I. U. BRONSTEIN

RECURRENCE, PERIODICITY, AND TRANSITIVITY IN DYNAMICAL SYSTEMS WITHOUT UNIQUENESS

(Presented by Academician P. S. Aleksandrov on 24 I 1963)

In this paper we consider certain properties of recurrence, periodicity, stability in the sense of Poisson, and transitivity in semigroups of multivalued mappings (s.m.m.) \((R, S, f)\) \((^1)\), where \(R\) is a uniform separated space \((^2)\).

\(1^\circ\). Definition 1 \((^3)\). A point \(p_0 \in R\) will be called a point of discontinuity for the s.m.m. \((R, S, f)\) if, for any \(s \in S\), the mapping \(f^s\) of the space \(R\) into itself, defined by the condition
\[ f^s(p)=f(p,s)\quad (p\in R), \]
is strongly discontinuous at the point \(p=p_0\) \((^4)\).

Definition 2. A set \(A \subseteq R\) will be called transitive \((^5)\) if
\[ A \subseteq \overline{f(p,S)} \]
for every point \(p \in A\).

A closed semi-invariant \((^1)\) transitive set is minimal semi-invariant. If all points of a minimal semi-invariant set \(\Sigma\) are points of discontinuity, then \(\Sigma\) is a transitive set.

Let \(\Sigma\) be an arbitrary minimal semi-invariant set and \(p \in \Sigma\). If the set \(F_1=\overline{f(p,S)}\subseteq\Sigma\) is semi-invariant, then \(\Sigma=F_1\); otherwise consider the set
\[ F_2=\overline{f(F_1,S)}\subseteq\Sigma . \]
Suppose that closed sets
\[ F_1 \subseteq F_2 \subseteq \cdots \subseteq F_\alpha \subseteq \cdots \subseteq \Sigma \]
have been constructed for all transfinite numbers \(\alpha\) less than some transfinite \(\beta\). If \(\beta\) is a limiting ordinal number, then define
\[ F_\beta=\bigcup_{\alpha<\beta}F_\alpha . \]
If \(\beta=\delta+1\), then define
\[ F_\beta=\overline{f(F_\delta,S)} . \]
In both cases, from \(\alpha<\beta\) it follows that \(F_\alpha\subseteq F_\beta\). Since the set of all closed subsets of \(\Sigma\) can be well ordered, there will be found some transfinite number \(\gamma\) such that
\[ F_{\gamma+1}=F_\gamma, \]
i.e.
\[ \overline{f(F_\gamma,S)}=F_\gamma; \]
then \(F_\gamma\) is a closed nonempty semi-invariant subset of \(\Sigma\). Therefore \(F_\gamma=\Sigma\).

Thus, to every minimal semi-invariant set \(\Sigma\) and point \(p\in\Sigma\) there corresponds some transfinite \(\gamma(\Sigma,p)\). Let
\[ \gamma(\Sigma)=\min_{p\in\Sigma}\gamma(\Sigma,p). \]
Examples show that, for any countable transfinite \(\alpha\), there is a minimal semi-invariant bicompact set \(\Sigma\) such that
\[ \gamma(\Sigma)\geq \alpha . \]

\(2^\circ\). Introduce the notation:
\[ T_p=f(p,S)\quad (p\in R); \]
\(D\) is the group of real numbers; \(D^+\) is the semigroup of nonnegative numbers; \(U\) is the filter of neighborhoods of the uniform structure of the space \(R\) \((^2)\).

Definition 3. We shall say that a point \(p\in R\) satisfies condition \(R_i\) \((i=1,2,\ldots,5)\), if for every \(\alpha\in U\) there exists a bicompact set \(K\subseteq S\) such that condition \(K_i\) \((i=1,2,\ldots,5)\) is fulfilled:

\(K_1.\)
\[ T_p\subseteq \alpha[f(p,sK)] \]
for every \(s\in S\).

\(K_2.\)
\[ T_p\subseteq \alpha[f(q,K)] \]
for every point \(q\in T_p\).

\(K_3.\) For arbitrary elements \(s_1\) and \(s_2\) of \(S\), there exists an element \(s_3\in s_2K\) such that
\[ f(p,s_1)\subseteq \alpha[f(p,s_3)] . \]

\(K_4.\) For arbitrary elements \(s_1\) and \(s_2\) of \(S\), there exists an element \(s_3\in s_2K\) such that
\[ f(p,s_1)\subseteq \alpha[f(p,s_3)] \]
and
\[ f(p,s_3)\subseteq \alpha[f(p,s_1)] . \]

\(K_5\). For any element \(s \in S\) and any point \(q \in T_p\) there exists an element \(s' \in K\) such that \(f(p,s) \subseteq a[f(q,s')]\).

