞. Deﬁne log s(n, ) hse(f, ) = lim sup . n→∞ n TOPOLOGICAL ENTROPY FOR SET-VALUED MAPS 3463

Since any (n, )-separated set is (n, 0)-separated for all 0 with 0 < 0 ≤ , we obtain 0 0 0 s(n, ) ≤ s(n, ) and so hse(f, ) ≤ hse(f, ) for 0 < ≤ . Then, the limit

lim hse(f, ) = sup hse(f, ) →0 >0 exists and the following definition is given. Definition 2.1. The separated topological entropy of f is defined by

hse(f) = lim hse(f, ). →0 As in the single-valued case ([8], [15]) one can also consider a topological entropy for set-valued maps via spanning sets. Given n ∈ N+, > 0 and E ⊂ X we say that E is (n, )-spanning (for f) if [ X = Bn[x, ]. x∈E Deﬁne r(n, ) = min{card(E): E is (n, )-spanning} and write r(n, , f) to empha- size f. It is possible that r(n, ) = ∞. Deﬁne log r(n, ) hsp(f, ) = lim sup . n→∞ n Since any (n, 0)-spanning set is (n, )-spanning for 0 < 0 ≤ , we obtain r(n, ) ≤ 0 0 0 r(n, ) and so hsp(f, ) ≤ hsp(f, ) for 0 < ≤ . It follows that the limit

lim hsp(f, ) = sup hsp(f, ) →0 >0 exists and the following definition is given. Definition 2.2. The spanning topological entropy of f is defined by

hsp(f) = lim hsp(f, ). →0 As the reader can see, these deﬁnitions emulate the similar ones for single-valued maps by Bowen [8] and Dinaburg [15]. In particular, they both reduce to the classical topological entropy in the single-valued case. We then write h(f) = hse(f) = hsp(f) for single-valued maps f. Notice that, unlike [8], [15], we do not assume any hypothesis of continuity for the involved maps. In the single-valued case, Ciklov´a[12] obtained a number of properties for the topological entropy without such a hypothesis (including the equality between the separated and spanning entropies, see Proposition 3.4 of p. 624 in [12]).

3. Properties. In this section we prove the main results of this paper dealing with properties of the separated and spanning entropies. Let f be a set-valued map on a metric space X. Given A ⊂ X we define [ f(A) = f(x). x∈A We say that A is invariant if f(A) ⊂ A. For such sets there is an induced set-valued map f|A defined by f|A(x) = f(x) for all x ∈ X. Our first property is about the separated entropy of invariant sets. Sm Theorem 3.1. Let f be a set-valued map on a metric space X. If X = i=1 Ai where each Ai is an invariant set of f, then hse(f) = max1≤i≤m hse(f|Ai ). 3464 D. CARRASCO-OLIVERA, R. METZGER ALVAN AND C. A. MORALES ROJAS

Proof. We assert that hse(f|A) ≤ hse(f) for any invariant set A ⊂ X. Indeed, for + f|A f any F ⊂ A, n ∈ N , > 0 and x ∈ F one has Bn [x, ] ∩ F = Bn[x, ] ∩ F . It follows that every (n, )-separated set F of f|A is (n, )-separated for f. Therefore, s(n, , f|A) ≤ s(n, , f) from which the assertion follows.

By the assertion we get hse(f) ≥ max0≤i≤m hse(f|Ai ). The reverse inequality follows as in Theorem 7.5 p. 172 of [35]. Our second property is about the natural inclusion order in the set of set-valued maps deﬁned by f ≤ g if and only if f(x) ⊂ g(x) for all x ∈ X. Theorem 3.2. Both the separated and spanning entropies reverse the inclusion order on the set of set-valued maps. Proof. Let f, g be two set-valued maps of a metric space X with f ≤ g. Let F be an (n, )-separated set of g. We claim that F is also (n, )-separated for f. Indeed, n n take distinct x, y ∈ F and sequences (xi)i=1,(yi)i=0 satisfying x0 = x, y0 = y, xi+1 ∈ f(xi) and yi+1 ∈ f(yi) for every i with 0 ≤ i ≤ n−1. Since f ≤ g, we obtain xi+1 ∈ g(xi) and yi+1 ∈ g(yi) for all i with 0 ≤ i ≤ n − 1. As F is (n, )-separated for g, we conclude that there is an i with 0 ≤ i ≤ n−1 such that d(xi, yi) > . Then, F is (n, )-separated for f as claimed. From this we obtain s(n, , g) ≤ s(n, , f) for every (n, ) ∈ N+×]0, ∞[, thus log s(n, , g) log s(n, , f) lim sup ≤ lim sup , ∀ > 0. n→∞ n n→∞ n

