MATHEMATICS

I. U. BRONSTEIN

ON DYNAMICAL SYSTEMS WITHOUT UNIQUENESS AS SEMIGROUPS OF MULTIVALUED MAPPINGS OF A TOPOLOGICAL SPACE

(Presented by Academician P. S. Aleksandrov, 25 I 1962)

Recently, in connection with the study of systems of differential equations with multivalued and discontinuous right-hand sides \((^{1,2})\), interest has again arisen in generalized dynamical systems \((^{3})\). On the other hand, in connection with the creation by Soviet and American mathematicians \((^{4,5})\) of the theory of general dynamical systems \((R,G)\), where \(G\) is an arbitrary topological group, systems of multivalued mappings of a space have been considered \((^{6})\) (also under very general assumptions). In addition, recently the theory of multivalued mappings of a topological space has begun to develop \((^{7,8})\).

In this note a concept is proposed of a dynamical system without uniqueness as a semigroup of multivalued mappings of a space.

1. Let \(R\) be a topological space, \(S\) a topological semigroup (i.e. a topological space with a binary associative multiplicative operation, continuous in the aggregate of its components) with identity \(e\); let \(f\) be a mapping assigning to each point \(p \in R\) and each element \(s \in S\) a nonempty bicompact set \(f(p,s) \subset R\).

The collection \((R,S,f)\) will be called a semigroup of multivalued mappings (s.m.m.) if the following three conditions are satisfied:

\(A_1.\) \(f(p,e)=p\) for every point \(p \in R\).

\(A_2.\) \(f[f(p,s_1),s_2]=f(p,s_1s_2)\) for any two elements \(s_1\) and \(s_2\) of \(S\) and any point \(p \in R\).

\(A_3.\) If \(f(p,s)=B\) \((p \in R,\ s \in S)\), then for any neighborhood \(U(B)\) of the set \(B\) in \(R\) there exist neighborhoods \(V(p)\) of the point \(p\) in \(R\) and \(W(s)\) of the element \(s\) in \(S\) such that \(f[V(p),W(s)] \subset U(B)\).

If \(f(p,s)\) is a point for all \(p \in R,\ s \in S\), then \((R,S,f)\) will be called a semigroup of single-valued mappings (s.s.m.). The set

\[ f(p,S)=\bigcup_{s \in S} f(p,s) \]

is called the funnel of the point \(p\). Let \(\{U_1,\ldots,U_n\}\) be a finite system of open sets in \(R\). Following \((^{8})\), by

\[ \langle U_1,\ldots,U_n\rangle \]

we shall denote the collection of all closed subsets \(A\) of \(R\) such that

\[ A \subseteq \bigcup_{i=1}^{n} U_i \quad\text{and}\quad A \cap U_i \ne \Lambda \quad (i=1,\ldots,n). \]

The collection of all \(\langle U_1,\ldots,U_n\rangle\) for arbitrary open sets \(U_1,\ldots,U_n\) forms a basis of the so-called finite \((^{8})\) topology in the set of all closed subsets of the space \(R\). We note that if the space \(R\) is metric, then the finite topology in the set of compact subsets of the space \(R\) is induced by the Hausdorff distance between two sets.

Theorem 1. If \(R\) is a \(T_1\)-separated space, and \(S=D^+\) \((D^-)\), where \(D^+\) \((D^-)\) is the additive semigroup of nonnegative (nonpositive) real numbers with the usual topology, then for any \(p \in R,\ s \in S\) and any open sets \(U_1,\ldots,U_n\) in \(R\) satisfying the condition

\(f(p,s)\in\langle U_1,\ldots,U_n\rangle\), there is a neighborhood \(W(s)\subset S\) such that \(f(p,s')\in\langle U_1,\ldots,U_n\rangle\) for all \(s'\in W(s)\).

Let us note that if \(S\ne D^+(D^-)\), then the assertion of Theorem 1 is, generally speaking, false.

If for the point \(p\in R\) and arbitrary \(s\in S\) the set \(f(p,s)\) is connected, then the funnel \(f(p,S)\) is connected. If \(S=D^+(D^-)\), then, by Theorem 1 and the work \(\left({}^7\right)\), the funnel of any point \(p\in R\) is connected.

