UDC 517.934

ON SECANT SETS OF DYNAMICAL SYSTEMS

V. A. BAIDOSOV

In the paper, for a space with a continuous group of transformations [1], the concept of a secant set is introduced by analogy with the concept of secant surfaces for dynamical systems with a one-parameter group of motions. Some properties of secant sets are considered.

§ 1. BASIC CONCEPTS AND NOTATION

1. By a dynamical system we shall mean here a continuous group of transformations of a topological space [1]. The groups under consideration are, generally speaking, noncommutative. A dynamical system determined by a continuous group of transformations \(W\) of a space \(R\) will be denoted by \((R, W)\). The space of systems will be assumed Hausdorff. Dynamical systems with Hausdorff space and a fixed group of motions form a \(K\)-class of systems [3].

Let a set \(X \subset R\) and a closed subgroup \(G \subset W\) be such that, for every \(x \in X\), the conditions \(w(x) \in X\) and \(w \in G\) are equivalent. Then the group \(G\) determines on \(X\) a dynamical system called a subsystem of the system \((R, W)\). If \(G\) is a normal divisor of \(W\), then the subsystem \((X, W)\) will be called normal.

Let \(V \subset W\) and \(M \subset R\). By \(V(M)\) we shall denote the set of all elements \(w(p)\), where \(w \in V\), \(p \in M\). A set \(X \subset R\) is said to generate \(R\) dynamically if \(W(X)=R\).

Let \(A\) be a topological group and \(B\) its subgroup. Setting \(b(a)=b\cdot a\), we define a system \((A,B)\), called a group system.

The kernel of ineffectiveness of the system \((R,W)\) is the set of all elements \(w\) for which \(w(p)=p\), whatever \(p \in R\). The kernel of ineffectiveness is a normal divisor of \(W\) [1].

A continuous mapping \(\gamma\) of the space of a dynamical system \((R,W)\) into the space of a system \((P,G)\) is called a homomorphism if, for some group homomorphism \(\gamma^{*}: W \to G\), the equality
\[ \gamma(w(p))=w^{*}(\gamma(p)), \]
holds, where \(w^{*}=\gamma^{*}(w)\), \(p \in R\), \(w \in W\). The homomorphism \(\gamma^{*}\) is called conjugate to \(\gamma\). Isomorphisms and automorphisms of dynamical systems are defined analogously.

1. The topology in Hausdorff spaces will be specified with the aid of the operation of convergence of directions [5]. A direction in \(R\) is a mapping into \(R\) of a partially ordered directed set \(\{\alpha\}\), which is written in the form \([p^{\alpha}]\), where \(p^{\alpha}\) is the image of the element \(\alpha\). The convergence of \([p^{\alpha}]\) to \(p\) is symbolically written \(\lim [p^{\alpha}]=p\). The set \(\{\alpha\}\) is called the carrier of the direction.

For binary relations we formulate criteria of continuity and completeness [6] with the aid of the operation \(\lim\). A binary relation \(\theta\) is continuous when

if and only if from \(\lim [p_1^\alpha]=p_1\), \(\lim [p_2^\alpha]=p_2\) and \(\theta(p_1^\alpha,p_2^\alpha)\) it follows that \(\theta(p_1,p_2)\).

Let \(\theta\) be an equivalence relation. The \(\theta\)-completion of a set \(X\subset R\) is the set of all elements belonging to \(\theta\)-classes having a nonempty intersection with \(X\). The equivalence \(\theta\) is called complete if the \(\theta\)-completion of an open set is always open.

For the completeness of an equivalence \(\theta\) it is sufficient that from \(\lim [p^\alpha]=p\) and \(\theta(p_1,p)\) it follow, for some direction \([p_1^\alpha]\) converging to \(p_1\), and for some \(\alpha_0\in\{ \alpha\}\), that \(\theta(p_1^\alpha,p^\alpha)\) for \(\alpha>\alpha_0\). The set of \(\theta\)-classes of the space \(R\) is denoted by \(R/\theta\). Let \(\psi:R\to R/\theta\) be the natural mapping taking each element into the \(\theta\)-class containing it.

