M. I. GRABAR

INDECOMPOSABLE MEASURES IN DYNAMICAL SYSTEMS

(Presented by Academician A. N. Kolmogorov on 17 VIII 1962)

Let us consider a dynamical system with compact phase space, i.e., a continuous one-parameter group \(\{S_t\}\) of homeomorphisms of a compact set \(R\) onto itself. As is known, a normalized measure invariant for the system \(\{S_t\}\) is called indecomposable if, for every measurable set invariant with respect to the system, this measure is equal either to \(0\) or to \(1\). The indecomposability of a measure with respect to a single homeomorphism \(S_t\) is defined analogously.

The following theorem establishes a connection between measures invariant and indecomposable for the whole system \(\{S_t\}\) and for its individual homeomorphisms. All measures are assumed to be normalized.

Theorem 1. \(1^\circ\). For any fixed \(t_0 \ne 0\) and any measure \(m\), invariant and indecomposable for \(\{S_t\}\), there exists a measure \(\mu\), invariant and indecomposable for \(S_{t_0}\), such that

\[ m(A)=\frac{1}{t_0}\int_0^{t_0}\mu(S_tA)\,dt \tag{1} \]

for any Borel set \(A \subset R\).

\(2^\circ\). The totality of all measures \(\mu\) for which the representation (1) holds for a given measure \(m\) is given by the formula

\[ \mu_t(A)=\mu(S_tA), \tag{2} \]

where \(\mu\) is any one of such measures.

\(3^\circ\). Every measure \(\mu\), invariant and indecomposable for \(S_{t_0}\), determines by formula (1) some measure \(m\), invariant and indecomposable for the system \(\{S_t\}\).

Proof. We first prove assertion \(3^\circ\). The fact that formula (1) defines some measure in the space \(R\) is obvious. Further:

\[ m(S_\tau A)=\frac{1}{t_0}\int_0^{t_0}\mu(S_tS_\tau A)\,dt =\frac{1}{t_0}\int_0^{t_0}\mu(S_{t+\tau}A)\,dt= \]

\[ =\frac{1}{t_0}\int_\tau^{t_0+\tau}\mu(S_tA)\,dt =\frac{1}{t_0}\int_0^{t_0}\mu(S_tA)\,dt =m(A), \]

since the function \(\mu(S_tA)\) has period \(t_0\) by virtue of the invariance of the measure \(\mu\) with respect to \(S_{t_0}\). This proves the invariance of the measure \(m\) with respect to the system \(\{S_t\}\). If now \(A\) is a set invariant for \(\{S_t\}\), then \(S_tA=A\) for every \(t\), and consequently,

\[ m(A)=\frac{1}{t_0}\int_0^{t_0}\mu(A)\,dt=\mu(A)= \left\{ \begin{array}{l} 0\\ 1 \end{array} \right\}, \]

for \(\mu\) is indecomposable.

We pass to assertion \(1^\circ\). Let \(m\) be any indecomposable measure in the system \(\{S_t\}\), and let \(E\) be its ergodic set \((^1)\). The set \(E\), being invariant for the system, is also invariant for \(S_{t_0}\). The same is true for the measure \(m\). Therefore the measure \(m\) is the limit of some sequence of linear combinations of measures indecomposable for \(S_{t_0}\). Consequently, there is a measure \(\mu\), indecomposable for \(S_{t_0}\), such that \(\mu(E)=1\). Then the measure \(m'\), defined from the measure \(\mu\) by formula (1), will coincide with the measure \(m\) on the set \(E\), since, obviously, \(m'(E)=1\), and an ergodic set determines only one indecomposable measure. \(1^\circ\) is proved.

Finally, suppose that

\[ m(A)=\frac{1}{t_0}\int_0^{t_0}\mu_1(S_tA)\,dt =\frac{1}{t_0}\int_0^{t_0}\mu_2(S_tA)\,dt, \tag{3} \]

where \(\mu_1,\mu_2\) are measures indecomposable for \(S_{t_0}\). Let \(\mathscr E_1\) and \(\mathscr E_2\) be the ergodic sets corresponding to these measures. Put \(E_1=\bigcup_t S_t(\mathscr E_1)\), \(E_2=\bigcup_t S_t(\mathscr E_2)\). Since \(\mu_1(\mathscr E_1)=1\) and \(\mu_2(\mathscr E_2)=1\), it follows, by (3), that \(m(E_1)=m(E_2)=1\), and consequently \(E_1\cap E_2\ne 0\). If \(x\in E_1\cap E_2\), then \(x=S_{t_1}x_1=S_{t_2}x_2\), where \(x_1\in\mathscr E_1\) and \(x_2\in\mathscr E_2\). Hence \(x_2=S_{t_1-t_2}x_1\), and this means that \(\mathscr E_2=S_t(\mathscr E_1)\) and, consequently, \(\mu_1(A)=\mu_2(S_tA)\), where \(t=t_1-t_2\). \(2^\circ\) is proved.

