Abstract
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.
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The topological entropy of autonomous dynamical systems on an invariant compact metric space was defined in [1]. Later, this concept was extended in [2] to dynamical systems defined on an arbitrary metric space.
1. 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 \(\mathcal {K}(X) \) be the set of compact subsets of \(X \), and let \(f:X\to X \) be a continuous mapping. Along with the original metric \(d \), we define an additional system of metrics \(d^{f}_n \), \(n\in \mathbb {N}\), on \(X \) by the formula
where \(f^{\circ i} \), \(i\in \mathbb {N}\), is the \(i \)th iteration of \(f \) and \(f^{\circ 0}\equiv \mathrm {id}_X\). Given a \(K\in \mathcal {K}(X) \), for any \(n\in \mathbb {N} \) and \(\varepsilon >0 \) we denote by \(N_d(K,f,\varepsilon ,n) \) the maximum number of points in \(K \) with pairwise \(d_n^f \)-distances greater than \(\varepsilon \). 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 \(x \in X\), \(\varepsilon >0 \), and \(n\in \mathbb {N} \), let \(B_{f}(x,\varepsilon ,n) \) be the open ball \(\{y\in K:d_n^{f}(x,y)<\varepsilon \}\) of radius \(\varepsilon \) centered at \(x \) in the space \((X,d^{f}_n) \). A set \(D\subset K \) is called an \((f,\varepsilon ,n) \)-cover of \(K \) if
Let \(S_d(K,f,\varepsilon ,n) \) be the minimum possible number of elements in an \((f,\varepsilon ,n)\)-cover of \(K \). Then the upper and lower topological entropies of \(f \) on \(K \) can be calculated by the formulas [3, p. 122 of the Russian translation]
Formulas (1) (or (2)) obviously imply the inequality \(\overline {h}_{\mathrm {top}}(K,f)\geq \underline {h}_{\mathrm {top}}(K,f) \). If \(K \) is \(f \)-invariant, i.e., \(f(K)\subset K \), 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 \(\Omega _2\) of sequences \( x=(x_1,x_2,x_3,\ldots )\), where \(x_i\in \{0,1\} \), with the metric
Note that the space \((\Omega _2,d_{\Omega _2}) \) is homeomorphic to the Cantor set on the interval \([0,1] \) with the metric induced by the standard metric of the real line. Let \( K_0\subset \Omega _2\) be the compact set defined by the condition
and let \(\sigma :\Omega _2\to \Omega _2 \) be the left shift mapping, \(\sigma ((x_1, x_2, x_3,\ldots ))=(x_2, x_3, x_4,\ldots )\). Then
Given a metric space \(\mathcal {M}\), a compact set \( K\subset X\), and a continuous mapping
consider the functions
In the present paper, for each mapping (3) the functions (4) and (5) are studied from the viewpoint of the Baire classification of functions. Recall that continuous functions \( M\rightarrow \mathbb {R}\) on a metric space \(\mathcal {M} \) are called functions of Baire class \(0\), and the functions of Baire class \( p\) are defined for each positive integer \(p \) as the functions that are pointwise limits of sequences of functions of Baire class \(p-1\).
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 \(h_{\mathrm {top}}(f) \). In was established in the paper [4] that for each mapping (3) the function
is of Baire class 2 on \(\mathcal {M} \). In [5], a family of homeomorphisms (3) with \(X=\mathcal {M}=\Omega _2 \) 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 \(\mathcal {M} \) is metrizable by a complete metric, then the set of points of lower semicontinuity of the function (6) contains a \( G_\delta \) set everywhere dense in \(\mathcal {M} \), while the paper [6] established that the set of points of lower semicontinuity itself is an everywhere dense \(G_\delta \) set in \(\mathcal {M} \). Further, let \(X=\Omega _2 \), and let \(\mathcal {M} \) be an arbitrary complete metric separable zero-dimensional space (for example, \(\Omega _2\)). In this case, for an arbitrary \(G_\delta \) set \(\mathcal {G} \) everywhere dense in \(\mathcal {M} \), 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 \(\mathcal {G} \). It turns our that the following assertion holds in the case of a noninvariant compact set \(K\).
Theorem 1.
For any \(K\in \mathcal {K}(X) \) and any mapping (3), the function (4) is of Baire class 3 and the function (5) is of Baire class 2 on \( \mathcal {M}\). If \( \mathcal {M}\) is metrizable by a complete metric, then the set of points of lower semicontinuity of the function (5) is an everywhere dense \(G_\delta \) set.