1. We indicate the connection between generalized dynamical systems (g.d.s.) \(\left({}^3,{}^9\right)\) and p.n.m. \((R,D^+,f)\), where \(R\) is a metric space. A g.d.s. \((R,D,f)\) determines two p.n.m. \((R,D^+,f)\) and \((R,D^-,f)\). Hence, by Theorem 1, it follows that axiom \(5^0\) of the work \(\left({}^9\right)\) is a consequence of the other axioms of that work.

Theorem 2. For \((R,D^+,f)\), where \(R\) is a compact metric space and \(f(R,t)=R\) for all \(t\geqslant0\), there exists a g.d.s. \((R,D,f^*)\) such that \(f^*(p,t)=f(p,t)\) for \(t\geqslant0,\ p\in R\).

If \(R\) is not compact, then the assertion of Theorem 2, generally speaking, loses its force.

Let us also note that if in axiom B of \(\left({}^6\right)\) the inclusion condition
\(f(p,g_1g_2)\subseteq f[f(p,g_1),g_2]\) is replaced by the condition of equality for any elements \(g_1\) and \(g_2\) of the group \(G\), then from this the single-valuedness of the mapping \(f(p,g)\) for all \(p\in R,\ g\in G\) will follow. Therefore it is natural to consider precisely semigroups of multivalued mappings.

1. Consider the set \(\mathscr C=\mathscr C(R,S,f)\), whose elements are the sets \(B=f(p,s)\) \((p\in R,\ s\in S)\), and endow \(\mathscr C\) with the topology of a subspace of the space \(\mathfrak X R\) \(\left({}^7\right)\): neighborhoods of a point \(B\in\mathscr C\) will be the collections \(U'\) of all \(B'\subseteq U(B)\), where \(U\) is an arbitrary neighborhood of the set \(B\) in \(R\), \(B'\in\mathscr C\). We denote the resulting space by \(\mathscr C_1\). A p.n.m. \((R,S,f)\) naturally induces on \(\mathscr C_1\) a p.o.m. \((\mathscr C_1,S,F_1)\).

Let us now endow the set \(\mathscr C\) with the finite \(\left({}^8\right)\) topology and denote the resulting space by \(\mathscr C_2\). If \(f(p,s)=B\) is a continuous mapping of \(R\times S\) into \(\mathscr C_2\), then the p.n.m. \((R,S,f)\) induces, analogously to the preceding case, a p.o.m. \((\mathscr C_2,S,F_2)\).

We shall call p.n.m. \((R,S,f)\) and \((R',S',f')\) isomorphic if there exist a homeomorphic mapping \(\varphi\) of the space \(R\) onto \(R'\) and an isomorphic mapping \(\varphi^*\) of the topological semigroup \(S\) onto \(S'\) such that
\(\varphi[f(p,s)]=f'[\varphi(p),\varphi^*(s)]\) for all \(p\in R,\ s\in S\). The consideration of p.o.m. \((\mathscr C_1,S,F_1)\) and \((\mathscr C_2,S,F_2)\) corresponding to the p.n.m. \((R,S,f)\) is useful in many cases; however, there exist nonisomorphic p.n.m. \((R,S,f)\) and \((R',S',f')\) such that the p.o.m. \((\mathscr C_1,S,F_1)\) and \((\mathscr C'_1,S',F'_1)\) are isomorphic, and the p.o.m. \((\mathscr C_2,S,F_2)\) and \((\mathscr C'_2,S',F'_2)\) are isomorphic.

1. In what follows the topological space \(R\) is assumed to be Hausdorff. A set \(A\subseteq R\) will be called semi-invariant (quasi-invariant) if \(f(A,s)\subseteq A\) \((f(A,s)\supseteq A)\) for all \(s\in S\). If \(f(A,s)=A\) \((s\in S)\), then the set \(A\) is called invariant. A set \(A\) will be called pseudo-invariant if for every point \(p\in A\) and every \(s\in S\) one has \(f(p,s)\cap A\ne\Lambda\). A set \(M\) is called minimal with respect to a certain property \(P\) if it is closed, nonempty, satisfies property \(P\), and there is no proper closed subset \(M'\subset M\) satisfying condition \(P\).

Theorem 3. Every semi-invariant (quasi-invariant, invariant, pseudo-invariant) bicompact set contains a minimal semi-invariant (quasi-invariant, invariant, pseudo-invariant) bicompact set.

If the semigroup \(S\) is commutative, then a bicompact set is minimal semi-invariant if and only if it is minimal invariant.

