Some Properties of the Topological Entropy of a Family of Dynamical Systems Defined on an Arbitrary Metric Space

We consider a family of dynamical systems defined on a noncompact metric space and continuously depending on a parameter varying in some metric space. For any such family, the topological entropy of the dynamical systems in the family is studied as a function of the parameter from the viewpoint of the Baire classification of functions.


BAIRE CLASS OF THE TOPOLOGICAL ENTROPY OF A FAMILY OF DYNAMICAL SYSTEMS IN THE CASE OF A NONINVARIANT COMPACT SET
Following [2], let us give a definition needed in what follows. Let (X, d) be a metric space, let K(X) be the set of compact subsets of X, and let f : X → X be a continuous mapping. Along with the original metric d, we define an additional system of metrics d f n , n ∈ N, on X by the formula where f •i , i ∈ N, is the ith iteration of f and f •0 ≡ id X . Given a K ∈ K(X), for any n ∈ N and ε > 0 we denote by N d (K, f, ε, n) the maximum number of points in K with pairwise d f n -distances greater than ε. The numbers are called the upper and lower topological entropies, respectively, of f on K.
Note that if the metric d is replaced by a metric generating the same topology as d, then the numbers (1) do not change [3, p. 121 of the Russian translation].
Let us also recall formulas for the upper and lower topological entropies, to be used below. For any x ∈ X, ε > 0, and n ∈ N, let B f (x, ε, n) be the open ball {y ∈ K : d f n (x, y) < ε} of radius ε centered at x in the space (X, d f n ). A set D ⊂ K is called an (f, ε, n)-cover of K if Formulas (1) (or (2)) obviously imply the inequality h top (K, f ) ≥ h top (K, f ). If K is f -invariant, i.e., f (K) ⊂ K, then the upper and lower topological entropies of f on K coincide [3, p. 122 of the Russian translation]. The following example shows that the numbers (1) do not necessarily coincide in the general case. Consider the set Ω 2 of sequences x = (x 1 , Note that the space (Ω 2 , d Ω2 ) is homeomorphic to the Cantor set on the interval [0, 1] with the metric induced by the standard metric of the real line. Let K 0 ⊂ Ω 2 be the compact set defined by the condition and let σ : Then Given a metric space M, a compact set K ⊂ X, and a continuous mapping consider the functions In the present paper, for each mapping (3) the functions (4) and (5)  We have already mentioned that the numbers (1) coincide if K is f -invariant; in that case, their common value is called the topological entropy of f and is denoted by h top (f ). In was established in the paper [4] that for each mapping (3) the function is of Baire class 2 on M. In [5], a family of homeomorphisms (3) with X = M = Ω 2 was constructed such that the function (6) is not of Baire class 1; consequently, the functions (4) and (5), generally speaking, are not of Baire class 1 either. It was shown in [4] that if the space M is metrizable by a complete metric, then the set of points of lower semicontinuity of the function (6) contains a G δ set everywhere dense in M, while the paper [6] established that the set of points of lower semicontinuity itself is an everywhere dense G δ set in M. Further, let X = Ω 2 , and let M be an arbitrary complete metric separable zero-dimensional space (for example, Ω 2 ). In this case, for an arbitrary G δ set G everywhere dense in M, the paper [7] presents the construction of a mapping (3) such that the set of points of lower semicontinuity of the corresponding function (6) coincides with G. It turns our that the following assertion holds in the case of a noninvariant compact set K.
Theorem 1. For any K ∈ K(X) and any mapping (3), the function (4) is of Baire class 3 and the function (5) is of Baire class 2 on M. If M is metrizable by a complete metric, then the set of points of lower semicontinuity of the function (5) is an everywhere dense G δ set.
Since the maximum and minimum of finitely many functions in a Baire class belong to the same class [9, Ch. IX, Sec. 37, III of the Russian translation], it obviously follows from these representations that the function µ → h top (K, f (µ, · )) is of Baire class 3 and the function µ → h top (K, f (µ, · )) is of Baire class 2 on M.
Since the function µ → h top (K, f (µ, · )) can be represented as the limit of a nondecreasing sequence of functions of Baire class 1, we see that its set of points of lower semicontinuity is an everywhere dense G δ set [10, Lemma 2]. The proof of the theorem is complete.
Note that if M is a complete metric space, then, by Baire's theorem [9, Ch. IX, Sec. 39, VI of the Russian translation], Theorem 1 implies that for each mapping (3) there exists an everywhere dense G δ set G ⊂ M such that the restrictions of the functions µ → h top (K, f (µ, · )) and µ → h top (K, f (µ, · )) to G are continuous.
There arises a natural question about the least Baire class containing the function (4). To answer this question, we construct metric spaces B and C. By definition, the points of B are all possible (countable) sequences µ = (µ k ) ∞ k=1 of positive integers. The distance between two points µ and ν is defined by the formula For each r ∈ N, let K r ∈ K(C) be the compact set K r = [0, 1] × {1, . . . , r} in C.
Theorem 2. Let M = B, X = C , and K = K r ; then there exists a mapping (3) such that the function (4) is everywhere discontinuous and does not belong to the second Baire class on M.
Proof. Given a sequence µ = (µ k ) ∞ k=1 ∈ B, we construct the sequence α(µ) with elements We use this sequence to construct a mapping f : B × C → C by setting By definition, the function f is continuous on B × C.
Le E ⊂ B be the set of sequences tending to infinity. Let us find the upper topological entropy of the mapping (9) for µ ∈ E.
for the mapping (9) for any r ∈ N.
To complete the proof of Theorem 2, we use the following assertion established in [11]: if a function µ → h top (K r , f (µ, · )) is of Baire class 2, then the intersection of closures of the sets h top (K r , f (E, · )) and h top (K r , f (B \ E, · )) is nonempty. By Lemmas 1 and 2, and consequently, the function µ → h top (K r , f (µ, · )) is not of Baire class 2. Since both E and B \ E are everywhere dense in B, it follows that this function is everywhere discontinuous on B. The proof of the theorem is complete.

