UDC 517.948.35

MATHEMATICS

A. M. STEPİN

ON PROPERTIES OF THE SPECTRA OF ERGODIC DYNAMICAL SYSTEMS WITH LOCALLY COMPACT TIME

(Presented by Academician A. N. Kolmogorov, 23 XI 1965)

In the work of Ya. G. Sinai \((^1)\), under a certain assumption called by him condition A, a number of results were obtained which should be regarded as analogues of spectral properties of ergodic dynamical systems with discrete spectrum. Condition A is natural for dynamical systems of probability-theoretic origin. At the same time examples are known in which it is not fulfilled \((^2)\). Therefore the question is of interest whether the results of Ya. G. Sinai carry over to systems that do not satisfy condition A. It has proved possible to answer this question in the case when the time in the dynamical system is a certain discrete commutative group. The present work therefore consists of two parts: in the first part we carry over the results of \((^1)\) to the case of dynamical systems with locally compact time; in the second we construct an example of a system not satisfying condition A, and investigate the spectral properties of this system.

1. Let \(G\) be a commutative separable locally compact group; \((X,\mu)\) a space with normalized measure such that \(L^2(X,\mu)\) is separable; \(g \to T_g\) a Haar-measurable representation of the group \(G\) by automorphisms of the space \((X,\mu)\); \(g \to U_g\) the unitary representation of \(G\) in \(L^2(X,\mu)\) conjugate with the representation \(g \to T_g\).

The spectral theorem for this case gives

\[ U_g=\int_{\hat G} \langle g,\chi\rangle E(d\chi),\quad \chi\in \hat G; \]

where \(\hat G\) is the character group of the group \(G\); \(E(\Delta)\) is a Borel spectral measure on \(\hat G\), whose values are projection operators in \(L^2(X,\mu)\).

Definition. A function \(f\in L^2(X,\mu)\) belongs to the class \(F^{k,l}\) if, for all integers \(k_1,l_1\), \(0\le k_1\le k\), \(0\le l_1\le l\), there exist complex generalized measures of bounded variation \(M_{k_1,l_1}\) on \(\hat G^{k_1+l_1}=\hat G\times\cdots\times \hat G\) \((k_1+l_1\) times) such that

\[ \begin{aligned} &M_{k_1,l_1}(\Delta_1\times\cdots\times \Delta_{k_1}\times \Delta'_1\times\cdots\times \Delta'_{l_1})=\\ &\qquad =\int_X E(\Delta_1)f\cdots E(\Delta_{k_1})f\,E(\Delta'_1)f\cdots E(\Delta'_{l_1})f\,d\mu; \end{aligned} \]

here \(\Delta_i,\Delta'_j\) are Borel subsets of \(\hat G\).

As in \((^3)\), we shall study properties of the measure \(M_{2,2}\). From the unitarity of \(U_g\) it follows that \(M_{2,2}(\chi_1,\chi_2;\chi'_1,\chi'_2)\) is concentrated on the subset \(\chi_1\chi_2=\chi'_1\chi'_2\). Moreover, \(M_{2,2}(\chi_1,\chi_2;\chi'_1,\chi'_2)\) is symmetric in the variables \(\chi_1,\chi_2\) and \(\chi'_1,\chi'_2\).

Let \(\psi_\Lambda(\chi,\chi')\) be the characteristic function of a measurable subset \(\Lambda\subset G^2\). Define a countably additive vector measure with values—

with values in \(L^2(X,\mu)\)

\[ X_{1,1}(\Lambda)=\int_{\hat G^2}\psi_\Lambda(\chi,\chi')\,E(d\chi)f\,E(d\chi')f. \]

Theorem 1. If \(\Lambda\subset\{(\chi,\chi'):\chi=\chi'\}\) and the representation \(g\to T_g\) is ergodic, then almost everywhere on \(X\), with respect to the measure \(\mu\),
\[ X_{1,1}(\Lambda)=M_{1,1}(\Lambda). \]

Corollary. Let the representation \(g\to T_g\) be ergodic and let \(\{\Lambda_k\}\) be a monotonically decreasing sequence of subsets of \(\hat G^2\) such that
\[ \Lambda=\bigcap_k\Lambda_k\subset\{(\chi,\chi):\chi=\chi'\}; \]
then
\[ \lim_{k\to\infty}X_{1,1}(\Lambda_k)=M_{1,1}(\Lambda) \]
almost everywhere.

In what follows, in part 1 the ergodicity of the representation \(g\to T_g\) is always assumed.

