UDC 517.933

DEFINING RELATIONS IN DYNAMICAL SYSTEMS

V. A. BAIDOSOV

By a dynamical system in this paper is meant a continuous group of transformations of a topological space [1]. The elements of the group are regarded as operations on the space of the system. By analogy with the theory of topological algebras, the concept is introduced of a dynamical system defined by a given topological space and by defining relations on it. In the article a number of results from A. I. Mal'cev’s paper [2] on free topological algebras are carried over to the case of dynamical systems.

§ 1. BASIC CONCEPTS AND NOTATION

A continuous group of transformations \(W\) of a topological space \(R\) will be called a dynamical system and denoted by \((R,W)\). The result of the action of an element \(w \in W\) on a point \(p \in R\) will be written in the form \(w(p)\). The mapping \(\xi: W \times R \to R\), which takes the element \((w,p)\) to \(w(p)\), is continuous. Let \((R,W)\) and \((P,G)\) be dynamical systems, \(\gamma: R \to P\) a continuous mapping, and \(\gamma^*: W \to G\) a group homomorphism satisfying the condition \(\gamma(w(r)) = w^*(\gamma(r))\), where \(w^*=\gamma^*(w)\), for all \(r \in R\) and \(w \in W\). The mapping \(\gamma\) is then called a homomorphism of the system \((R,W)\) into \((P,G)\), and \(\gamma^*\) is called the homomorphism conjugate to \(\gamma\). For simplicity we shall agree to write the image of \(w\) under the conjugate homomorphism in the form \(w^*\). In the case when \(\gamma\) is a homeomorphism and \(\gamma^*\) is an isomorphism, the systems are called isomorphic (or similar).

If \(\gamma(R)=P\), \(\gamma^*(W)=G\), then one speaks of a homomorphism onto the system \((P,G)\). Let \(M\) be a subset of the space of the dynamical system \((R,W)\) and \(V \subset W\). The set of elements \(w(r)\), where \(w \in V\), \(r \in M\), will be denoted by \(V(M)\). The set \(W(M)\) is called the trajectory, or orbit, of \(M\). If \(W(M)=R\), then \(M\) is said to dynamically generate \(R\). The set \(M\) is called invariant if \(W(M)=M\). The action of the group \(W\) then defines on \(M\) a dynamical system \((M,W)\), called a subsystem of the system \((R,W)\).

Let \(A\) be a topological group and \(B\) a subgroup of the abstract group \(A\). Define the action of the topological group \(B\) on \(A\) by putting \(b(a)=b \cdot a\), where \(b \in B\), \(a \in A\). We obtain a dynamical system, called by E. A. Barbashin a group system.

Let \(\{(R_\alpha,W_\alpha)\}\) be a set of dynamical systems, indexed by the elements of the set \(I=\{\alpha\}\). We regard all the groups \(W_\alpha\) as isomorphic to one another. Let \(H=\prod_{\alpha \in I} R_\alpha\). The elements \(h \in H\) are functions of \(\alpha\) on \(I\) with values in \(\bigcup_{\alpha \in I} R_\alpha\), such that \(h(\alpha) \in R_\alpha\). The mapping \(\pi_\alpha: H \to R_\alpha\), de-

defined by the condition \(\pi_a(h)=h(a)\), is called the projection of \(H\) onto \(R_a\). Let \(W\) be a group topologically isomorphic to the groups \(W_a\), and under the corresponding isomorphisms let an element \(w\in W\) go over into \(w_a\in W_a\).

Define the action of \(W\) on \(H\) by putting

\[ (wh)(a)=w_a(h(a)). \]

Then a dynamical system \((H,W)\) is defined on \(H\)—the direct product of the systems \((R_a,W_a)\). We have \(\pi_a(wh)=w_a(\pi_a(h))\). Hence the projections are homomorphisms of the system \((H,W)\) onto \((R_a,W_a)\).

§ 2. DEFINING RELATIONS

We shall regard a dynamical system \((R,W)\) as a topological algebra on the space \(R\) with the set of operations \(W\). From this point of view, dynamical systems with isomorphic groups of motions form one primitive class [2]. The monomials in this algebra contain only one variable and have the form \(w(x)\), where \(w\in W\). However, the indicated analogy with algebras is not complete, since the set of operations in the case of a dynamical system will be a topological space. We shall define the notion of a \(K\)-class of dynamical systems by analogy with the corresponding notion for algebras [2]. \(K\)-classes of systems are taken within the class of systems with isomorphic groups of motions. Where this does not lead to confusion, we shall denote the groups of motions by the same letter.