Each of the properties \(R_i\) \((i=1,2,\ldots,5)\) of a point \(p \in R\) in the case of an ordinary dynamical system \((^6)\) \((S=D,\ R\) a metric space) is equivalent to the recurrence of the motion issuing from the point \(p\).

Properties \(R_3\) and \(R_4\) (in the case where the space \(R\) is metric and \(S=D^+\)) were introduced by B. M. Budak \((^7)\) under the names \((+)\)-recurrence and strong \((+)\)-recurrence.

Theorem 1.

\[ \begin{gathered} R_4 \to R_3 \to R_1\\ \uparrow \qquad \downarrow\\ R_5 \to R_2 \end{gathered} \]

(an arrow denotes logical implication).

Examples show that the conditions \(R_4\) \((R_3,\ R_5,\ R_5)\) are stronger than the corresponding conditions \(R_3\) \((R_1,\ R_3,\ R_2)\), while the conditions \(R_2\) and \(R_3\) are independent. It is unknown whether condition \(R_2\) \((R_5)\) follows from condition \(R_4\).

Theorem 2. If \(\Sigma\) is a bicompact minimal semi-invariant set, all of whose points are points of continuity, then any point \(p \in \Sigma\) satisfies condition \(R_2\).

There exist bicompact invariant transitive sets no point of which satisfies condition \(R_2\).

Theorem 3. If a point \(p\) satisfies condition \(R_2\) and the space \(R\) is complete, then \(\overline{T}_p\) is a bicompact transitive set.

Examples show that in Theorem 3 condition \(R_2\) cannot be replaced by condition \(R_3\), even if all points of the space are points of discontinuity.

From Theorems 2 and 3 there follows a corollary which is a generalization of well-known theorems of Birkhoff \((^6)\):

Corollary. Let \(R\) be a complete space all of whose points are points of discontinuity. In order that the set \(\Sigma \subseteq R\) be a bicompact minimal semi-invariant set, it is necessary and sufficient that all points \(p \in \Sigma\) satisfy condition \(R_2\).

In the proof of the following theorem, essential use is made of the fact that the set of all bicompact subsets of a bicompact set, endowed with the finite topology, is bicompact \((^8)\).

Theorem 4. In complete spaces all of whose points are points of discontinuity, the conditions \(R_2\) and \(R_5\) \((R_1\) and \(R_3)\) are equivalent. Under the same assumptions, condition \(R_4\) is stronger than \(R_2\), and \(R_2\) is stronger than \(R_3\).

Thus Theorems 2 and 3 are a generalization of the corresponding results of B. M. Budak \((^7)\) and M. I. Minkevich \((^9)\).

An example of E. A. Barbashin \((^3)\) shows that there exist bicompact minimal semi-invariant sets all of whose points are points of discontinuity and no point of which satisfies condition \(R_4\).

\(3^\circ\). Let \((R,S,f)\) be an s.t.s. and \(p \in R\). Introduce the notation:
\[ S_1(p)=\mathcal{E}\{s \in S: f(p,s)\in p\}. \]
It is easy to see that \(S_1(p)\) is a closed subsemigroup of the semigroup \(S\).

Definition 4. We shall say that a point \(p \in R\) satisfies condition \(O_i\) \((i=1,2,3)\) if there exists a bicompact set \(K \subseteq S\) such that condition \(K_i'\) \((i=1,2,3)\) is fulfilled:

\(K_1'\). \(T_p=f(p,K)\).

\(K_2'\). \(T_p=f(p,K)\), and for any point \(q \in T_p\) there exists an element \(s \in S\) for which \(p \in f(q,s)\).

\(K_3'\). \(T_p=f(q,K)\) for any point \(q \in T_p\).

Condition \(O_2\) (for \(S=D^+\)) was introduced by M. I. Minkevich \((^{10})\). If a point \(p\) satisfies condition \(O_2\), then the funnel \(T_p\) is called closed \((^{10})\). Each of the conditions \(O_i\) \((i=1,2,3)\) in the case of an ordinary dynamical system \((S=D)\) means periodicity of the motion issuing from the point \(p\).

