MATHEMATICS

M. I. GRABAR’

ON THE EQUATION $\dfrac{d\varphi}{dt}=f-\alpha$ ON THE PHASE SPACE OF A DYNAMICAL SYSTEM

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

Let $\{S_t\}$ be a dynamical system, or measurable flow, in a Lebesgue space $R$ with normalized measure $m$. We shall assume that the measure $m$ is indecomposable for the system $\{S_t\}$. Denote by $L_m^2(R)$ the space of all complex-valued functions defined on $R$ with square-integrable modulus with respect to the measure $m$, and consider the equation

\[ \varphi(S_t x)=\varphi(x)+\int_0^t [f(S_\tau x)-\alpha]\,d\tau, \tag{1} \]

where $f(x)$ is a given function from $L_m^2(R)$, and $\alpha$ is its mean value, i.e.
\[ \alpha=\int_R f(x)\,dm, \]
while $\varphi(x)$ is the required function, also from $L_m^2(R)$. Here (1) must hold for any $t$ at the points of some set $M\subset R$, invariant with respect to the system $\{S_t\}$, of full measure.

We note that, under well-known restrictions on the function $f(S_t x)$, on the set $M$ equation (1) is equivalent to

\[ \frac{d\varphi(S_t x)}{dt}=f(S_t x)-\alpha. \tag{1'} \]

In the present note, conditions are given for the existence of a solution of equation (1), and some examples of the application of this equation to the study of dynamical systems are considered (see $(^{2-4})$).

I. It is not difficult to show that, by virtue of the indecomposability of the measure $m$, the function $\varphi(x)$ is determined by equation (1) up to a constant term. Therefore, in the class of functions from $L_m^2(R)$ with zero mean value, the solution is unique. In particular, it follows from this that $\varphi(x)$ is real if $f(x)$ is real.

Theorem 1. If there exists a function $\varphi_0(x)\in L_m^2(R)$ such that, for some $t_0\ne 0$ and almost all $x\in R$

\[ \varphi_0(S_{t_0}x)=\varphi_0(x)+\int_0^{t_0} f^*(S_\tau x)\,d\tau, \quad \text{where } f^*(x)=f(x)-\alpha, \tag{2} \]

then there exists a solution of equation (1).

Proof. Obviously, one may assume that the mean value of $\varphi_0(x)$ is equal to 0. Then, in the space of functions from $L_m^2(R)$ orthogonal to 1, (2) will be equivalent to

\[ U_{t_0}\varphi_0=\varphi_0+\int_0^{t_0} U_\tau f^*\,d\tau, \tag{3} \]

where $U_t$ is the unitary operator corresponding to $S_t$. Let $A$ be self-adjoint-

the infinitesimal operator of the group \(\{U_t\}\), \(E(\lambda)\) is its spectral function, and \(g_1,\ldots,g_k\) are generating vectors for \(A\). Then

\[ \varphi_0=\sum_{k=1}^{\infty}\int_{-\infty}^{\infty} b_k(\lambda)\,dE(\lambda)g_k, \qquad f^*=\sum_{k=1}^{\infty}\int_{-\infty}^{\infty} a_k(\lambda)\,dE(\lambda)g_k, \]

where

\[ \sum_{k=1}^{\infty}\int_{-\infty}^{\infty}|b_k(\lambda)|^2\,d\sigma_k(\lambda)<\infty, \qquad \sum_{k=1}^{\infty}\int_{-\infty}^{\infty}|a_k(\lambda)|^2\,d\sigma_k(\lambda)<\infty, \]

where \(\sigma_k(\lambda)=(E(\lambda)g_k,g_k)\). Further,

\[ U_{t_0}\varphi_0=\sum_{k=1}^{\infty}\int_{-\infty}^{\infty} e^{i\lambda t_0} b_k(\lambda)\,dE(\lambda)g_k, \]

\[ \int_0^{t_0} U_\tau f^*\,d\tau = \sum_{k=1}^{\infty}\int_{-\infty}^{\infty} \frac{e^{i\lambda t_0}-1}{i\lambda}\,a_k(\lambda)\,dE(\lambda)g_k. \]