As is known \((^2)\), if a measure \(m\), indecomposable and invariant with respect to the system \(\{S_t\}\), is decomposable with respect to some \(S_{t_0}\), where \(t_0\ne0\), then the system \(\{S_t\}\) has at least one nonzero eigenfrequency. The following theorem refines this assertion.

Theorem 2. Let \(t_0\ne0\) be fixed, and let \(m\) be any measure invariant and indecomposable for the system \(\{S_t\}\). If the measure \(m\) is decomposable for \(S_{t_0}\), then there is an integer \(k\ne0\) such that \(\lambda=2k\pi/t_0\) is an eigenfrequency of the system \(\{S_t\}\). If, on the other hand, the measure \(m\) is indecomposable for \(S_{t_0}\), then for \(k\ne0\) there are no numbers of the form \(2k\pi/t_0\) among the eigenfrequencies of the system \(\{S_t\}\).

Proof. If the measure \(m\) is decomposable for \(S_{t_0}\), then any measure \(\mu\), invariant and indecomposable for \(S_{t_0}\) and connected with \(m\) by formula (1), will be distinct from \(m\). The latter means that the measure \(\mu\) cannot be invariant for all \(S_t\). Denote by \(I(\mu)\) the set of all numbers \(t\) such that \(\mu\) is invariant with respect to \(S_t\). \(I(\mu)\) is a closed subgroup of the group of all real numbers.

The group property is obvious. Let us prove closedness. If \(t_n\in I(\mu)\) and \(t_n\to t\), then for any continuous function \(f(x)\)

\[ \int_R f(x)\,d\mu = \int_R f(S_{t_n}x)\,d\mu \to \int_R f(S_tx)\,d\mu = \int_R f(x)\,d\mu . \]

Consequently, \(t\in I(\mu)\).

Since, as was already noted, \(I(\mu)\) cannot contain all real numbers, \(I(\mu)\) is a set of numbers of the form \(n\tau_0\), where \(\tau_0\ne0\) and \(n\) is an integer. In particular, there is an integer \(k\ne0\) such that \(t_0=k\tau_0\). This means that \(S_{t_0}=(S_{\tau_0})^k\), and consequently the measure \(\mu\) will be indecomposable also for \(S_{\tau_0}\). Let \(\mathscr E\) be the ergodic set of the homeomorphism \(S_{\tau_0}\) corresponding to this measure. Put \(E=\bigcup_t S_t(\mathscr E)\). Then, by (1), \(m(E)=1\), since \(\mu(\mathscr E)=1\). We now prove that if \(S_t(\mathscr E)\cap\mathscr E\ne0\), then \(t=n\tau_0\), where \(n\) is an integer. Let \(x_1\in S_t(\mathscr E)\cap\mathscr E\). Then \(x_1=S_tx_0\), where \(x_0\in\mathscr E\). Since the individual measures of the points \(x_0\) and \(x_1\) coincide with the measure \(\mu\), for any continuous-

continuous function \(f(x)\)

\[ \int_R f(S_t x)\,d\mu = \lim_{T\to\infty}\frac{1}{T}\int_0^T f(S_t S_\tau x_0)\,d\tau = \lim_{T\to\infty}\frac{1}{T_0}\int_0^T f(S_\tau S_t x_0)\,d\tau = \]

\[ = \lim_{T\to\infty}\frac{1}{T}\int_0^T f(S_\tau x_1)\,d\tau = \int_R f(x)\,d\mu . \]

This means that \(t\in I(\mu)\) and, consequently, \(t=n\tau_0\). Let now \(x\in E\). Then, if \(x=S_{t_1}x_1=S_{t_2}x_2\), where \(x_1,x_2\in\mathcal E\), then \(x_2=S_{t_1-t_2}x_1\) and, consequently, \(t_1-t_2=n\tau_0\). Therefore there exists a number \(\alpha(x)\), least in absolute value, such that \(x=S_{\alpha(x)}\tilde x\), where \(\tilde x\in\mathcal E\). It is easy to verify that

\[ \alpha(S_t x)=\alpha(x)+t+n\tau_0, \tag{4} \]

where \(n\) is an integer. Finally, put, for \(x\in E\),

\[ \varphi(x)=e^{\,i\frac{2\pi\alpha(x)}{\tau_0}} . \]

Then, by virtue of (4),

\[ \varphi(S_t x)=e^{i\lambda t}\varphi(x), \qquad \text{where } \lambda=\frac{2\pi}{\tau_0}=\frac{2k\pi}{t_0}. \]

Thus, \(\varphi(x)\) is an eigenfunction of the system \(\{S_t\}\) with eigenfrequency \(\lambda=2k\pi/t_0\).