It is easy to see that \(O_3\) implies \(O_2\), and \(O_2\) implies \(O_1\). Examples show that \(O_2\) is stronger than \(O_1\). Let us show that \(O_3\) is stronger than \(O_2\).

Example 1. As the space \(R\), take the set of points of three-dimensional space \(XYZ\) lying on the following curves: a) on the circles
\(x^2+y^2=1,\ z=0\) and \(x^2+y^2=4,\ z=0\); b) on the spiral
\(x=(2-e^\varphi)\sin\varphi,\ y=(2-e^\varphi)\cos\varphi,\ z=0\) \((-\infty<\varphi\leq 0,\ \varphi\) is the polar angle); c) on the semicircles lying in the half-plane \(x=0,\ z\geq 0\) and subtending the segments \([1;\,2-e^{-2\pi n}]\) \((n=1,2,\ldots)\) of the \(Y\)-axis as diameters; d) on the circle lying in the plane \(x=0\) and subtending the segment \([1,2]\) of the \(Y\)-axis as diameter.

Prescribe along the curves lying in the plane \(z=0\) uniform motion in the positive direction with unit velocity, and along the curves lying in the plane \(x=0\) uniform motion with the same velocity, starting from the point \(p=(0;\,1;\,0)\) in the direction of increasing \(z\). In this way we define the p.n.o. \((R,D^+,f)\). The point \(p\) satisfies condition \(O_2\), but does not satisfy condition \(O_3\).

The constructed example shows that Theorem 1 (§ 2) of the work of M. I. Minkevich \({}^{(10)}\) is false.

In the same work it is asserted (Theorem 8, § 2) that in a closed funnel (when \(S=D^+\)) the sections repeat periodically, starting from some time \(T\geq 0\). The following example shows that the indicated theorem is also false.

Example 2. Let the circle \(L_1\) have length \(\pi\), and let the circle \(L_2\) have length 1, lie in the same plane, and touch the circle \(L_1\) from within at the point \(p\). Prescribe on these circles uniform motion in one and the same direction with unit velocity. In the resulting system the point \(p\) satisfies condition \(O_3\), but the sections do not repeat, owing to the incommensurability of the numbers 1 and \(\pi\).

However, the following theorem holds.

Theorem 5. Let \(S=D^+\), \(p\in R\). The sections of the funnel \(T_p\) repeat periodically, starting from some \(T\geq 0\), if at least one of the following conditions is satisfied:

1) the point \(p\) satisfies condition \(O_3\), and the semigroup \(S_1(p)\) is multigenic \({}^{(11)}\);

2) the point \(p\) satisfies condition \(O_1\) and \(S_1(p)\) contains a nonempty interval.

4°. Consider the following property of a point \(p\), analogous to the property of Poisson stability:

P. For any neighborhood \(U(p)\) and any point \(q\in T_p\), there exists an element \(s\in S\) such that \(f(q,s)\cap U(p)\ne \Lambda\).

Theorem 6. In a transitive set all points satisfy condition P.

For nontransitive minimal bicompact sets the assertion of Theorem 6, generally speaking, does not hold.

Definition 5. We shall say that a semi-invariant set \(\Sigma\) is regionally transitive \({}^{(5)}\), if for any two open in \(\Sigma\) sets \(U\) and \(V\) there exists an element \(s\in S\) such that
\(U\cap f(V,s)\ne \Lambda\).

A transitive set is regionally transitive. If the funnel \(T_p\) of every point \(p\in\Sigma\) contains an interior (relative to \(\Sigma\)) point and the set \(\Sigma\) is regionally transitive, then \(\Sigma\) is transitive.

There exist regionally transitive bicompact minimal semi-invariant sets that are not transitive.

Definition 6. A point \(p\in R\) will be called orbitally stable if for any neighborhood \(\alpha\in U\) there exists a neighborhood \(\beta\in U\) such that from \(q\in\beta(p)\) it follows that \(T_q\subseteq \alpha(T_p)\).

Theorem 7. If all points of the set \(T_p\) are points of unrest and at the same time points of orbital stability, and the point \(p\) satisfies condition P, then \(\overline{T}_p\) is a transitive set.

Theorem 8. In order that a semi-invariant bicompact set \(\Sigma\) be transitive, it is necessary and sufficient that it be regionally transitive and that all points \(p \in \Sigma\) be orbitally stable.

There exist bicompact semi-invariant nontransitive sets all of whose points are orbitally stable.

In conclusion, the author expresses his gratitude to Prof. V. V. Nemytskii for discussion of the work and valuable comments.

Received
11 I 1963

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