Taking limits as → 0 above we obtain hse(g) ≤ hse(f). The proof that the spanning entropy reverses the set-valued map order is similar Recall that a selection of a set-valued map f : X → 2X is any map s : X → X satisfying s(x) ∈ f(x) for all x ∈ X. Selections always exist under the axiom of choice. A direct consequence of Theorem 3.2 is the following.

Corollary 3.3. If f is a set-valued map on a metric space, then hse(f) ≤ h(s) for every selection s of f. Every single-valued map on a metric space satisﬁes r(n, ) ≤ s(n, ) ≤ r(n, 2 ). It is natural to expect the same inequalities in the set-valued case. However, we only obtain the ﬁrst of these inequalities as reported below. The proof of the second in the single-valued case depends on the fact that the maps dn in (1) are metrics, a fact which is false in general by Example 4.5. Lemma 3.4. If f is a set-valued map on a metric space, then r(n, ) ≤ s(n, ) for every n ∈ N+ and > 0. Proof. If s(n, ) = ∞ there is nothing to prove. Then, we can assume s(n, ) < ∞. Hence, there is an (n, )-separated set E such that card(E) = s(n, ). In particular card(E) < ∞. If we prove that E is (n, )-spanning, then r(n, ) ≤ card(E) = s(n, ) and we are done. Otherwise, we can arrange ! [ y ∈ K \ Bn[x, ] . (2) x∈E In particular, y∈ / E and so card(E ∪ {y}) > card(E) because card(E) < ∞. If x ∈ E,(2) implies Bn[x, ] ∩ (E ∪ {y}) = Bn[x, ] ∩ E = {x}. Moreover, if there were some x ∈ Bn[y, ] ∩ E then y ∈ Bn[x, δ] for some x ∈ E against (2). Therefore, TOPOLOGICAL ENTROPY FOR SET-VALUED MAPS 3465

Bn[y, ] ∩ E = ∅ which proves Bn[y, ] ∩ (E ∪ {y}) = {y}. Thus, E ∪ {y} is (n, )- separated. By (2) we have y ∈ K thus E ∪{y} ⊂ K contradicting card(E) = s(n, ). This proves the result. Our third property compares the separated and spanning entropies. Theorem 3.5. The spanning entropy is less than or equal to the separated entropy. Proof. The proof is a direct consequence of Lemma 3.4. The next property gives a suﬃcient condition for the separated and spanning entropies of a set-valued dynamical system to be equal. Theorem 3.6. Both the separated and spanning entropies coincide when the maps dn in (1) are metrics for all n large. Proof. The proof is similar to the single-valued case [35]. Indeed, by Theorem 3.5 it suﬃces to show that hse(f) ≤ hsp(f) and, for this, we only need to prove that s(n, ) ≤ r(n, 2 ) for any n large and any > 0. This is done as follows. + Take N ∈ N such that dn in (1) is a metric for n ≥ N. Let F and E be an (n, )-separated set and an (n, 2 )-spanning set respectively for n ≥ N and > 0. Then, there is a map φ : F → E satisfying x ∈ Bn[φ(x), 2 ] for x ∈ F . This 0 0 map is injective. Indeed, if φ(x) = φ(x ) for some x, x ∈ F , then dn(x, x) ≤ 0 dn(x, φ(x)) + dn(φ(x), x ) ≤ because n ≥ N (and so dn is metric). This implies 0 0 x ∈ Bn[x, ] ∩ F and so x = x . It follows that card(F ) ≤ card(E) which proves s(n, ) ≤ r(n, 2 ). We observe that the hypothesis of the above result are not valid in general (e.g. Example 4.5). Next we discuss the invariance of the entropies under topological conjugacy. Observe that, for any pair of sets X and Y , every map H : X → Y induces a map H : 2X → 2Y given by H(A) = {H(a): a ∈ A} for A ⊂ X. About this map we have the following proposition. Proposition 3.7. Let f and g be set-valued maps of metric spaces X and Y respectively. If there is a uniformly continuous surjective map H : X → Y satisfying H ◦ f ≤ g ◦ H, then h∗(f) ≥ h∗(g) for ∗ = se, sp.