Definition. A class of dynamical systems with isomorphic groups of motions will be called a \(K\)-class if

\(K_1)\) the direct product of \(K\)-systems is a \(K\)-system;

\(K_2)\) a subsystem of a \(K\)-system is a \(K\)-system.

In particular, a \(K\)-class is the class of systems with a Hausdorff space and a fixed group of motions.

In what follows we shall assume that the \(K\)-class contains the dynamical system consisting of a single fixed point.

Definition. By a system of relations defined on a space \(X\) by a group \(W\) we shall mean a collection of formal expressions of the form \(w_1(x_1)=w_2(x_2)\), where \(w_1,w_2\in W\), \(x_1,x_2\in X\).

Let \(\Sigma\) be a system of relations defined on a topological space \(X\) by a group \(W\). We shall say that a continuous mapping \(f\) of the space \(X\) into the space of a dynamical system \((P,W^*)\) with a group of motions \(W^*\) isomorphic to \(W\) preserves the relations from \(\Sigma\) if there exists an isomorphism \(f^*:W\to W^*\) such that for every relation \(w_1(x_1)=w_2(x_2)\) from \(\Sigma\), in \((P,W^*)\) one has

\[ f^*(w_1)[f(x_1)] = f^*(w_2)[f(x_2)]. \]

Definition. A dynamical system \((R,W)\) with a given continuous mapping \(\sigma:X\to R\) is called definable in the class of \(K\)-systems with group of motions \(W\) by the space \(X\) and the defining relations from \(\Sigma\), if

\(F_1)\) the mapping \(\sigma\) preserves the relations from \(\Sigma\);

\(F_2)\) \(R\) is dynamically generated by the set \(\sigma(X)\);

\(F_3)\) for every continuous mapping \(\bar{\gamma}\) of the space \(X\) into the space \(P\) of an arbitrary dynamical system of the class \(K\) preserving the relations from \(\Sigma\), there exists a continuous homomorphism \(\gamma\) of the system \((R,W)\) into \((P,W)\) satisfying the condition \(\gamma(\sigma(x))=\bar{\gamma}(x)\) for \(x\in X\).

We shall say that the homomorphism \(\gamma\) is generated by the mapping \(\bar\gamma\). The theorem on specifying a system by defining relations, proved by A. I. Mal'cev in [2] for topological algebras, carries over to dynamical systems. A. I. Mal'cev’s proof carries over almost without change to the case of dynamical systems.

Theorem 1. For any topological space \(X\), any \(K\)-class of dynamical systems with group of motions \(W\), and any system of defining relations \(\Sigma\) on \(X\), a dynamical system with properties \(F_1, F_2\), and \(F_3\) exists and is determined by these properties up to isomorphisms that carry the space \(\sigma(X)\) onto itself.

Without giving the proof, we shall only describe the construction used in it. The set of dynamical systems \((R_\alpha,W_\alpha)\) of class \(K\) for which there exist continuous mappings \(\sigma_\alpha:X\to R_\alpha\) satisfying the conditions \(F_1\) and \(F_2\) is nonempty, since it contains a system whose space consists of a single point. It follows from \(F_2\) that the cardinality of \(R_\alpha\) does not exceed the cardinality of the set \(W\times X\). We shall say that the systems \((R_\alpha,W_\alpha)\) and \((R_\beta,W_\beta)\) are isomorphic over \(X\) if there exists a topological isomorphism \(\varphi:R_\alpha\to R_\beta\) such that \(\varphi\sigma_\alpha=\sigma_\beta\).

Consider the direct product \((H,W)\) of all systems nonisomorphic over \(X\) that satisfy the conditions \(F_1\) and \(F_2\). Define a mapping \(\sigma:X\to H\) so that \(\pi_\alpha(\sigma(x))=\sigma_\alpha(x)\). The subsystem \((R,W)\) in \((H,W)\) dynamically generated by the set \(\sigma(X)\) will be the required system [2].