Proof. For any \(\varepsilon >0 \) and \(n\in \mathbb {N} \), the function \(\mu \mapsto n^{-1}\ln N_d(K,f(\mu ,\cdot \thinspace ),\varepsilon ,n) \) is lower semicontinuous [8] and the function \(\mu \mapsto n^{-1}\ln S_d(K,f(\mu ,\cdot \thinspace ),\varepsilon ,n) \) is upper semicontinuous [4]; consequently [9, Ch. IX, Sec. 37, XI of the Russian translation], there exist sequences of continuous functions \(\mu \mapsto \varphi _d^m(K;\mu ,\varepsilon ,n) \) and \(\mu \mapsto \psi _d^m(K;\mu ,\varepsilon ,n)\), \(m\in \mathbb {N} \), on \(\mathcal {M} \) such that
Hence we obtain the representations
by (7) and
by (8).
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 \(\mu \mapsto \overline {h}_{\mathrm {top}}(K,f(\mu ,\cdot \thinspace )) \) is of Baire class 3 and the function \(\mu \mapsto \underline {h}_{\mathrm {top}}(K,f(\mu ,\cdot \thinspace )) \) is of Baire class 2 on \(\mathcal {M} \).
Since the function \(\mu \mapsto \underline {h}_{\mathrm {top}}(K,f(\mu ,\cdot \thinspace ))\) 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 \(G_\delta \) set [10, Lemma 2]. The proof of the theorem is complete.
Note that if \(\mathcal {M}\) 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 \(G_\delta \) set \(G\subset \mathcal {M} \) such that the restrictions of the functions \(\mu \mapsto \underline {h}_{\mathrm {top}}(K,f(\mu ,\cdot \thinspace )) \) and \(\mu \mapsto \overline {h}_{\mathrm {top}}(K,f(\mu ,\cdot \thinspace )) \) 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 \( \mathcal {B}\) and \(\mathcal {C} \). By definition, the points of \(\mathcal {B} \) are all possible (countable) sequences \(\mu =(\mu _k)_{k=1}^\infty \) of positive integers. The distance between two points \(\mu \) and \(\nu \) is defined by the formula
Note that the space \((\mathcal {B},d_{\mathcal {B}})\) is homeomorphic to the set of irrational numbers on the interval \([0,1]\) with the metric induced by the natural metric of the real line. The points of \(\mathcal {C} \) are all possible pairs \((x,i) \), where \(x\in [0,1] \) and \(i\in \mathbb {N} \). The distance between points \((x,i) \) and \((y,j) \) is defined by the formula
For each \(r\in \mathbb {N}\), let \(K_r\in \mathcal {K}(\mathcal {C})\) be the compact set \( K_r=[0,1]\times \{1,\ldots ,r\}\) in \(\mathcal {C} \).
Theorem 2.
Let \(\mathcal {M}=\mathcal {B} \), \( X=\mathcal {C}\), 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 \(\mathcal {M} \).
Proof. Given a sequence \(\mu =(\mu _k)_{k=1}^\infty \in \mathcal {B}\), we construct the sequence \(\alpha (\mu ) \) with elements \(\alpha _k(\mu )=\mu _{[\log _2(k+1)]}\) (where \([\cdot ] \) is the integer part of a number). Consider the sequence \((f_k) \) of mappings of \(\mathcal {B}\times [0,1] \) into \([0,1] \) given by
We use this sequence to construct a mapping \(f:\mathcal {B}\times \mathcal {C}\to \mathcal {C} \) by setting
By definition, the function \(f \) is continuous on \(\mathcal {B}\times \mathcal {C}\).
Le \(\mathcal {E}\subset \mathcal {B} \) be the set of sequences tending to infinity. Let us find the upper topological entropy of the mapping (9) for \(\mu \in \mathcal {E} \).
Lemma 1.
If \(\mu \in \mathcal {E} \), then
for the mapping (9) for any \(r\in \mathbb {N} \).
Proof. Fix an \(\varepsilon \!\in \! (0,1) \) and a \(\mu \!\in \!\mathcal {E} \). Then there exists a number \(k_0(\varepsilon )\!>\!r \) such that \({1}/{\alpha _k(\mu )}\!<\!\varepsilon /2\) for each \(k\geq k_0(\varepsilon ) \).