Theorem 2. Let \(\Xi_1,\Xi_2\) be measurable subsets of the diagonal in \(\hat G^2\), and
\[ \Xi=\Xi_1\times\Xi_2=\{(\chi_1,\chi_2;\chi_1',\chi_2'):(\chi_1,\chi_1')\in\Xi_1,\;(\chi_2,\chi_2')\in\Xi_2\}; \]
then
\[ M_{2,2}(\Xi)=M_{1,1}(\Xi_1)M_{1,1}(\Xi_2). \tag{1} \]

The proof of this theorem differs only slightly from the corresponding proof in (3). It is enough to establish the decomposition (1) for sets \(\Xi_i,\ i=1,2\), of the form
\[ \{(\chi,\chi'):\chi=\chi',\ \chi\in\Delta_i\}, \]
where \(\Delta_i\) are Borel sets. Choose in \(G\) a countable everywhere dense subset \(\Pi\) such that, if \(g\in\Pi\), then \(g^k\in\Pi\). Let
\[ \delta_0=\{e^{i\varphi}:0\le \varphi<\pi\},\qquad \delta_1=\{e^{i\varphi}:\pi\le \varphi<2\pi\}. \]
Put
\[ \Delta_{i_1\ldots i_a}=\{\chi:\chi(g_a)\in\delta_{i_a},\ a=1,\ldots,k\},\qquad i_k=0,1; \]
the \(\Delta_{i_1\ldots i_a}\) do not intersect and
\[ \bigcup_{i_1\ldots i_k}\Delta_{i_1\ldots i_k}=\hat G. \]

Consider
\[ K_n^i=\bigcup_{i_1\ldots i_n}(\Delta_i\cap\Delta_{i_1\ldots i_n})\times (\Delta_i\cap\Delta_{i_1\ldots i_n})\in\hat G^2,\qquad i=1,2. \]
It is easy to verify that the sequence \(K_n^i\) decreases and
\[ \bigcap_n K_n^i=\Xi_i. \]
By the corollary to Theorem 1,
\[ M_{2,2}(K_n^1\times K_m^2)\to M_{1,1}(\Xi_1)\cdot M_{1,1}(K_m^2) \quad\text{as } n\to\infty. \]
Since \(M_{2,2}\) is countably additive, we have
\[ \lim_{n\to\infty}M_{2,2}(K_n^1\times K_m^2)=M_{2,2}(\Xi_1\times K_m^2). \]
Passing to the limit in \(m\), we obtain (1).

Observe that
\[ M_{1,1}(\{(\chi,\chi'):\chi=\chi',\ \chi\in\Delta\})=\sigma_f(\Delta), \]
where
\[ \sigma_f(\Delta)=(E(\Delta)f,f). \]
Thus, on the subset \(\chi_1=\chi_1',\ \chi_2=\chi_2'\) in \(\hat G^4\), the measure \(M_{2,2}(\chi_1,\chi_2;\chi_1',\chi_2')\) reduces to
\[ \sigma_f(\chi_1)\cdot\sigma_f(\chi_2). \]
The same is valid for the subset \(\chi_1=\chi_2',\ \chi_2=\chi_1'\), by virtue of the symmetry of the measure \(M_{2,2}\). From Theorem 2 it follows:

Theorem 3. If \(f\in F^{2,2}\) and
\[ \sigma_f(\Delta)=(E(\Delta)f,f), \]
then there exists a subspace \(H\subset L^2(X,\mu)\), invariant with respect to the group \(U_g\), such that the maximal spectral type \(U\) of the group \(U_g\) in \(H\) is subordinated to the type
\[ \sigma_f*\sigma_f. \]

The proof differs almost not at all from the proof given in (1) for the case of real time.

Theorem 4. If the vectors from \(F^{2,2}\) are everywhere dense in \(L^2(X,\mu)\) (condition A), then the maximal spectral type \(\sigma\) of the representation \(g\to U_g\) subordinates its convolution
\[ \sigma*\sigma. \]

2. We shall construct a representation \(g\to T_g\) for which the hypothesis and the conclusion of Theorem 4 do not hold. As \(G\) take the discrete group of dyadic-rational numbers \((\bmod\,1)\); \((X,\mu)\) is the direct product of the circle \(Y\) with Lebesgue measure and the two-point set \(Z\) with measures \((1/2,1/2)\). Let
\[ \varepsilon_n=1/2^n\in G. \]
Put
\[ T_{\varepsilon_0}=E \]
—the identity transformation;
\[ T_{\varepsilon_k}:(y,i)\to (y+\varepsilon_{k-1},\alpha(y,\varepsilon_k)i), \]
where \(y\in Y,\ i=1,-1;\ k=1,2,\ldots\). We require that the collection of functions \(\alpha(y,\varepsilon_k)\), taking the values \(1\) and \(-1\), satisfy the conditions
\[ \alpha(y,\varepsilon_{k+1})\alpha(y+\varepsilon_k,\varepsilon_{k+1})=\alpha(y,\varepsilon_k), \tag{2} \]
\[ \alpha(y,\varepsilon_k)=-1\quad\text{for }0<y<1/2^{k-1}. \tag{3} \]

Lemma 1. A system of functions satisfying (2), (3) exists.