BAIRE CLASS OF THE TOPOLOGICAL ENTROPY OF A FAMILY OF DYNAMICAL SYSTEMS ON A NONCOMPACT METRIC SPACE
Following [2], we define the upper and lower topological entropies of a mapping f : X → X as the numbers respectively. The following example shows that the two numbers (13) may be distinct. Let us construct a space A as follows. The points of A are all possible pairs (x, i), where x ∈ Ω 2 and i ∈ N, and the metric is defined by the formula Consider the sequence Proof. For an arbitrary r ∈ N, let H r ∈ K(A) be the compact set H r = Ω 2 × {1, . . . , r}. Each compact set K ∈ K(A) is contained in H r for some r; consequently, Let p ∈ N, and let Q be an (f A , 1/p, t 2k )-cover of H r with minimum number of elements. Then Q is an (f A , 1/p, t 2k+1 )-cover of H r by the definition of f A . Since the points (x, i) ∈ A, where x = (x 1 , . . . , x t 2k +p , 0, 0, . . .) and i ∈ {1, 2, . . . , r}, form an (f A , 1/p, t 2k )-cover of H r , it follows that the number of elements in Q does not exceed r2 t 2k +p . Therefore, and hence h top (f A ) = 0.
Take one point (y x , 1) ∈ H 1 in the preimage of every point (x, t 2k+1 ), x ∈ R k , under the mapping f is not less than the cardinality of R k , which is 2 (2k+2)! , for each ε < 1, and hence The proof of the lemma is complete. For the mapping (3), consider the functions µ If X is a compact metric space, then the numbers (13) are equal to the topological entropy of f . Therefore, the functions (14) and (15) are of Baire class 2 by [4] but in general not of Baire class 1 by [5].  (14) is everywhere discontinuous and does not belong to the second Baire class on M.
Proof. Each compact set K ∈ K(C) is contained in K r for some r, and consequently, the topological entropy of an arbitrary continuous mapping f : C → C satisfies the relation By Lemmas 1 and 2, we obtain the chain of inequalities for the family (9). Consequently, the function µ → h top (f (µ, · )) is not of Baire class 2 [11], and since the sets E and B \ E are everywhere dense in B, it follows that this function is everywhere discontinuous on B. The proof of the theorem is complete.
Recall that a metric space X is said to be locally compact if each of its points has a compact neighborhood [12, p. 315 of the Russian translation]. A locally compact space X is said to be countable at infinity [12, p. 316 of the Russian translation] if it is a union of a countably many compact sets. The space R n , the above-defined space C, and, in general, any locally compact space with countable base are examples of such spaces [12, p. 316; 13, p. 254]. Proof. Since X is countable at infinity, it follows that there exists an increasing sequence {U s } ∞

s=1
of relatively compact open sets that form a cover of X and satisfy U s ⊂ U s+1 for all s ∈ N [12, p. 316 of the Russian translation]. Every compact set K ⊂ X is contained in U s0 for some s 0 , because otherwise the cover of K by the sequence {U s } ∞ s=1 would not contain a finite subcover, which contradicts the compactness of K. Thus, for each continuous mapping f : X → X. Using formula (7), we obtain Since the maximum and minimum of finitely many functions in some Baire class belong to the same class [9, Ch. IX, Sec. 37, III of the Russian translation], it follows that the function µ → h top (f (µ, · )) is of Baire class 3 and the function µ → h top (f (µ, · )) is of Baire class 2 on M.
Since the function µ → h top (f (µ, · , )) can be represented as the limit of a nondecreasing sequence of functions of Baire class 1, it follows that its set of points of lower semicontinuity is an everywhere dense G δ set provided that M is metrizable by a complete metric [10, Lemma 2]. The proof of the theorem is complete.

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