Let \(A_{k_0(\varepsilon )} \) be an \((f(\mu ,\cdot \thinspace ),\varepsilon /2,k_0(\varepsilon ))\)-cover of \(K_r \) with minimum number of elements. Let us prove that \( A_{k_0(\varepsilon )}\) is an \((f(\mu ,\cdot \thinspace ),\varepsilon ,k_0(\varepsilon )+i)\)-cover of \(K_r \) for each \(i\in \mathbb {N}\cup \{0\} \).
By the definition of \(A_{k_0(\varepsilon )} \), for each point \((x,l)\in K_r \) there exists an element \((x_0,l)\in A_{k_0(\varepsilon )}\) such that \((x,l)\in B_{f(\mu ,\cdot \thinspace )}((x_0,l),\varepsilon /2,k_0(\varepsilon )) \).
If \( f^{\circ (k_0(\varepsilon )-1)}(\mu ,(x,l)), f^{\circ (k_0(\varepsilon )-1)}(\mu ,(x_0,l))\in [0,1-\varepsilon /2]\times \mathbb {N} \), then
for each \(i\in \mathbb {N}\cup \{0\} \), because the interval \([0,1-\varepsilon /2] \) is invariant with respect to the mapping \( f_{k_0(\varepsilon )+i}(\mu ,\cdot \thinspace ) \) for all \(i\in \mathbb {N}\cup \{0\} \).
If \( f^{\circ (k_0(\varepsilon )-1)}(\mu ,(x,l)), f^{\circ (k_0(\varepsilon )-1)}(\mu ,(x_0,l))\in [1-\varepsilon /2,\thinspace 1]\times \mathbb {N} \), then
for each \(i\in \mathbb {N}\cup \{0\} \) , because the interval \([1-\varepsilon /2,\thinspace 1]\) is invariant with respect to the mappings \( f_{k_0(\varepsilon )+i}(\mu ,\cdot \thinspace ) \) for all \(i\in \mathbb {N}\cup \{0\} \).
If either \(f^{\circ (k_0(\varepsilon )-1)}(\mu ,(x,l))\in [0,1-\varepsilon /2]\times \mathbb {N}\) and \(f^{\circ (k_0(\varepsilon )-1)}(\mu ,(x_0,l))\in [1-\varepsilon /2,1]\times \mathbb {N} \) or \(f^{\circ (k_0(\varepsilon )-1)}(\mu ,(x_0,l))\in [0,1-\varepsilon /2]\times \mathbb {N} \) and \(f^{\circ (k_0(\varepsilon )-1)}(\mu ,(x,l))\in [1-\varepsilon /2,1]\times \mathbb {N} \), then
for each \(i\in \mathbb {N}\cup \{0\} \).
It follows from inequalities (10)–(12) that \((x,l)\in B_{f(\mu ,\cdot \thinspace )}((x_0,l),\varepsilon ,k_0(\varepsilon )+i) \) for each \(i\in \mathbb {N}\cup \{0\} \), and hence the set \(A_{k_0(\varepsilon )} \) is an \((f(\mu ,\cdot \thinspace ),\varepsilon ,k_0(\varepsilon )+i)\)-cover of \(K_r \) for each \(i\in \mathbb {N}\cup \{0\} \). Thus, the estimate
holds for \(n\geq k_0(\varepsilon ) \), which implies that
The proof of the lemma is complete.
Now let us estimate the upper topological entropy of the mapping (9) for \(\mu \notin \mathcal {E} \).
Lemma 2.
If \(\mu \notin \mathcal {E} \), then
for the mapping (9).
Proof. Let \(\mu \notin \mathcal {E} \). Then there exists a subsequence \((\mu _{k_j})_{j=1}^{\infty }\subset (\mu _k)_{k=1}^{\infty }\) and a positive integer \(q \) such that \(\mu _{k_j}=q \) for all \(j\in \mathbb {N} \).