If \(\tau_1\) and \(\tau_2\) are two topologies on one and the same set \(M\) and the identity mapping \(M_{\tau_1}\to M_{\tau_2}\) is continuous, then \(\tau_1\) is said to be stronger than \(\tau_2\). The quotient space \(R/\theta\) is the set \(R/\theta\) with the strongest of the topologies for which the natural mapping \(\psi\) is continuous. The open sets in \(R/\theta\) are the sets \(\psi(X)\), where \(X\) has an open \(\theta\)-completion in \(R\). Completeness of the equivalence \(\theta\) is necessary and sufficient for the openness of the mapping \(\psi\).

If \(\theta\) is a complete and continuous equivalence, then \(R/\theta\) belongs to the class \(T_2\). Conversely, if an open and continuous mapping \(\psi:R\to P\) is given and \(P\) belongs to the class \(T_2\), then \(\psi\) defines on \(R\) a complete and continuous equivalence \(\theta\), and \(R/\theta\) will be homeomorphic to \(P\) [6].

1. An equivalence \(\theta\) on the space of a dynamical system \((R,W)\) is called a congruence if from \(\theta(p,p_1)\), for all \(w\in W\), it follows that \(\theta(wp,wp_1)\). A complete and continuous congruence \(\theta\) induces on the quotient space \(R/\theta\) a dynamical system \((R/\theta,W)\). In this case
   \[ w(\psi(p))=\psi(w(p)). \]
   The system \((R/\theta,W)\) will be called the quotient system by the congruence \(\theta\).

2. Let the set \(X\) dynamically generate the space of the system \((R,W)\). We shall say that the topology \(R\) is free with respect to \(X\) in the class \(K\), if it is stronger than any topology \(\tau\) on \(R\), compatible with \(X\), for which \((R_\tau,W)\) is a dynamical system of the class \(K\) [3].

In what follows, by \(K\) we shall mean the class of all systems with a Hausdorff space.

§ 2. SECANT SETS

1. Definition. A partial dynamical system will mean a pair \((R,W)\), where \(R\) is a topological space, \(W\) is a topological group, and for certain elements \(p\in R\), \(w\in W\) the element \(w(p)\in R\) is defined. Moreover, the following conditions are satisfied:

\(D_1\). The element \(e(p)\), where \(e\) is the identity of the group \(W\), is defined for all \(p\in R\) and is equal to \(p\).

\(D_2\). Let the element \(w_2(p)\) be defined. Then, if one of the two elements \((w_1w_2)(p)\) or \(w_1(w_2(p))\) is defined, the other is also defined, and
\[ (w_1w_2)(p)=w_1(w_2(p)). \]

\(D_3\). If the directions \([w^\alpha]\) and \([p^\alpha]\) converge respectively to \(w\) and \(p\), and the elements \(w^\alpha(p^\alpha)\) are defined for all \(\alpha\), then the element \(w(p)\) is defined and
\[ \lim [w^\alpha(p^\alpha)]=w(p). \]

\(D_4\). If the element \(w(p)\) is defined and \(\lim [p^\alpha]=p\), then there exist a direction \([w^\alpha]\) converging to \(w\), and an element of the directed set \(\alpha_0\), such that for \(\alpha>\alpha_0\) the elements \(w^\alpha(p^\alpha)\) will be defined.

Let \((X,W)\) and \((Y,G)\) be partial dynamical systems. A continuous mapping \(\gamma:X\to Y\) will be called a homomorphism of the system \((X,W)\) into \((Y,G)\), if there exists a continuous group homomorphism \(\gamma^*:W\to G\) such that,

that for every defined element \(w(x)\) the element \(\gamma^*(w)(\gamma(x))\) is also defined and is equal to \(\gamma(w(x))\). Obviously, ordinary dynamical systems are partial systems.

Let \(X\) be a subset of the space of a dynamical system \((R,W)\). A partial action of the group \(W\) is defined on \(X\) in a natural way. Consider the pair \((X,W)\). It is easy to see that the conditions \(D_1\), \(D_2\), and \(D_3\) for \((X,W)\) are always satisfied. The condition \(D_4\) may fail to hold. If it holds, we obtain a partial dynamical system \((X,W)\).