Proof. First we prove hse(f) ≥ hse(g). We claim that for every > 0 there is δ > 0 such that for every n ∈ N+ and every (n, )-separated set F of g there is an (n, δ)-separated set F 0 of f such that card(F ) = card(F 0). Since H is uniformly continuous, for every > 0 there is δ > 0 such that whenever a, b ∈ X one has d(H(a),H(b)) > =⇒ d(a, b) > δ. Let F be an (n, )-separated set of g. Since H is onto, we can choose an injective map φ : F → X such that H ◦ φ = Id where Id is the identity. Let us prove that F 0 = φ(F ) is (n, )-separated for f. 0 n n Take distinct x, y ∈ F and sequences (xi)i=0,(yi)i=0 such that x0 = x, y0 = y, xi+1 ∈ f(xi) and yi+1 ∈ f(yi) for all i with 0 ≤ i ≤ n − 1. Since x 6= y, we have H(x) 6= H(y). In fact, x = φ(a) and y = φ(b) for some a, b ∈ F . As x 6= y one has a 6= b thus H(x) = H(φ(a)) = a 6= b = H(φ(b)) = H(y). n n Now, the sequences (H(xi))i=0 and (H(yi))i=0 satisfy H(x0) = H(x), H(y0) = H(y), H(xi+1) ∈ H(f(xi)) ⊂ g(H(xi)) and H(yi+1) ∈ H(f(yi)) ⊂ g(H(yi)) for all i with 0 ≤ i ≤ n − 1. As H(x),H(y) ∈ F and F is (n, )-separated for g, there is an 3466 D. CARRASCO-OLIVERA, R. METZGER ALVAN AND C. A. MORALES ROJAS

i0 with 0 ≤ i0 ≤ n − 1 such that d(H(xi0 ),H(yi0 )) > . Then, d(xi0 , yi0 ) > δ and the claim follows because card(F 0) = car(φ(F )) = card(F ). It follows from the claim that for every > 0 there is δ > 0 such that s(n, , g) ≤ s(n, δ, f) and so log s(n, , g) log s(n, δ, f) lim sup ≤ lim sup ≤ hse(f). n→∞ n n→∞ n

Letting → ∞ we obtain hse(g) ≤ hse(f). Next we prove hsp(f) ≥ hsp(g). We claim that for every > 0 there is δ > 0 such that for every n ∈ N+ and every (n, δ)-spanning set E0 of f there is an (n, )- spanning set E of g such that card(E) ≤ car(E0). Fix > 0 and take δ > 0 such that d(a, b) ≤ δ =⇒ d(H(a),H(b)) ≤ . Now ﬁx n ∈ N+ and an (n, δ)-spanning set E0 of f, i.e., [ X = Bn[x, δ]. x∈E0 Because H is surjective one has [ Y = H(Bn[x, δ]). x∈E0