Consider a system \((R,W)\), defined in the class \(K\) by the space \(X\), the system of relations \(\Sigma\), and the mapping \(\sigma:X\to R\). By \(\Sigma^*\) denote the system of all relations \(w_1(x_1)=w_2(x_2)\) for which, in \((R,W)\), the equality \(w_1(\sigma(x_1))=w_2(\sigma(x_2))\) holds. It is clear that \(\Sigma\) is contained in \(\Sigma^*\). We shall call the system \(\Sigma^*\) the extension of the system \(\Sigma\). The space \(X\) and the extended system of relations \(\Sigma^*\) define in \(K\) the same dynamical system \((R,W)\). The fulfillment of conditions \(F_1\) and \(F_2\) is obvious. Note that every mapping \(\bar\gamma\) of the space \(X\) into the space of the system \((P,W)\) that preserves the relations from \(\Sigma\) also preserves the relations from the extended system. Indeed, for any relation \(w_1(x_1)=w_2(x_2)\) in \((R,W)\), the equality \(w_1(\sigma(x_1))=w_2(\sigma(x_2))\) holds. Applying the homomorphism \(\gamma\) generated by the mapping \(\bar\gamma\), we obtain \(w_1^*\bar\gamma(x_1)=w_2^*\bar\gamma(x_2)\). Hence condition \(F_3\) follows.

Let \((R,W)\) be a dynamical system and let \(X\) be a subset of \(R\) that dynamically generates \(R\). To each equality \(w_1(x_1)=w_2(x_2)\) in \(R\) we assign the relation \(w_1(x_1)=w_2(x_2)\) on \(X\). Denote by \(\Sigma\) the system of relations thus obtained.

Let \(K\) be some class containing \((R,W)\).

Consider the dynamical system \((P,W)\), defined in \(K\) by the space \(X\), the relations from \(\Sigma\), and some mapping \(\sigma:X\to P\). The identity mapping \(\bar\gamma:X\to R\) generates a homomorphism \(\gamma\) of the system \((P,W)\) into \((R,W)\). We have
\[ \gamma(w(\sigma(x)))=w^*(x). \]
Putting \(w=e\), we obtain \(\gamma(\sigma(x))=x\). Hence it follows that the mapping \(\sigma:X\to\sigma(X)\) is a homeomorphism.

Let us show that the homomorphism \(\gamma\) is one-to-one. Let
\[ \gamma(w_1(\sigma(x_1)))=\gamma(w_2(\sigma(x_2))). \]
Then the relation \(w_1(x_1)=w_2(x_2)\) belongs to \(\Sigma\). Therefore
\[ w_1(\sigma(x_1))=w_2(\sigma(x_2)). \]

The system \((P,W)\) topologically contains \(X\) and differs from \((R,W)\) only in the topology of the space. By analogy with the corresponding notion for topological algebras [2], the topology \(P\) will be called free in

class \(K\) relative to \(X\). The characteristic property of the free topology is as follows. If a topology \(\tau\) on \(R\) induces on \(X\) the given topology, and the system \((R_\tau,W)\) is a dynamical system of class \(K\), then the identity mapping \(P\) onto \(R_\tau\) will be continuous. In particular, if the system \((R,W)\) is defined in the class \(K\) by the space \(X\), relations from \(\Sigma\), and the mapping \(\sigma:X\to R\), then the topology of \(R\) will be free relative to \(\sigma(x)\). Each relation \(w_1(x_1)=w_2(x_2)\) from \(\Sigma\) defines the relation \(w_1(\sigma(x_1))=w_2(\sigma(x_2))\) on \(\sigma(X)\). The space \(\sigma(X)\), together with the relations obtained on it, defines in \(K\) the very same system \((R,W)\).

Let the space of the system \((P,W)\) be dynamically generated by the set \(Z\). We shall call the topology \(P\) initial relative to \(Z\) if the sets of the form \(V(U)\), where \(V\) is a region from \(W\), and \(U\) is a relative region in \(Z\), form an open base of the space \(R\).