2. Definition. A subset \(X\) of the space of a dynamical system \((R,W)\) will be called a secant set of this system if:

\(C_1.\) The operations \(W\) define a partial dynamical system on \(X\).

\(C_2.\) \(X\) dynamically generates \(R\).

\(C_3.\) The topology \(R\) is free with respect to \(X\) [3].

Theorem 1. Let \(X\) be a secant set of a dynamical system \((R,W)\) of class \(K\), and let \((X,W)\) be the partial dynamical system defined on \(X\) by the operations from \(W\). Then for every homomorphism \(\gamma\) of the partial system \((X,W)\) into an arbitrary dynamical system \((P,W)\) of class \(K\), there exists a homomorphism \(\gamma\) of the system \((R,W)\) into \((P,W)\) such that \(\gamma(r)=\gamma(r)\) for \(r\in X\).

Proof. Equalities of the form \(w_1(x_1)=w_2(x_2)\), where \(x_1,x_2\in X\), define a system of relations \(\Sigma\) on \(X\) [3]. Since the topology \(R\) is free with respect to \(X\), the space \(X\) and the relations from \(\Sigma\) define again, in class \(K\), the system \((R,W)\) [3]. Observing that homomorphisms of the partial system \((X,W)\) are maps preserving the relations from \(\Sigma\), we immediately obtain what was required.

Theorem 2. Every partial dynamical system \((X,W)\) can be embedded in some ordinary dynamical system \((R,W)\) so that \(X\) will be a secant set in \(R\). The system \((R,W)\) is determined by this uniquely up to isomorphisms identical on \(X\).

Proof. Consider the free dynamical system \((Y,W)\) defined in class \(K\) by the space \(X\) [3]. According to [3], \(Y=W\times X\), and the action of \(W\) on \(Y\) is given by the condition \(w(w_1,x)=(ww_1,x)\). Define on \(Y\) a binary relation \(\theta\), putting \(\theta((w_1,x_1),(w_2,x_2))\) if in the system \((X,W)\) the equality \(x_1=w_1^{-1}w_2(x_2)\) holds. It is easy to verify that \(\theta\) is a congruence of the system \((Y,W)\). We show that \(\theta\) is a continuous and complete congruence. Let \(\theta((w_1^\alpha,x_1^\alpha),(w_2^\alpha,x_2^\alpha))\), \(\lim[(w_1^\alpha,x_1^\alpha)]=(w_1,x_1)\), and \(\lim[(w_2^\alpha,x_2^\alpha)]=(w_2,x_2)\). Then in \((X,W)\) we have \(x_1^\alpha=(w_1^\alpha)^{-1}w_2^\alpha(x_2^\alpha)\), and, passing to the limit, \(x_1=w_1^{-1}w_2(x_2)\). Hence \(\theta((w_1,x_1),(w_2,x_2))\). It follows that \(\theta\) is continuous.

Next, let \(\theta((w_1,x_1),(w_2,x_2))\) and \(\lim[(w_1^\alpha,x_1^\alpha)]=(w_1,x_1)\). Then \(x_2=w_2^{-1}w_1(x_1)\), and, by property \(D_4\), there exist a direction \([\overline{w}_1^\alpha]\) converging to \(w_2^{-1}w_1\), and an element \(\alpha_0\), such that for \(\alpha>\alpha_0\), \(\overline{w}_1^\alpha(x_1^\alpha)=\overline{x}_1^\alpha\in X\). Obviously, \(\lim[\overline{x}_1^\alpha]=x_2\). Observing that \(\lim[w_1^\alpha(\overline{w}_1^\alpha)^{-1}]=w_2\), we obtain \(\lim[(w_1^\alpha(\overline{w}_1^\alpha)^{-1},\overline{x}_1^\alpha)]=(w_2,x_2)\). It is easy to see that for \(\alpha>\alpha_0\),
\[ \theta\bigl((w_1^\alpha(\overline{w}_1^\alpha)^{-1},\overline{x}_1^\alpha),(w_1^\alpha,x_1^\alpha)\bigr). \]
From the criterion of completeness of equivalence in § 1 follows the completeness of \(\theta\).