We assert that H(Bn[x, δ]) ⊂ Bn[H(x), ]. Take y ∈ Bn[x, δ]. It follows that n n there are sequences (xi)i=0 and (yi)i=0 satisfying x0 = x, y0 = y, xi+1 ∈ f(xi), yi+1 ∈ f(yi) and d(xi, yi) ≤ δ for all i with 0 ≤ i ≤ n − 1. Then, the sequences n n (H(xi))i=0 and (H(yi))i=0 satisfy H(x0) = H(x), H(y0) = H(y), H(xi+1) ∈ H(f(xi)) ⊂ g(H(xi) and H(yi+1) ∈ H(f(xi)) ⊂ g(H(yi)) for all i with 0 ≤ i ≤ n−1. Moreover, we also have d(H(xi),H(yi)) ≤ for all i with 0 ≤ i ≤ n−1 by the choice of δ. Therefore H(y) ∈ Bn[H(x), ] proving the assertion. It follows from the assertion that [ Y = Bn[H(x), ]. x∈E0 Then, E = H(E0) is (n, )-spanning for g. Clearly card(E) ≤ car(E0) and the claim follows. The claim implies that for every > 0 there is δ > 0 such that r(n, δ, f) ≥ r(n, , g) for every n ∈ N+. Indeed, ﬁx > 0 and take δ as in the claim. We can assume that r(n, δ, f) < ∞. Otherwise we are done. Then, there is an (n, δ)- spanning set of f such that card(E0) = r(n, δ, f). By the claim we can choose an (n, )-spanning set E of g such that card(E0) ≥ car(E). Then, r(n, δ, f) = card(E0) ≥ car(E) ≥ r(n, , g). From this we obtain that for every > 0 there is δ > 0 such that log r(n, δ, f) log r(n, , g) hsp(f) ≥ lim sup ≥ lim sup . n→∞ n n→∞ n

As is arbitrary, we get hsp(f) ≥ hsp(g). We say that the set-valued maps f and g of the respective metric spaces X and Y are topologically conjugated if there is a uniform homeomorphism H : X → Y such that g ◦ H = H ◦ f. Correspondly, two metrics d and d0 of X are said to be uniformly equivalent if both Id :(X, d) → (Xd0) and Id :(X, d0) → (X, d) are uniformly continuous, where Id(x) = x is the identity. TOPOLOGICAL ENTROPY FOR SET-VALUED MAPS 3467

Our next property is the invariance of the separated and spanning entropy under conjugacies or equivalent metrics. Theorem 3.8. The separated and spanning entropies are invariant under topological conjugacy. Moreover, both entropies are independent from uniformly equivalent metrics. Proof. The first part is a direct consequence of Proposition 3.7 while the second follows from the first. Remark 3.9. Theorem 3.8 implies that the topological entropy is also an invariant for any single-valued map, whether continuous or not. This extends the single- valued result in Proposition 3.7 of p. 625 in Ciklová[12]. We define the composition g ◦ f of set-valued maps f, g of X by (g ◦ f)(x) = g(f(x)) ∀x ∈ X. We define f 0 by f 0(x) = {x} for all x ∈ X. Inductively we define f k = f ◦ f k−1 for k ≥ 1. Lemma 3.10. For every set-valued map f of a metric space one has f k f + dn ≤ dkn ∀n, k ∈ N . Proof. We can assume that k ∈ N+. Fix n ∈ N+ and x, y ∈ X. Given γ > 0 kn kn there are sequences (xi)i=0 and (yi)i=0 such that x0 = x, y0 = y, xi+1 ∈ f(xi), f yi+1 ∈ f(yi) and d(xi, yi) ≤ dkn(x, y) + γ for all i with 0 ≤ i ≤ kn − 1. Define xî = xki andy î = yki for all i with 0 ≤ i ≤ n. Then, the resulting sequences n n k k (ˆxi)i=0 and (ˆyi)i=0 satisfyx ˆ0 = x, yˆ0 = y,x î+1 ∈ f (ˆxi),y î+1 ∈ f (ˆyi) and f f k f d(ˆxi, yî) ≤ dkn(x, y) + γ for all i with 0 ≤ i ≤ n − 1. Then, dn (x, y) ≤ dn(x, y) + γ. As γ is arbitrary we obtain the result. Lemma 3.10 implies the following corollary. Corollary 3.11. For any set-valued map f of a metric space X one has f f k + Bkn[x, ] ⊂ Bn [x, ], ∀x ∈ X, n ∈ N , > 0. We say that a set-valued map f of a metric space X is uniformly continuous if for every > 0 there is δ > 0 such that d(f(x), f(y)) < whenever x, y ∈ X satisfy d(x, y) < δ. This definition reduces to the usual uniform continuity in the single-valued case. The next lemma is a kind of converse of the above corollary in the uniformly continuous case. Lemma 3.12. Let f be a uniformly continuous set-valued map on a metric space. Then, for every > 0 there is δ > 0 such that f k f + Bn [x, δ] ⊂ Bkn[x, ] ∀x ∈ X, n ∈ N , k ∈ N.