Thus, (3) gives

\[ \sum_{k=1}^{\infty}\int_{-\infty}^{\infty} \bigl(e^{i\lambda t_0}-1\bigr)b_k(\lambda)\,dE(\lambda)g_k = \sum_{k=1}^{\infty}\int_{-\infty}^{\infty} \frac{e^{i\lambda t_0}-1}{i\lambda}\,a_k(\lambda)\,dE(\lambda)g_k, \]

whence, for any \(k\),

\[ \int_{\infty}^{\infty} \bigl(e^{i\lambda t_0}-1\bigr)b_k(\lambda)\,dE(\lambda)g_k = \int_{-\infty}^{\infty} \frac{e^{i\lambda t_0}-1}{i\lambda}\,a_k(\lambda)\,dE(\lambda)g_k. \]

This equality, in turn, means that almost everywhere with respect to the measure \(d\sigma_k(\lambda)\),

\[ \bigl(e^{i\lambda t_0}-1\bigr)b_k(\lambda) = \frac{e^{i\lambda t_0}-1}{i\lambda}\,a_k(\lambda). \]

Thus, for any \(k\), for almost all \(\lambda\) (with respect to the measure \(d\sigma_k(\lambda)\)), either \(b_k(\lambda)=a_k(\lambda)/i\lambda\), or \(\lambda=2\pi n/t_0\), where \(n\) is an integer. Hence, for \(\lambda\ne0\),

\[ \left|\frac{a_k(\lambda)}{\lambda}\right|^2 \le |b_k(\lambda)|^2+\frac{|a_k(\lambda)|^2t_0^2}{4\pi^2 n^2} \le |b_k(\lambda)|^2+t_0^2|a_k(\lambda)|^2, \]

and, consequently, by the continuity of \(\sigma_k(\lambda)\) at zero,

\[ \sum_{k=1}^{\infty}\int_{-\infty}^{\infty} \left|\frac{a_k(\lambda)}{\lambda}\right|^2\,d\sigma_k(\lambda)<\infty. \tag{4} \]

From (4) follows the existence of a function \(\varphi(x)\in L_m^2(R)\) such that

\[ \varphi = \frac{1}{i}\sum_{k=1}^{\infty}\int_{-\infty}^{\infty} \frac{a_k(\lambda)}{\lambda}\,dE(\lambda)g_k = \frac{1}{i}A^{-1}f^*. \]

As was shown in (3), it follows from this that, for any fixed \(t\), for almost all \(x\in R\),

\[ \varphi(S_t x)=\varphi(x)+\int_0^t f^*(S_\tau x)\,d\tau. \tag{5} \]

With the aid of a well-known device (1), one can alter the function \(\varphi(x)\) on a set of measure \(0\) so that equality (5) holds for any \(t\) at the points of some invariant set of full measure with respect to the system \(\{S_t\}\). This means precisely that \(\varphi(x)\) satisfies equation (1).

Theorem 2. In order that equation (1) have a solution in \(L_m^2(R)\), it is necessary and sufficient that

\[ \int_R \left(\frac{1}{t}\int_0^t f(S_\tau x)\,d\tau-\alpha\right)^2 dm = O\left(\frac{1}{t^2}\right). \tag{6} \]

Proof. Suppose (6) is satisfied. Clearly, in \(L_m^2(R)\) (6) is equivalent to

\[ \left\|\int_0^t U_\tau f^*\,d\tau\right\|\le C, \tag{6'} \]

where \(C\) does not depend on \(t\). If, for any \(t_0\ne 0\), we put

\[ g(x)=\int_0^{t_0} f^*(S_\tau x)\,d\tau, \]

then

\[ \left\|\sum_{k=0}^n U_{t_0}^k g\right\|\le C \]

for all \(n\). From Browder’s results \({}^{(5)}\) it follows that in this case there exists \(\varphi_0\in L_m^2(R)\) such that

\[ U_{t_0}\varphi_0=\varphi_0+g. \]

By the definition of \(g\), this equality is equivalent to (2), and by Theorem 1 equation (1) has a solution.

