UDC 519.54 + 517.51

MATHEMATICS

I. U. BRONSTEIN

ON THE THEORY OF DISTAL MINIMAL SETS AND DISTAL FUNCTIONS

(Presented by Academician I. G. Petrovskii, March 11, 1966)

1°. Let \((X,\rho)\) be a compact metric space; \(J\) the group of real numbers; \((X,J,\pi)\) a dynamical system \((^1)\) (a group of transformations \((^2)\)). The system \((X,J,\pi)\) is called distal \((^3)\) if, for any two distinct points \(x\) and \(y\) of \(X\), there is a number \(d>0\) such that \(\rho(xt,yt)>d\) for all \(t\in J\). Every equicontinuous \((^2)\) (uniformly Lyapunov stable \((^1)\)) system is, obviously, distal.

In the work of Auslander, Hahn, and Markus \((^4)\) it was shown that on every nilmanifold \((^5)\) \(X\) one can define a dynamical system \((X,J,\pi)\) under which \(X\) is a distal minimal set, but, generally speaking, is not equicontinuous.

In the present note it is proved that on every torus \(T^\alpha\) (finite-dimensional or infinite-dimensional) one can define a distal dynamical system under which \(T^\alpha\) is a minimal set. An example is given of a distal non-equicontinuous system on the three-dimensional torus \(T^3\), under which \(T^3\) is a minimal set. Further, it is proved that if \(\varphi(t)\) and \(\psi(t)\) \((t\in J)\) are Bohr almost-periodic functions \((^6)\), then the function

\[ \varphi\left(\int_0^t \psi(\tau)\,d\tau\right) \]

is distal \((^{7,8})\), but, generally speaking, is not Bohr almost-periodic.

2°. In what follows, by \(T\) we shall denote the factor group of the group \(J\) by the subgroup generated by the number \(2\pi\).

Theorem 1. A dynamical system defined on the \(n\)-dimensional torus \(T^n\) by the system of differential equations

\[ dx_1/dt=\varphi_1, \]

\[ dx_i/dt=\varphi_i(x_1,\ldots,x_{i-1})\quad (i=2,\ldots,n), \tag{1} \]

where \(\varphi_i\in J\), \(\varphi_i(x_1,\ldots,x_{i-1})\) are functions periodic with period \(2\pi\) in each variable and is distal. For the system (1) there exist numbers \(\gamma_2,\ldots,\gamma_n\) such that the system of equations

\[ dx_1/dt=\varphi_1, \]

\[ dx_i/dt=\varphi_i(x_1,\ldots,x_{i-1})+\gamma_i\quad (i=2,\ldots,n) \tag{1′} \]

defines a dynamical system under which \(T^n\) is a minimal set.

Proof. The proof of Theorem 1 is carried out by induction on \(n\). For \(n=1\) the assertion is obvious. Suppose the assertion is true for \(n=k\), and let \(a=(x_1,\ldots,x_{k+1})\) and \(b=(y_1,\ldots,y_{k+1})\) be two distinct points of the torus \(T^{k+1}\). If there is an index \(i\) such that \(1\le i\le k\) and \(x_i\ne y_i \pmod {2\pi}\), then the points \(a\) and \(b\) are distal by virtue of the induction hypothesis. If, however, \(x_i=y_i \pmod {2\pi}\) for all \(i\), \(1\le i\le k\), then \(x_{k+1}\ne y_{k+1}\pmod {2\pi}\), and in this case the distance between the points \(at\) and \(bt\) does not depend on \(t\in J\).

By virtue of the induction assumption we may suppose that the system of equations (1) defines on the \(k\)-dimensional torus \(T^k=\{(x_1,\ldots,x_k)\}\) a dynamical system for which \(T^k\) is a distal minimal set. It is easy to prove that the intersection of the closure of the trajectory of the point \(p=(0,\ldots,0)\in T^{k+1}\) with the circle \((0,\ldots,0,x_{k+1})\) is a subgroup of this group. To complete the proof of the second assertion of the theorem it suffices to use the following proposition.

Lemma. Let \(X\) be a compactum and let \(p\) be a homomorphic mapping of a distal dynamical system \((X,J,\pi)\) onto a system \((Y,J,\rho)\), in which \(Y\) is a minimal set. If there exists a point \(x\in X\) such that the closure of the trajectory of the point \(x\) contains the set \(p^{-1}(p(x))\), then \(X\) is a minimal set for \((X,J,\pi)\).