Proof. Fix > 0. Since f is uniformly continuous, there are 0 < δk < δk−1 < ··· < δ1 < δ0 = such that for every r satisfying 0 ≤ r ≤ k and every a, b ∈ X one has

d(a, b) ≤ δr =⇒ d(f(b), f(a)) < δr−1. (3)

Let us prove that δ = δk satisﬁes the conclusion of the lemma. Fix x ∈ X, n ∈ N+ and k ∈ N. We can assume that k ∈ N+. 3468 D. CARRASCO-OLIVERA, R. METZGER ALVAN AND C. A. MORALES ROJAS

f k n n Take y ∈ Bn [x, δ], i.e., there are sequences (xi)i=0 and (yi)i=0 such that x0 = 0, k k y0 = y, xi+1 ∈ f (xi), yi+1 ∈ f (yi) and d(xi, yi) ≤ δ for all i with 0 ≤ i ≤ n − 1. i k−1 i k−1 Given i with 0 ≤ i ≤ n − 1 we construct the sequences (xj)j=0 and (yj)j=0 as follows: i i i i Define x0 = xi and y0 = yi. Then, d(x0, y0) = d(xi, yi) ≤ δ = δk. Applying (3) i i i i i i there are x1 ∈ f(x0) and y1 ∈ f(y0) such that d(x1, y1) < δk−1. Again by (3) there i i i i i i are x2 ∈ f(x1) and y2 ∈ f(y1) such that d(x2, y2) < δk−2. Repeating the process i i i i i i we obtain xj and yj for 1 ≤ j ≤ k − 1 such that xj+1 ∈ f(xj), yj+1 ∈ f(yj) for all j i i with 0 ≤ j ≤ k − 2 and d(xj, yj) ≤ δk−j for all i with 0 ≤ j ≤ k − 1. This completes the construction. kn kn Next we define the sequences (ˆxl)l=0 and (ˆyl)l=0 by i i xîk+j = xj andy îk+j = yj for all i, j with 0 ≤ i ≤ n − 1 and 0 ≤ j ≤ k − 1 (resp.).

It follows from these choices thatx ˆ0 = x,y ˆ0 = y,x ˆl+1 ∈ f(ˆxl),y ˆl+1 ∈ f(ˆyl) and f d(ˆxl, yˆj) ≤ for all l with 0 ≤ l ≤ kn − 1. Then, y ∈ Bkn[x, ] and the lemma follows. Our next property is a power inequality for set-valued maps corresponding to the power formula h(f k) = k · h(f) for single-valued maps f. Theorem 3.13. Every uniformly continuous set-valued map f of a metric space satisﬁes k + h∗(f) ≤ h∗(f ) ≤ k · h∗(f) ∀k ∈ N where ∗ = sp, se. Proof. We ﬁrst prove the result for ∗ = se. Fix k ∈ N+. Let n ∈ N+, > 0 and F be an (n, )-separated set for f k. By Corollary 3.11 we obtain f f k F ∩ Bkn[x, ] ⊂ F ∩ Bn [x, ] = {x} ∀x ∈ F, hence F is (kn, )-separated for f. This implies s(n, , f k) ≤ s(kn, , f) and so k hse(f ) ≤ k · hse(f). k To prove hse(f) ≤ hse(f ), let > 0 and take δ from Lemma 3.12. Take n ∈ N and let F be (kn, )-separated set of f. Then, Lemma 3.12 implies

f k f F ∩ Bn [x, δ] ⊂ F ∩ Bkn[x, ] = {x} ∀x ∈ F, hence F is (n, δ)-separated for f k. This implies s(kn, , f) ≤ s(n, δ, f k). As s(n, , f) k k is increasing in n, we get s(n, , f) ≤ s(n, δ, f ). Then, hse(f) ≤ hse(f ). Next we prove the result for ∗ = sp. Fix k ∈ N+. Let n ∈ N+, > 0 and E be an (kn, )-spanning set for f. By Corollary 3.11 we obtain [ f [ f k X = Bkn[x, ] ⊂ Bn [x, ] x∈E x∈E and so E is (n, )-spanning for f k. This implies r(n, , f k) ≤ r(kn, , f) hence k hsp(f ) ≤ k · hsp(f). k To prove hsp(f) ≤ hsp(f ), let > 0 and take δ from Lemma 3.12. Take n ∈ N and let E be a (n, δ)-spanning for f k. Then, Lemma 3.12 implies