Consider the continuous mapping \(\psi:W\times Z\to P\), defined by the condition \(\psi(w,z)=w(z)\). It is easy to see that \(\psi\) will be open if and only if the topology \(P\) is initial relative to \(Z\). Let \(\tau\) be another topology on \(P\), for which \((P_\tau,W)\) is a system of class \(K\). If \(\tau\) induces on \(Z\) the former topology, then the mapping \(\varphi:W\times Z\to P_\tau\), defined as \(\psi\), will be continuous. Hence we obtain that the identity mapping \(P\) onto \(P_\tau\) is also continuous.

Consequently, the topology initial relative to \(Z\) will be free relative to \(Z\).

Let \((R,W)\) be a system defined in \(K\) by the space \(X\), relations from \(\Sigma\), and the mapping \(\sigma:X\to R\).

Theorem 2. If a continuous mapping \(\bar{\gamma}\) of the space \(X\) into the space of the system \((P,W)\), preserving the relations from \(\Sigma\), is an open mapping of \(X\) onto \(\bar{\gamma}(X)\), and the topology of \(P\) is initial relative to \(\bar{\gamma}(X)\), then the homomorphism \(\gamma\) generated by \(\bar{\gamma}\) will be an open mapping.

Proof. By assumption, the set \(\bar{\gamma}(X)\) dynamically generates \(P\). Hence \(\gamma(R)=P\). Since \(\gamma(\sigma(x))=\bar{\gamma}(x)\), from the openness of \(\bar{\gamma}\) and the continuity of \(\sigma\) we obtain that the mapping \(\gamma:\sigma(X)\to\bar{\gamma}(X)\) is open.

Consider the mapping \(\xi:W\times\sigma(X)\to R\), sending \((w,\sigma(x))\) to \(w(\sigma(x))\), and the mapping \(\psi:W\times\sigma(X)\to P\), sending \((w,\sigma(x))\) to \(w^{*}\gamma(\sigma(x))\). The mapping \(\psi\) is open. Further,
\[ \gamma\bigl(\xi(w,\sigma(x))\bigr)=\gamma\bigl(w(\sigma(x))\bigr)=w^{*}\bigl(\gamma(\sigma(x))\bigr)=\psi(w,\sigma(x)). \]

That is, \(\gamma\xi=\psi\). From the openness of \(\psi\) and the continuity of \(\xi\) we conclude that \(\gamma\) is an open mapping. This is what was required to prove.

Let us now note that the topology of the space of a dynamical system is initial relative to this space. Hence we immediately obtain

Corollary. If a continuous mapping \(\bar{\gamma}:X\to P\), preserving the relations from \(\Sigma\), is open and \(\bar{\gamma}(X)=P\), then the homomorphism \(\gamma\) of the system \((R,W)\) onto \((P,W)\) generated by it is also open.

§ 3. FREE DYNAMICAL SYSTEMS

Definition. A dynamical system \((R,W)\), defined in the class of \(K\)-systems with the motion group \(W\) by the space \(X\), the continuous mapping \(\sigma:X\to R\), and the empty set of defining relations on \(X\), will be called a free dynamical system.

From Theorem 1 there immediately follow the existence and uniqueness of free systems. Let us formulate property \(F_3\) for free systems.

\(F_3)\) For every continuous mapping \(\bar\gamma\) of a space \(X\) into the space \(P\) of an arbitrary dynamical system \((P, W)\) of class \(K\), there exists a continuous homomorphism \(\gamma\) of the system \((R, W)\) into \((P, W)\) such that
\[ \gamma(\sigma(x))=\bar\gamma(x). \]
We note that the extension \(\Sigma^*\) of the empty system of relations \(\Sigma\) is nonempty, since it contains at least all identical relations of the form \(w(x)=w(x)\).

Theorem 3. Every dynamical system of class \(K\) is a continuous and open homomorphic image of a free dynamical system of the same class.

Proof. Let \((P, W)\) be a system of class \(K\), and let \((R, W)\) be the free dynamical system determined in \(K\) by the space \(P\) and by some mapping \(\sigma:P\to R\). The identity mapping \(\bar\gamma:P\to P\) generates a homomorphism \(\gamma:R\to P\) of dynamical systems. According to the corollary of Theorem 2, the homomorphism \(\gamma\) is open.

Let \(K\) be a class of systems with a Hausdorff space and a group of motions \(W\).