Consider the quotient space \(R=Y/\theta\). Let \(\psi:Y\to R\) be the natural mapping and \(X^* = e\times X\). Note that,

that the mapping \(\psi:X^*\to\psi(X^*)\) is one-to-one. Indeed, from
\(\psi((e,x_1))=\psi((e,x_2))\) it follows that \(x_1=x_2\). Let us show that \(\psi:X^*\to\psi(X^*)\) is a homeomorphism. Consider in \(X^*\) an open set \(U^*\). It has the form \((e,U)\), where \(U\) is open in \(X\). It is easy to show that from condition \(D_3\) for a partial system the following property follows. If \(x\in U\), then there exist a neighborhood \(U_1\) of the point \(x\) and a neighborhood \(V\) of the identity of the group \(W\) such that, if \(x_1\in U_1\), \(w\in V\), and the element \(w(x_1)\) is defined, then \(w(x_1)\in U\). For each point \(x\in U\) choose neighborhoods \(U_x\) and \(V_x\) with the indicated properties. The set
\[ S=\bigcup_{x\in U}(V_x,U_x) \]
will be open in \(Y\). Moreover, \(S\cap X^*=U^*\). We shall show that
\[ \psi(U^*)=\psi(S)\cap\psi(X^*). \]
It is obvious that \(\psi(U^*)\subset\psi(S)\cap\psi(X^*)\). Let us prove the reverse inclusion.

Let \(p\in\psi(S)\cap\psi(X^*)\). Then
\[ p=\psi((w,x))=\psi((e,x_1)), \]
where \((w,x)\in S\), \((e,x_1)\in X^*\). In the system \((X,W)\) we obtain \(x_1=w(x)\). By the definition of \(S\), \(x_1\in U\). Hence \((e,x_1)\in U^*\) and \(p\in\psi(U^*)\). The equality is proved.

Since \(\psi(S)\) is open in \(R\), \(\psi(U^*)\) is open in the relative topology \(\psi(X^*)\). Hence we obtain that the mapping \(\psi:X^*\to\psi(X^*)\) is a homeomorphism.

Now consider the factor-system \((R,W)\). It is easy to see that the transformations from \(W\) define on \(\psi(X^*)\) a partial dynamical system isomorphic to \((X,W)\). Further, \(\psi(X^*)\) dynamically generates \(R\). Note that the topology of \(R\) is the strongest among the topologies for which the canonical mapping \(\psi\) is continuous. On the other hand, if a topology \(\tau\) on \(R\) is compatible with \(\psi(X^*)\) and \((R_\tau,W)\) is a dynamical system, then it is easy to see that the mapping \(\psi:Y\to R_\tau\) will be continuous. Consequently, the topology of the factor space is stronger than \(\tau\). Thus the topology \(R\) is free relative to \(\psi(X^*)\). The uniqueness of \((R,W)\) follows from the fact that this system is determined in the class \(K\) by the space \(X\) and the system of relations uniquely determined by the partial system \((X,W)\). The theorem is proved.

§ 3. SECANT SUBSYSTEMS

1. Definition. A subsystem \((X,G)\) of a dynamical system \((R,W)\) will be called a secant subsystem if \(X\) is a secant set in \((R,W)\) and \(G\) is a subgroup in \(W\).

Consider some dynamical system \((X,G)\) and a topological group \(W\) containing \(G\) as a closed subgroup. Define on \(X\) a partial action of the group \(W\), putting \(w(x)\) to be defined only in the case \(w\in G\). In this case \(w(x)\) is determined from \((X,G)\). We obtain a partial dynamical system \((X,W)\), which is verified without difficulty. Taking Theorem 2 into account, we have

Theorem 3. For every dynamical system \((X,G)\) and topological group \(W\) containing \(G\) as a closed subgroup, there exists a dynamical system \((R,W)\) having \((X,G)\) as a secant subsystem. The system \((R,W)\) is determined uniquely up to automorphisms identical on \(X\).