[ f k [ f X = Bn [x, δ] ⊂ Bkn[x, ] x∈E x∈E and so E is (kn, )-spanning for f. This implies r(kn, , f) ≤ r(n, δ, f k). But again k r(n, , f) is increasing in n, so hsp(f) ≤ hsp(f ). TOPOLOGICAL ENTROPY FOR SET-VALUED MAPS 3469

Next we prove the following lemma. Lemma 3.14. Let f be a uniformly continuous set-valued map of a metric space + X. Then, for every n ∈ N and > 0 there is δ > 0 such that dn(x, y) < whenever x, y ∈ X satisfy d(x, y) < δ.

Proof. Fix > 0. By uniform continuity there are 0 < δn < ··· < δ1 < δ0 = such that d(f(w), f(z)) < δi−1 whenever d(z, w) < δi and 1 ≤ i ≤ n. Now take δ = δn. From this lemma we obtain the following property. Lemma 3.15. If f is a uniformly continuous set-valued map on a compact metric space X, then s(n, ) < ∞ for every n ∈ N and > 0. + Proof. Otherwise there are n ∈ N , > 0 and a sequence Fk of (n, )-separated sets satisfying card(Fk) → ∞ as k → ∞. For these n and we fix δ > 0 as in Lemma 3.14. By compactness there are k large and distinct points x, y ∈ Fk with d(x, y) < δ. Then, Lemma 3.14 implies y ∈ Bn[x, ] ∩ Fk which contradicts that Fk is (n, )-separated. We say that a set-valued map f of a metric space X is equicontinuous if for every > 0 there is δ > 0 such that for all x, y ∈ X with d(x, y) < δ there are sequences (xi)i∈N and (yi)i∈N such that x0 = x, y0 = y, xi+1 ∈ f(xi), yi+1 ∈ f(yi) and d(xi, yi) < for every i ∈ N. This definition is the natural extension of the corresponding definition in the single-valued case [9]. Another related concept but in the set-valued setting is Definition 3.1 in [22]. The last property is a generalization of a well known fact in the single-valued case. Theorem 3.16. Both separated and spanning entropies vanish for equicontinuous set-valued maps on compact metric spaces. Proof. Equicontinuity is equivalent to the property that for every > 0 there is δ > 0 such that B[x, δ] ⊂ Bn[x, ], for every x ∈ X and n ∈ N. This implies s(n, ) ≤ s(1, δ) for all n ∈ N+. On the other hand, it is obvious that s(1, δ) > 0 and we also have s(1, δ) < ∞ by Lemma 3.15. Then, hse(f) = 0 and so hsp(f) = 0 by Theorem 3.5.

4. Examples. In this section we present some related examples. For this we need the following facts. Lemma 4.1. If f is a set-valued map on a metric space X, then

d2(a, b) = max(d(a, b), d(f(a), f(b))), ∀a, b ∈ X. Proof. We derive the result from the two assertions below:

• If d(f(a), f(b)) ≤ d(a, b), then d2(a, b) = d(a, b). • If d(a, b) < d(f(a), f(b)), then d2(a, b) = d(f(a), f(b)).

It follows from the deﬁnition of d2 that

d2(a, b) = inf{max(d(a, b), d(a1, b1)) : a1 ∈ f(a), b1 ∈ f(b)}.