We note that the topology in Hausdorff spaces can be described with the aid of the operation of convergence of nets \(\lim\) [2]. A net in a space \(R\) 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 \(\alpha\). The convergence of \([p^\alpha]\) to \(p\) is symbolically written as \(\lim[p^\alpha]=p\).

Theorem 4. In order that a dynamical system \((R, W)\) of class \(K\) be a free system, determined in \(K\) by some space \(X\) and mapping \(\sigma:X\to R\), it is necessary and sufficient that \((R, W)\) possess a homomorphism onto the group system \((W, W)\).

Proof. If \((R, W)\) is free, then the mapping \(\bar\gamma\) sending \(X\) to the identity of the group \(W\) generates the required homomorphism
\[ \gamma(w(\sigma(x)))=w\cdot\bar\gamma(x)=w. \]

Conversely, let \(\gamma\) be a homomorphism of \((R, W)\) onto \((W, W)\), and let \(X=\gamma^{-1}(e)\). We shall show that \((R, W)\) is a free system determined in \(K\) by the space \(X\) and by the identity mapping of \(X\) into \(R\). First, \(X\) dynamically generates \(R\). Let \(p\in R\) and \(\gamma(p)=w\). Then
\[ \gamma(w^{-1}(p))=w^{-1}\gamma(p)=e. \]
Hence \(w^{-1}(p)=x\in X\). Therefore \(p=w(x)\).

The representation of a point \(p\in R\) in the form \(p=w(x)\), where \(x\in X\), is unique. Indeed, from \(w_1(x_1)=w_2(x_2)\), applying \(\gamma\), we obtain \(w_1=w_2\). Hence \(x_1=x_2\). Let the net \([p^\alpha]\) converge to \(p^0\), \(p^\alpha=w^\alpha(x^\alpha)\), \(p^0=w^0(x^0)\). Since the homomorphism \(\gamma\) is continuous, it follows that
\[ \lim[\gamma(p^\alpha)]=\gamma(p^0) \quad\text{and}\quad \lim[w^\alpha]=w^0. \]
Finally,
\[ \lim[x^\alpha]=\lim[(w^\alpha)^{-1}(p^\alpha)]=(w^0)^{-1}(p^0)=x^0. \]

It follows from this that \(R\) is homeomorphic to the direct product \(W\times X\). But then the topology of \(R\) will be initial with respect to \(X\), and hence also free. From § 2 we obtain that \((R, W)\) is a free dynamical system determined in \(K\) by the space \(X\). This is what was required to prove.

From the theorem just proved we directly obtain

Corollary. A free dynamical system determined by a space \(X\) in the class of systems with a Hausdorff space and a group

of motions \(W\), is the direct product of the system \((X, W)\), where \(W\) acts identically on \(X\), and the group system \((W, W)\).

Examples. a) Let \(K\) be the class of systems with Hausdorff space and a group of motions isomorphic to the group of real numbers. For systems in \(K\) whose space is locally compact and satisfies the second axiom of countability, the notion of freeness coincides with the notion of rectifiability. This follows from the work of E. A. Barbashin [3] and from Theorem 4.

b) Let \((R, W)\) be a free system generated, in some class \(K\), by a space \(X\) consisting of a single point. Consequently, \(R\) consists of only one trajectory. If \(K\) contains \((W, W)\), then it will be a free system. Otherwise \((R, W)\) will be a continuous and one-to-one image of the system \((W/A, W)\) on some factor group \(W/A\). Let \(\psi : W \to W/A\) be the natural mapping. The motions in \((W/A, W)\) are defined by the condition \(\omega(\psi(w_1))=\psi(ww_1)\).

For example, if \(W\) is the group of real numbers and \(K\) does not contain \((W, W)\), then the systems in \(K\) have only periodic trajectories. In this case the free system under consideration will be isomorphic to the system of rotations of the circle.

References

1. Pontryagin L. S. Continuous Groups. 2nd ed., Moscow, 1954.
2. Mal'tsev A. I. Free topological algebras. Izvestiya AN SSSR, Ser. Mat., 21, 1957, pp. 171–198.
3. Barbashin E. A. On homomorphisms of dynamical systems. II, Mat. Sb., 29 (71), 1951, pp. 501–518.

Received by the editors
July 24, 1965

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