Let now the measure \(m\) be indecomposable for \(S_{t_0}\). Suppose that there exists an eigenfunction \(\varphi(x)\) of the system \(\{S_t\}\) with eigenfrequency \(\lambda=2k\pi/t_0\), where \(k\ne 0\) is an integer. Put \(\tau_0=t_0/k\). Then \(S_{t_0}=(S_{\tau_0})^k\) and, consequently, \(m\) will also be indecomposable for \(S_{\tau_0}\). Since, obviously, \(\varphi(S_{\tau_0}x)=\varphi(x)\), it follows that \(\varphi(x)\) is constant almost everywhere (with respect to the measure \(m\)), which is possible only when \(k=0\). Theorem 2 is proved.

The following theorem is also adjacent to Theorem 2.

Theorem 3. If, for some \(t_0\ne 0\), the homeomorphism \(S_{t_0}\) has a nonconstant and measurable \((m)\) eigenfunction with eigenfrequency \(\alpha\), then either this function is also an eigenfunction for the system \(\{S_t\}\), with frequency \(\lambda=\alpha+2k\pi/t_0\), where \(k\) is an integer, or the system \(\{S_t\}\) will have an eigenfrequency \(\lambda=2k\pi/t_0\) with an integer \(k\ne 0\).

Proof. Let \(\varphi(x)\) be an eigenfunction of \(S_{t_0}\) with frequency \(\alpha\). Then almost everywhere with respect to the measure \(m\)

\[ \varphi(S_{t_0}x)=e^{i\alpha t_0}\varphi(x). \tag{5} \]

Assume first that the measure \(m\) is indecomposable for \(S_{t_0}\). Since it follows from (5) that \(|\varphi(S_{t_0}x)|=|\varphi(x)|\), i.e. \(|\varphi(x)|\) is a function invariant with respect to \(S_{t_0}\), almost everywhere \(|\varphi(x)|=\mathrm{const}\), and one may assume that \(|\varphi(x)|=1\). Further one may assume that (5) holds for all \(x\in R\). Then, for arbitrary \(t\),

\[ \varphi(S_t S_{t_0}x)=\varphi(S_{t_0}S_t x)=e^{i\alpha t_0}\varphi(S_t x). \tag{6} \]

From (5) and (6) it follows that the function \(\varphi(S_t x)/\varphi(x)\) is invariant with respect to \(S_{t_0}\), and therefore, for every \(t\), almost everywhere

\[ \varphi(S_t x)=z(t)\varphi(x), \tag{7} \]

where \(z(t)\) does not depend on \(x\). From (7) it is easy to obtain that for any \(t_1\) and \(t_2\)

\(z(t_1+t_2)=z(t_1)z(t_2)\). Let us note further that equality (7) in the space \(L_m^2(R)\) is equivalent to

\[ U_t\varphi=z(t)\varphi, \tag{8} \]

where \(U_t\) is the unitary operator corresponding to \(S_t\). It follows from (8) that \((U_t\varphi,\varphi)=z(t)(\varphi,\varphi)\), and, consequently, \(z(t)\) is a continuous function of \(t\). Finally, since \(|\varphi(x)|=1\), we have \(|z(t)|=1\), and therefore \(z(t)=e^{i\lambda t}\), where \(\lambda\) is a real number. Thus (7) takes the form

\[ \varphi(S_t x)=e^{i\lambda t}\varphi(x), \tag{9} \]

and comparison of (9) and (5) shows that \(\lambda=\alpha+2k\pi/t_0\). The first assertion of the theorem is proved. As for the second, it follows directly from Theorem 2 if the measure \(m\) is decomposable for \(S_{t_0}\).

Remark. If the system \(\{S_t\}\) is strictly ergodic, i.e., admits a unique invariant normalized measure, then Theorem 1 of Note \(^3\) follows in an obvious way from Theorem 2.

Received
17 VIII 1962

REFERENCES

\(^1\) V. V. Nemytskii, V. V. Stepanov, Qualitative Theory of Differential Equations, Moscow—Leningrad, 1949, pp. 529—531. \(^2\) V. A. Rokhlin, Uspekhi Mat. Nauk, 4, no. 2 (30) (1949). \(^3\) M. I. Grabar’, Dokl. Akad. Nauk SSSR, 95, no. 1 (1954).