1. The existence of secant subsystems is closely connected with homomorphisms onto group systems. We first prove the following theorem.

Theorem 4. Let \(\gamma\) be a homomorphism of the system \((R,W)\) onto the group system \((H,H)\). If the conjugate homomorphism \(\gamma^*:W\to H\) is open, then \(\gamma\) is also open.

Proof. Consider an open set \(U\) in \(R\). We shall show that \(\gamma(U)\) is open in \(H\). Let \(h\in\gamma(U)\) and \(h=\gamma(p)\), where \(p\in U\). Choose a neighborhood \(V\) of the identity of the group \(W\) such that the condition is satisfied

\(V(p)\subset U\). We obtain \(\gamma(V(p))\subset \gamma(U)\), or \(\gamma^*(V)\cdot \gamma(p)\subset \gamma(U)\), i.e., \(\gamma^*(V)\cdot h\subset \gamma(U)\). But \(\gamma^*(V)\cdot h\) is a neighborhood of the element \(h\). Hence it follows that \(\gamma(U)\) is an open set. This is what was required to prove.

Theorem 5. A dynamical system \((R,W)\) having a normal secant subsystem with motion group \(G\), where \(G\) is a normal divisor of \(W\), possesses an open homomorphism onto the group system \((W/G,W/G)\).

Proof. Let \((X,G)\) be a normal secant subsystem of the system \((R,W)\). Define on \(W/G\) the dynamical system \((W/G,W)\) by putting
\(\omega(\psi(\omega_1))=\psi(\omega\omega_1)\) [1], where \(\psi\) is the natural mapping of \(W\) onto \(W/G\). Consider the mapping \(\bar{\gamma}:X\to W/G\) taking \(X\) to the identity element \(e^*\). \(\bar{\gamma}\) is a homomorphism of the partial system \((X,W)\) into \((W/G,W)\). Indeed,
\[ \bar{\gamma}(g(x))=e^*=\psi(g\cdot g^{-1})=g(\psi(g^{-1}))=g(\psi(x))= \]
\[ =g(\bar{\gamma}(x)). \]
Thus there exists a homomorphism \(\gamma\) of the system \((R,W)\) into \((W/G,W)\) which coincides on \(X\) with \(\bar{\gamma}\). It is easy to see that \(\gamma(R)=W/G\). Finally, from the equality \(\gamma(\omega(p))=\psi(\omega)\cdot\gamma(p)\) it follows that \(\gamma\) is a homomorphism of the system \((R,W)\) onto \((W/G,W/G)\), having \(\psi\) as its conjugate homomorphism. The openness of \(\gamma\) follows from Theorem 4. This is what was required to prove.

Let now the system \((R,W)\) possess a homomorphism \(\gamma\) onto the system \((W/G,W/G)\). Consider the set \(X=\gamma^{-1}(e^*)\), where \(e^*\) is the identity in \(W/G\). The pair \((X,G)\) is a subsystem in \((R,W)\). Indeed, from the equality \(\gamma(w(x))=\gamma^*(w)\), where \(\gamma^*\) is the conjugate homomorphism, it follows that \(w(x)\subset X\) is equivalent to \(w\in G\). In addition, \(X\) dynamically generates \(R\). Let \(p\in R\) and \(\gamma(p)=w^*=\gamma^*(w)\). Then \(\gamma(w^{-1}(p))=(w^*)^{-1}w^*=e^*\), and \(w^{-1}(p)\in X\).

In the case where \(W\) has the first axiom of countability, we prove the theorem converse to the preceding one.

Theorem 6. Let a system \((R,W)\) possess a homomorphism \(\gamma\) onto the group system \((W/G,W/G)\) with open conjugate homomorphism \(\gamma^*\). Suppose also that \(W\) satisfies the first axiom of countability. Then \((R,W)\) has a normal secant subsystem of the form \((X,G)\).