Now let \(\varphi\in L_m^2(R)\) be a solution of equation (1). In \(L_m^2(R)\) this means that

\[ U_t\varphi=\varphi+\int_0^t U_\tau f^*\,d\tau, \]

whence

\[ \left\|\int_0^t U_\tau f^*\,d\tau\right\| \le \|U_t\varphi\|+\|\varphi\|=2\|\varphi\|. \]

Thus (6′) is satisfied, and hence so is (6).

Remark. By virtue of the indecomposability of the measure \(m\), it follows from the ergodic theorem that the left-hand side of (6) tends to zero as \(t\to\infty\) for any function \(f\in L_m^2(R)\).

II. Let now \(\{\widetilde S_t\}\) be another dynamical system in \(R\), obtained from the system \(\{S_t\}\) by the change of time \(d\tau=F(S_t x)\,dx\), where \(F(x)\) is a real measurable function, bounded with respect to the measure \(m\), such that \(F(x)\ge \varepsilon>0\). The measure \(\widetilde m\), defined by the formula \(d\widetilde m=F(x)\,dm\), is known to be invariant and indecomposable with respect to the system \(\{\widetilde S_t\}\). Assuming that both measures \(m\) and \(\widetilde m\) are normalized, from Theorem 2 and the results of the remark \({}^{(2)}\) the following follows.

Theorem 3. If the function \(F(x)\) satisfies condition (6), then the systems \(\{S_t\}\) and \(\{\widetilde S_t\}\), with measures \(m\) and \(\widetilde m\), are isomorphic.

III. Consider the system of differential equations:

\[ \frac{dx_k}{dt}=f_k(x_1,\ldots,x_p),\qquad k=1,2,\ldots,p, \tag{7} \]

where the \(f_k\) are real continuous functions with period \(2\pi\) in each variable. Suppose that system (7) possesses a positive indecomposable integral invariant \(M(x_1,\ldots,x_p)\), having period \(2\pi\) in each variable, and that uniqueness of solutions holds for it. In this case system (7) defines on the \(p\)-dimensional torus

with cyclic coordinates, taken modulo \(2\pi\), a dynamical system with indecomposable measure \(dm=M(x_1,\ldots,x_p)\,dx_1,\ldots,dx_p\). Put

\[ \alpha_k= \frac{ \displaystyle \int_{0}^{2\pi}\cdots\int_{0}^{2\pi} f_k M\,dx_1\ldots dx_p }{ \displaystyle \int_{0}^{2\pi}\cdots\int_{0}^{2\pi} M\,dx_1\ldots dx_p }. \]

Then the following holds:

Theorem 4. If, for some \(k\),

\[ \int_{0}^{2\pi}\cdots\int_{0}^{2\pi} \left( \frac{1}{t}\int_{0}^{t} f_k\,d\tau-\alpha_k \right)^2 M\,dx_1\ldots dx_p = O\left(\frac{1}{t^2}\right), \tag{8} \]

then \(\alpha_k\) is an eigenfrequency of system (7).

Proof. By Theorem 2 there exists a real function \(\varphi_k(x_1,\ldots,x_p)\), of period \(2\pi\) in each variable and of class \(L_m^2\), such that along almost all trajectories of system (7)

\[ \frac{d\varphi_k}{dt}=f_k-\alpha_k. \]

Put \(\Phi_k(x_1,\ldots,x_p)=e^{i(x_k-\varphi_k)}\). Then \(\Phi_k\) will be an eigenfunction of system (7) with frequency \(\alpha_k\). Indeed,

\[ \frac{d\Phi_k}{dt}=i\alpha_k\Phi_k. \]

Remark. If condition (8) is fulfilled for all \(k=1,2,\ldots,p\), then the discrete part of the spectrum of system (7) contains all frequencies of the form \(n_1\alpha_1+\ldots+n_p\alpha_p\), where \(n_1,\ldots,n_p\) are integers.

Received
17 VIII 1962

REFERENCES

1. V. A. Rokhlin, UMN, 4, no. 2 (30), (1949).
2. M. I. Grabar, DAN, 109, no. 2 (1956).
3. M. I. Grabar, DAN, 109, no. 3 (1956).
4. M. I. Grabar, DAN, 109, no. 4 (1956).
5. F. Browder, Proc. Am. Math. Soc., 9, no. 5 (1958).