Theorem 1 can obviously be generalized also to infinite-dimensional tori.

We now show that on any torus \(T^n\) \((n\geqslant 3)\) one can define a dynamical system for which \(T^n\) is a distal, not uniformly continuous, minimal set. By virtue of Theorem 1 it suffices to construct an example of such a system on \(T^3\).

Example. In \((^1)\), on p. 424, it is shown that there exist an irrational number \(\mu\) and a function \(\Phi(x_1,x_2)\), continuous on \(T^2\), such that for any \(m>0\) and \(\varepsilon>0\) one can specify \(a\in T\) and \(t\in J\) for which

\[ \left|\int_0^t \Phi(\tau,\mu\tau)\,d\tau-\int_0^t \Phi(\tau,a+\mu\tau)\,d\tau\right|>m,\qquad |a|<\varepsilon \pmod {2\pi}. \tag{2} \]

By virtue of Theorem 1 there exists a number \(\gamma\) such that the system of equations

\[ \begin{aligned} dx_1/dt&=1,\\ dx_2/dt&=\mu,\\ dx_3/dt&=\Phi(x_1,x_2)+\gamma \end{aligned} \tag{3} \]

defines on \(T^3\) a distal dynamical system for which \(T^3\) is a minimal set. From condition (2) it follows that the system (3) is not uniformly continuous.

From the results of the paper \((^9)\) the following assertion follows.

Theorem 2. If the torus \(T^2\) is a distal minimal set for the system \((T^2,J,\pi)\), then this system is uniformly continuous.

However, the torus \(T^2\) may be a distal, not uniformly continuous, minimal set for a discrete system.

3°. A continuous real function \(\varphi(t)\) \((t\in J)\) is called distal \((^7,^8)\) if the closure of the trajectory of the point \(\varphi(t)\) in the Bebutov system \((^{10})\) is a compact distal minimal set. It is known \((^7)\) that the set of all distal functions forms an algebra. It is easy to prove that the limit of a uniformly convergent sequence of distal functions is a distal function. From \((^{11})\) it follows that, for any distal function \(\Phi(t)\), the limit

\[ \lim_{T\to\infty}\frac{1}{2T}\int_{-T}^{+T}\Phi(t+a)\,dt \]

exists uniformly in \(a\in J\).

Denote by \(B\) the set of all Bohr almost-periodic functions.

Theorem 3. If \(\psi\in B\), then the set

\[ B_\psi=\left\{\varphi\left(\int_0^t \psi(\tau)\,d\tau\right)\mid \varphi\in B\right\} \]

forms an algebra of distal functions.

Proof. It is known \((^{6})\) that for every function \(\psi \in B\) there exist a function \(F(x_1,\ldots,x_n,\ldots)\), continuous on \(T^\infty\), and numbers \(\lambda_1,\ldots,\lambda_n,\ldots\) such that
\[ \psi(t)=F(\lambda_1 t,\ldots,\lambda_n t,\ldots). \]
The system
\[ \begin{aligned} dx_i/dt&=\lambda_i \qquad (i=1,\ldots,n,\ldots),\\ dy/dt&=F(x_1,\ldots,x_n,\ldots) \end{aligned} \]
is distal. Therefore, for any function \(\varphi(y)\) periodic with period \(2\pi\), the function
\[ \varphi\left(\int_0^t F(\lambda_1\tau,\ldots,\lambda_n\tau,\ldots)\,d\tau\right) = \varphi\left(\int_0^t \psi(\tau)\,d\tau\right) \]
is distal. To complete the proof it suffices to use the remarks made at the beginning of this section.

The example constructed above shows that functions of the form
\[ \varphi\left(\int_0^t \psi(\tau)\,d\tau\right), \]
\((\varphi,\psi\in B)\), generally speaking, are not almost periodic in the sense of Bohr. Moreover, they are not, generally speaking, almost periodic in the sense of Levitan \((^{6})\), as the following assertion shows.

Theorem 4. If a function almost periodic in the sense of Levitan is distal, then it is almost periodic in the sense of Bohr.

In conclusion we note that, using Theorem 1, one can also indicate other classes of distal functions.

Institute of Mathematics with Computing Center
Academy of Sciences of the USSR

Received
2 III 1966

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