Proof. It is enough to show that the topology of \(R\) is free relative to \(X=\gamma^{-1}(e^*)\). Consider in \(R\) a direction \([p^\alpha]\) converging to \(p\). Let \(\gamma(p^\alpha)=\overline{w}^{\,\alpha}\), \(\gamma(p)=\overline{w}\), where \(\overline{w}^{\,\alpha},\overline{w}\in W/G\). Let \(\{V_n\}\) be a base at the identity of the group \(W\). From the convergence of the direction \([\overline{w}^{\,\alpha}]\) to \(\overline{w}\) it follows that for some \(\alpha_0\), when \(\alpha>\alpha_0\), the complete preimages
\(K^\alpha=(\gamma^*)^{-1}(\overline{w}^{-1}\overline{w}^{\,\alpha})\)
will have nonempty intersections with certain neighborhoods \(V_n\). Let \(\alpha>\alpha_0\). By \(w^\alpha\) denote an element of \(K^\alpha\) belonging to the neighborhood with the largest number. The direction \([w^\alpha]\) will converge to the identity. Indeed, for any neighborhood \(V_n\) there is an \(\alpha_n\) such that, for \(\alpha>\alpha_n\), all intersections \(K^\alpha\cap V_n\) are nonempty. Hence, for \(\alpha>\alpha_n\), \(w^\alpha\in V_n\). Let \(\overline{w}=\gamma^*(w)\) and \(w_1^\alpha=w\cdot w^\alpha\). Then
\((w_1^\alpha)^{-1}(p^\alpha)=x^\alpha\in X\), where \(X=\gamma^{-1}(e^*)\). Indeed,
\[ \gamma\bigl((w_1^\alpha)^{-1}(p^\alpha)\bigr)=(\overline{w}^{\,\alpha})^{-1}\overline{w}\,\overline{w}^{-1}\overline{w}^{\,\alpha}=e^*. \]
Further,
\(\lim [x^\alpha]=w^{-1}(p)=x\in X\). Hence,
\(p^\alpha=w_1^\alpha(x^\alpha)\), \(p=w(x)\), where \(\lim [w_1^\alpha]=w\), \(\lim [x^\alpha]=x\). Hence it follows that the direction \([p^\alpha]\) will converge to \(p\) in any topology \(R\) compatible with \(X\) for which \((R,W)\) remains a dynamical system. Thus the topology of \(R\) is free relative to \(X\). This is what was required to prove.

We note that a special case of a normal secant subsystem is given by subsystems with the identity motion group. If a system \((R,W)\) has a secant subsystem of this type, then it is a free dynamical system [3].

1. The question of embedding a discrete dynamical system, i.e., a system with an infinite cyclic group of motions, into a continuous one-parameter system was studied by N. P. Zhidkov [4]. He constructed a dynamical system defined by differential equations and having the given discrete system as a global section, i.e., as a secant subsystem.

The connection between secant subsystems and homomorphisms onto group systems was first found and studied by E. A. Barbashin for one-parameter dynamical systems [2]. Let \(W\) be the group of real numbers, \(Z\) the group of integers, and \(K = W/Z\). The existence of homomorphisms onto the group systems \((W, W)\) and \((K, K)\) determines, respectively, the rectifiability and harmonicity of a dynamical system.

References

1. Pontryagin L. S. Continuous Groups. 2nd ed., Moscow, 1954.

2. Barbashin E. A. On homomorphisms of dynamical systems. II. Mat. Sb., 29 (71), 1951, pp. 501–518.

3. Baidosov V. A. Differential equations, 2, No. 12, 1547–1552, 1966.

4. Zhidkov N. P. Some properties of discrete dynamical systems. Uch. zap. un-ta, 163, Mathematics, issue 6, 1952, pp. 31–59.

5. Maltsev A. I. Free topological algebras. Izv. Akad. Nauk SSSR, Ser. Math., 21, 171–198, 1957.

6. Maltsev A. I. On the general theory of algebraic systems. Mat. Sb., 35 (77), 1954, pp. 3–20.

Received by the editors
December 24, 1965

Sverdlovsk Branch of the Mathematical
Institute named after V. A. Steklov