In particular, d2 ≥ d. First we prove the ﬁrst assertion. If γ > 0, there are a1 ∈ f(a), b1 ∈ f(b) such that d(a1, b1) < d(a, b) + γ. For this particular choice one 3470 D. CARRASCO-OLIVERA, R. METZGER ALVAN AND C. A. MORALES ROJAS has max(d(a, b), d(a1, b1)) < d(a, b) + γ so d2(a, b) < d(a, b) + γ. As γ is arbitrary, d2(a, b) ≤ d(a, b) hence d2(a, b) = d(a, b). For the second assertion, if d(a, b) < d(f(a), f(b)), then max(d(a, b), d(a1, b1)) = d(a1, b1) for all a1 ∈ f(a), b1 ∈ f(b). Then, d2(a, b) = d(f(a), f(b)). Using this lemma we obtain the following proposition. Proposition 4.2. Let f be set-valued map on a metric space X. If there are a, b, c ∈ X satisfying 1 d(f(a), f(b)) = d(f(b), f(c)) = 0 and max(d(a, b), d(b, c)) < d(f(a), f(c)), 2 then d2 is not a metric.

Proof. It follows at once from Lemma 4.1 that d2(a, b) = d(a, b) and d2(b, c) = d(b, c). On the other hand, 1 1 d(a, c) ≤ d(a, b) + d(b, c) < d(f(a), f(c)) + d(f(a), f(c)) = d(f(a), f(c)). 2 2

From this and Lemma 4.1 we obtain d2(a, c) = d(f(a), f(c)). Then,

d2(a, c) = d(f(a), f(c)) 1 1 = d(f(a), f(c)) + d(f(a), f(c)) 2 2 > d(a, b) + d(b, c)

= d2(a, b) + d2(b, c) and so d2 does not satisfies the triangle inequality. Then, d2 is not a metric and the result follows. Now we present some examples. The first is the so-called subdifferential of the Euclidean norm f(x) = |x| of R (see p. 215 in [28]). Example 4.3. Consider the set-valued map ∂f of R defined by   {1} if x > 0 ∂f(x) = [−1, 1] if x = 0  {−1} if x < 0 1 1 By putting a = 2 , b = 0 and c = − 2 in Proposition 4.2 we obtain that d2 is not a metric. The second example is the following. Example 4.4. It is easy to see that the set-valued map f of [0, 1] defined by  1 {2x} if 0 ≤ x < 2  1 f(x) = [0, 1] if x = 2  1 {2x − 1} if 2 < x ≤ 1 1 is uniformly continuous. Nevertheless, by putting a < b = 2 < c with d(a, c) small in Proposition 4.2 we obtain that d2 is not a metric in this case as well. The next example is the following.

Example 4.5. For every k ∈ N+ there is a set-valued map f on the sphere Sk = k+1 k+1 {x ∈ R : kxk = 1} of R for which dn is not a metric for every n ≥ 2. TOPOLOGICAL ENTROPY FOR SET-VALUED MAPS 3471

Proof. Select three distinct points a, b, c ∈ Sk and three subsets A, B, C ⊂ Sk satisfying the following three hypotheses: • d(A, B) = d(B,C) = 0; 1 • max(d(a, b), d(b, c)) < 2 d(A, C); •{ a, b, c} ∩ (A ∪ B ∪ C) = 0. Since a 6= b 6= c 6= a, the set-valued map f of Sk defined by   A if x = a  B if x = b f(x) =  C if x = c  {x} if x∈ / {a, b, c} is well-defined. It follows from the two first hypothesis and Proposition 4.2 that d2 is not a metric for this f. Using the third hypothesis we obtain dn = d2 for every n ≥ 2, and so, dn is not a metric for n ≥ 2. This finishes the construction. Another example is the following one. Example 4.6. Let f be a set-valued map on a metric space X satisfying x ∈ f(x) for all x ∈ X. It follows that d(f(x), f(y)) ≤ d(x, y) for all x, y ∈ X. Then, d2 = d is a metric by Lemma 4.1. Now fix δ > 0 and define the set-valued map f on R2 by 2 f(x) = B[x, δ] for all x ∈ R . Then, f is not single-valued but d2 is a metric. These maps have both separating and spanning entropies equal to zero (see Example 4.7). To finish we present some example where the separated and spanning entropies can be computed. Example 4.7. Let f be a set-valued map on a metric space X satisfying x ∈ f(x) for all x ∈ X. Then, hse(f) = hsp(f) = 0. Indeed, in this case the identity Id is a continuous selection of f so hse(f) ≤ h(Id) = 0 by Corollary 3.3. This can be also proved by noticing that all such set-valued maps are equicontinuous and then both entropies vanish by Theorem 3.16. In particular, given δ ≥ 0 the set-valued map f(x) = B[x, δ] for x ∈ X has zero separated and spanning topological entropies. For the next three examples we will consider the unit circle 1 S = {z ∈ C : |z| = 1}. Define g : S1 → S1 by g(z) = z2. It follows that h(g) = log 2. Furthermore g is an expanding map, i.e., there is λ > 1 such that kDg(z)k ≥ λ for all z ∈ S1. Example 4.8. Endow the unit interval [0, 1] with the Euclidean metric. Define the set-valued map f of [0, 1] by  1 {2x}, if 0 ≤ x < 2  1 f(x) = {0, 1}, if x = 2  1 {2x − 1}, if 2 < x ≤ 1.