Further, \(f_k(\mu ,x)=f_{2^{k_j}-1}(\mu ,x)=t_q(x) \) for all \(j\in \mathbb {N} \), \(k\in \{2^{k_j}-1,\ldots ,2^{k_{j}+1}-2\}\), and \(x\in [0,1] \), where
The affine order-preserving transformation \( \varphi \) mapping the interval \(I_q=[1-1/q,\thinspace 1] \) onto the interval \([0,\thinspace 1] \) takes the mapping \(t_q|_{I_q}:I_q\to I_q \) to the mapping \(g=\varphi \circ t_q|_{I_q}\circ \varphi ^{-1}:[0,\thinspace 1]\to [0,\thinspace 1] \) given by the formula
It was established in the monograph [3, p. 502 of the Russian translation] that the topological entropy of \(g \) is \(\ln 2 \); consequently, there exists an \(\varepsilon _0<1/q \) such that
for every \(\varepsilon <\varepsilon _0\). For each \(n\in \mathbb {N} \), consider a set \(\{a_1,\ldots ,a_{N_d([0,\thinspace 1],g,\varepsilon ,n)}\} \) of points on the interval \([0,\thinspace 1] \) with pairwise \(d_n^g \)-distances greater than \(\varepsilon >0 \).
Let \(\varepsilon <\varepsilon _0 \) and \(n=2^{k_{j}+1}-2^{k_{j}}-1 \). Then the \(d_{2^{k_{j}}+n-1}^{f(\mu ,\cdot \thinspace )}\)-distance between any preimages of two arbitrary points \((\varphi ^{-1}(a_i),2^{k_{j}}-1) \) and \((\varphi ^{-1}(a_m),2^{k_{j}}-1)\), \(i\ne m \), under the mapping \(f^{\circ (2^{k_{j}}-2)}(\mu ,\cdot \thinspace )\) is greater than \(\varepsilon /q>0\), and consequently,
Hence we obtain the estimates
The proof of the lemma is complete.
To complete the proof of Theorem 2, we use the following assertion established in [11]: if a function \(\mu \mapsto \overline {h}_{\mathrm {top}}(K_r,f(\mu ,\cdot \thinspace )) \) is of Baire class 2, then the intersection of closures of the sets \( \overline {h}_{\mathrm {top}}(K_r,f(\mathcal {E},\cdot \thinspace )) \) and \(\overline {h}_{\mathrm {top}}(K_r,f(\mathcal {B} \setminus \mathcal {E},\cdot \thinspace )) \) is nonempty. By Lemmas 1 and 2,
and consequently, the function \(\mu \mapsto \overline {h}_{\mathrm {top}}(K_r,f(\mu ,\cdot \thinspace )) \) is not of Baire class 2. Since both \(\mathcal {E} \) and \(\mathcal {B}\setminus \mathcal {E}\) are everywhere dense in \(\mathcal {B} \), it follows that this function is everywhere discontinuous on \( \mathcal {B}\). The proof of the theorem is complete.
2. 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 \(f:X\to X\) as the numbers
respectively. The following example shows that the two numbers (13) may be distinct. Let us construct a space \(\mathcal {A}\) as follows. The points of \(\mathcal {A}\) are all possible pairs \( (x,i)\), where \(x\in \Omega _2 \) and \(i\in \mathbb {N} \), and the metric is defined by the formula
Consider the sequence
of continuous self-mappings of \( \Omega _2\) and define a continuous mapping \(f_{\mathcal {A}}:\mathcal {A}\to \mathcal {A}\) by setting
Lemma 3.
\(\underline {h}_{\mathrm {top}}(f_{\mathcal {A}}) <\overline {h}_{\mathrm {top}}(f_{\mathcal {A}}) \) .
Proof. For an arbitrary \(r\in \mathbb {N} \), let \(H_r\in \mathcal {K}(\mathcal {A})\) be the compact set \(H_r=\Omega _2\times \{1,\ldots ,r\}\). Each compact set \(K\in \mathcal {K}(\mathcal {A})\) is contained in \(H_r \) for some \(r \); consequently,
Let \(p\in \mathbb {N}\), and let \(Q \) be an \((f_{\mathcal {A}},{1}/{p},t_{2k})\)-cover of \(H_r \) with minimum number of elements. Then \(Q \) is an \((f_{\mathcal {A}},{1}/{p},t_{2k+1})\)-cover of \(H_r \) by the definition of \(f_{\mathcal {A}} \). Since the points \((x,i)\in {\mathcal {A}} \), where \(x=(x_1,\ldots ,x_{t_{2k}+p},0,0,\ldots )\) and \(i\in \{1,2,\ldots ,r\}\), form an \((f_{\mathcal {A}},{1}/{p},t_{2k})\)-cover of \(H_r \), it follows that the number of elements in \(Q \) does not exceed \(r2^{t_{2k}+p} \). Therefore,
and hence \(\underline {h}_{\mathrm {top}}(f_{\mathcal {A}})=0\).