Let us note that in a bicompact space the closure of a quasi-invariant set is quasi-invariant; if, however, the space is not bicompact, then this assertion is, generally speaking, false.

In what follows it is assumed that \(S=D^{+}\). Introduce the concept of a motion \(\varphi(p,t)\) \((t\ge 0)\) analogously to how this was done in \((^9)\), and denote

\[ \varphi(p,D^{+})=\bigcup_{t\ge 0}\varphi(p,t). \]

It is easy to prove the existence of a motion \(\varphi(p,t)\) passing through an arbitrary point \(q\in\varphi(p,D^{+})\).

Theorem 4. In order that a set \(A\) be pseudo-invariant, it is necessary and sufficient that for every point \(q\in A\) there exist a motion \(\varphi(q,t)\) such that \(\varphi(q,D^{+})\subseteq A\).

In order that a bicompact set \(A\) be a minimal pseudo-invariant set, it is necessary and sufficient that for every point \(q\in A\) and every motion \(\varphi(q,t)\) such that \(\varphi(q,D^{+})\subseteq A\), one have

\[ \overline{\varphi(q,D^{+})}=A. \]

The notions of a bicompact minimal pseudo-invariant set and a bicompact minimal quasi-invariant set coincide. However, there exist pseudo-invariant sets that are neither semi-invariant nor quasi-invariant.

1. Let \(p\in R\) and let \(E\) be an open or closed subset of \(R\). Introduce the functions \((t\ge 0)\):

\[ v_{1}(p,t,E)= \begin{cases} 1, & \text{if } f(p,t)\subseteq E,\\ 0, & \text{if } f(p,t)\cap(R\setminus E)\ne\Lambda; \end{cases} \]

\[ v_{2}(p,t,E)= \begin{cases} 1, & \text{if } f(p,t)\cap E\ne\Lambda,\\ 0, & \text{if } f(p,t)\cap E=\Lambda. \end{cases} \]

We note that \(v_{2}(p,t,E)=1-v_{1}(p,t,R\setminus E)\). From Theorem 1 and \((^8)\) it follows that the functions under consideration are measurable. Introduce the notation:

\[ P_i(p,E)=\lim_{T\to\infty}\frac{1}{T}\int_{0}^{T} v_i(p,t,E)\,dt \quad (i=1,2); \]

if the indicated limits exist.

A bicompact set \(Z_{1}(p)\) \(\bigl(Z_{2}(p)\bigr)\) will be called the center of strong (weak) attraction of the funnel of the point \(p\), if for every neighborhood \(U(Z_{1})\) \(\bigl(U(Z_{2})\bigr)\) one has

\[ P_{1}[p,U(Z_{1})]=1 \quad (P_{2}[p,U(Z_{2})]=1). \]

Theorem 5. If \(\overline{f(p,D^{+})}\) is bicompact, then there exists a unique minimal center of strong attraction \(Z_{1}(p)\), and the set \(Z_{1}(p)\) is quasi-invariant. Under the same condition there exists at least one minimal center of weak attraction \(Z_{2}(p)\). All \(Z_{2}(p)\) belong to \(Z_{1}(p)\).

Let \(\{\varphi_{\alpha}(p,t)\}\) be the totality of all motions issuing from the point \(p\). Introduce, as usual \((^{10})\), the concept of the probability \(P[\varphi_{\alpha}(p,t)]\) that the motion \(\varphi_{\alpha}(p,t)\) is in the set \(E\) as \(t\to+\infty\), and the concept of the center of attraction of a motion.

Theorem 6. If \(\overline{\varphi_{\alpha}(p,D^{+})}\) is bicompact, then there exists a unique minimal center of attraction \(Z[\varphi_{\alpha}(p,t)]\), and the set \(Z[\varphi_{\alpha}(p,t)]\) is quasi-invariant.

Let us note that \(Z[\varphi_{\alpha}(p,t)]\) is a center of weak attraction of the funnel of the point \(p\), but is not necessarily minimal. There exist \(Z_{2}(p)\) that are not centers of attraction for any motion \(\varphi_{\alpha}(p,t)\). All \(Z[\varphi_{\alpha}(p,t)]\) belong to \(Z_{1}(p)\).

The closure of the union of all \(Z[\varphi_\alpha(p,t)]\), generally speaking, is a proper subset of the set \(Z_1(p)\).

Institute of Physics and Mathematics
Academy of Sciences of the MSSR

Received
23 I 1962

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