It follows that hse(f) = log 2. Proof. Let H : [0, 1] → S1 be defined by H(x) = e2πxi. It follows that H is a continuous surjection. One can check easily that g ◦ H = H ◦ f thus we have hse(f) ≥ hse(g) = h(g) = log 2 by Proposition 3.7. On the other hand, the map 1 2x, if 0 ≤ x ≤ 2 s(x) = 1 2x − 1, if 2 < x ≤ 1. 3472 D. CARRASCO-OLIVERA, R. METZGER ALVAN AND C. A. MORALES ROJAS is a selection of f. It follows that hse(f) ≤ h(s) by Corollary 3.3. As h(s) = log 2, we obtain hse(f) ≤ log 2 whence hse(f) = log 2. Example 4.9. Given δ > 0 we define the set-valued map f of S1 by f(z) = B[g(z), δ] for all z ∈ S1. Since g is expanding, there is n ∈ N+ depending on δ only such that f n(z) = S1 for all z ∈ S1. In particular, z ∈ f n(z) for all z ∈ S1 n and so hse(f ) = 0. On the other hand, f is clearly uniformly continuous so n hse(f) ≤ hse(f ) = 0 by Theorem 3.13. Therefore, hse(f) = 0. From this and Theorem 3.5 we obtain hsp(f) = 0. The following is an example of a genuine (i.e. not single-valued) map for which the separated and spanning entropies coincide and the common value is positive. Example 4.10. Let f be the set-valued map on [0, 1] defined in Example 4.8. By Theorem 3.5 one has hsp(f) ≤ hse(f) = log 2. On the other hand, we have already seen that hsp(f) ≥ hsp(g). As hsp(g) = hse(g) = log 2 we conclude that hsp(f) = hse(f) = log 2. The last example is the following. Example 4.11. The set-valued map f on [0, 1] defined in Example 4.4 has both spanning and separating entropies equal to log 2. Proof. Clearly f is bigger than the set-valued map in Example 4.8 which, in turns, has spanning entropy less than or equal to log 2. Then, hsp(f) ≤ log 2 by Theorem 3.5. On the other hand, by taking suitable closed intervals I1, I2 at each side of 1 T −n we obtain a compact invariant set A = f (I2 ∪ I2) for which f|A has 2 n∈N topological entropy equals to log 2. Then, log 2 ≤ hsp(f) by Theorem 3.1 and we are done. Similarly we obtain hse(f) = log 2.

5. Conclusions. In this paper we used the single-valued approach of separated and spanning sets [8], [15] to deﬁne the spanning and separated topological entropies for set-valued maps. We proved that these entropies satisfy some properties resembling to the single-valued case. These include the kind of sub-additivity property in Theorem 3.1 (similar to the single-valued case), that they reverse natural inclusion orders for set-valued maps (not available in the single-valued case), that the spanning entropy is less than the separated one (and that they both coincide when the induced semimetrics are metrics), that they are topological invariants (similar to the single-valued case), that they satisfy a power inequality closely related to the power formula in the single-valued case, and that they both vanish for equicontinuous set-valued maps (again as in the single-valued case). We also computed them in some genuine (i.e. not single-valued) examples.

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