Let us establish the inequality \(\overline {h}_{\mathrm {top}}(f_{\mathcal {A}})\geq 0.5\ln 2\), whence Lemma 3 will follow. In the space \(\Omega _2 \), consider the set \(R_k \), \(k\in \mathbb {N}\cup \{0\} \), of points of the form
Take one point \((y_x,1)\in H_1 \) in the preimage of every point \((x,t_{2k+1}) \), \(x\in R_k \), under the mapping \(f_{\mathcal {A}}^{\circ (t_{2k+1}-1)}\). If \(x^{\prime }\ne x^{\prime {}\prime }\), \(x^{\prime }, x^{\prime {}\prime }\in R_k\), then
Thus, \(N_{d_{\mathcal {A}}}(H_1,f_{\mathcal {A}},\varepsilon ,t_{2k+2})\) is not less than the cardinality of \(R_k \), which is \(2^{(2k+2)!} \), for each \(\varepsilon <1 \), and hence
The proof of the lemma is complete.
For the mapping (3), consider the functions
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].
Theorem 3.
If \(\mathcal {M}=\mathcal {B} \) and \( X=\mathcal {C}\), then there exists a mapping (3) such that the function (14) is everywhere discontinuous and does not belong to the second Baire class on \(\mathcal {M} \).
Proof. Each compact set \(K\in \mathcal {K}(\mathcal {C}) \) is contained in \(K_r \) for some \(r \), and consequently, the topological entropy of an arbitrary continuous mapping \(f:\mathcal {C}\to \mathcal {C} \) satisfies the relation
By Lemmas 1 and 2, we obtain the chain of inequalities
for the family (9). Consequently, the function \(\mu \mapsto \overline {h}_{\mathrm {top}}(f(\mu ,\cdot \thinspace )) \) is not of Baire class 2 [11], and since the sets \(\mathcal {E} \) and \(\mathcal {B}\setminus \mathcal {E}\) are everywhere dense in \(\mathcal {B} \), it follows that this function is everywhere discontinuous on \( \mathcal {B}\). 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 \(\mathbb {R}^n\), the above-defined space \(\mathcal {C}\), and, in general, any locally compact space with countable base are examples of such spaces [12, p. 316; 13, p. 254].
Theorem 4.
Let \(X \) be a locally compact space countable at infinity. Then, for any space \(\mathcal {M} \) and any mapping (3), the function (14) is of Baire class 3 on \( \mathcal {M}\) and the function (15) is of Baire class 2 on \( \mathcal {M}\). If \( \mathcal {M}\) is metrizable by a complete metric, then, for each mapping (3), the set of points of lower semicontinuity of the function (15) is a \( G_\delta \) set everywhere dense in \(\mathcal {M} \).
Proof. Since \(X \) is countable at infinity, it follows that there exists an increasing sequence \(\{U_s\}_{s=1}^\infty \) of relatively compact open sets that form a cover of \(X \) and satisfy \(\overline {U}_s\subset U_{s+1} \) for all \(s\in \mathbb {N} \) [12, p. 316 of the Russian translation]. Every compact set \(K\subset X \) is contained in \(U_{s_0} \) for some \(s_0 \), because otherwise the cover of \(K \) by the sequence \(\{U_s\}_{s=1}^\infty \) would not contain a finite subcover, which contradicts the compactness of \(K\). Thus,
for each continuous mapping \(f:X\to X \). Using formula (7), we obtain
By (8), we have
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 \( \mu \mapsto \overline {h}_{\mathrm {top}}(f(\mu ,\cdot \thinspace )) \) is of Baire class 3 and the function \(\mu \mapsto \underline {h}_{\mathrm {top}}(f(\mu ,\cdot \thinspace )) \) is of Baire class 2 on \(\mathcal {M} \).
Since the function \(\mu \mapsto \underline {h}_{\mathrm {top}}(f(\mu ,\cdot \thinspace ,\thinspace ))\) 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 \(G_\delta \) set provided that \(\mathcal {M}\) is metrizable by a complete metric [10, Lemma 2]. The proof of the theorem is complete.
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Vetokhin, A.N. Some Properties of the Topological Entropy of a Family of Dynamical Systems Defined on an Arbitrary Metric Space. Diff Equat 57, 975–983 (2021). https://doi.org/10.1134/S0012266121080012
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DOI: https://doi.org/10.1134/S0012266121080012