The transformations \(T_{\varepsilon_k}\) are automorphisms of the space \((X,\mu)\) and
\[ T_{\varepsilon_{k+1}}^{2}=T_{\varepsilon_k}. \]
If
\[ g=\sum_k i_k\varepsilon_k,\qquad i_k=0,1, \]
then put
\[ T_g=\prod_k T_{\varepsilon_k}^{\,i_k}. \]
The mapping \(g\to T_g\) is a measurable representation of \(G\) in \((X,\mu)\). It is ergodic.

Let \(H_1\) be the subspace of \(L^2(X,\mu)\) consisting of functions \(f(x)=f(y,i)\) such that \(f(y,1)=f(y,-1)\); and let \(H_{-1}\) be the subspace of \(L^2(X,\mu)\) consisting of functions for which \(f(y,1)=-f(y,-1)\). Then
\[ L^2(X,\mu)=H_1\oplus H_{-1}. \]
The spectrum of the group \(U_g\) in the invariant subspace \(H_1\) is discrete. The eigenvalues are precisely those characters \(\chi\in \widehat{G}\) for which
\[ \chi(\varepsilon_k)\to 1 \quad\text{and}\quad \chi(\varepsilon_1)=1. \]

We shall determine the structure of the spectrum of the group \(U_g\) in the invariant subspace \(H_{-1}\). If \(y=0,i_1\ldots i_n,\ldots\) is the binary expansion of \(y\in Y\), then, by definition, put
\[ 0i_1\ldots i_n=\{(y,1): y=0,i_1\ldots i_n\ldots\}, \]
\[ 1i_1\ldots i_n=\{(y,-1): y=0,\bar i_1\ldots \bar i_n\ldots\}, \]
where \(\bar i=0\) if \(i=1\), and \(\bar i=1\) if \(i=0\).

Let \(\{\lambda_n\}\) be a sequence of complex numbers such that
\[ |\lambda_n|=1,\qquad \lambda_0=1,\qquad \lambda_{n+1}^2=\lambda_n. \]
Introduce functions \(\varphi_{\lambda_n}\) satisfying the conditions
\[ U_{\varepsilon_n}\varphi_{\lambda_n}=\lambda_n\varphi_{\lambda_n},\qquad \varphi_{\lambda_n}=1\quad\text{for }x\in 0\ldots 0\ (n\text{ times}). \]

Suppose that \(f\in L^2(X,\mu)\) is a normalized eigenfunction of the group \(U_g\):
\[ U_{\varepsilon_n}f=\lambda_n f,\qquad \lambda_{n+1}^2=\lambda_n,\qquad \lambda_0=1,\qquad |f|=1. \]

Then there exist complex numbers \(c_n\), \(|c_n|=1\), such that
\[ c_n\varphi_{\lambda_n}\to f \]
as \(n\to\infty\). From the definition of the functions \(\varphi_{\lambda_n}\) it follows that
\[ \|c_{n+1}\varphi_{\lambda_{n+1}}-c_n\varphi_{\lambda_n}\|^2 = \frac12 |c_{n+1}\lambda_{n+1}-c_n\lambda_1|^2 + \frac12 |c_{n+1}-c_n|^2. \]
Hence \(\lambda_{n+1}\to\lambda_1\), but the sequence \(\{\lambda_n\}\) cannot have limit \(-1\). Thus the eigenfunction \(f\) must be even. Therefore the spectrum of the group \(U_g\) in \(H_{-1}\) is continuous.

Let us prove its simplicity. The vector
\[ h_1(x)=1,\quad\text{if }x\in 0;\qquad h_1(x)=-1,\quad\text{if }x\in 1 \]
is cyclic in \(H_{-1}\). In general, put
\[ h_n(x)=1,\quad\text{if }x\in 0\ldots 0\ (n\text{ times}); \]
\[ h_n(x)=-1,\quad\text{if }x\in 1\ldots 1\ (n\text{ times}); \]
\[ h_n(x)=0\quad\text{at the remaining points}. \]

Suppose that, by linearly combining the shifts \(U_g h_1\), one can obtain the function \(h_n\); then in the same way one can also obtain \(h_{n+1}\). Consider the shifts
\[ U_{\varepsilon_{n+1}}^{\,k}h_n,\qquad k=0,1,\ldots,2^n-1. \]
Note that for \(n>1\)
\[ \operatorname{supp}h_n\cap \operatorname{supp}U_{\varepsilon_{n+1}}h_n = 0\ldots 01\cup 1\ldots 10 \quad (n+1\text{ times}) \]
and on
\[ \underbrace{0\ldots 01}_{n+1}\cup \underbrace{1\ldots 10}_{n+1} \]
the functions \(h_n\) and \(U_{\varepsilon_{n+1}}h_n\) differ in sign;
\[ \operatorname{supp}h_n\cap \operatorname{supp}U_{\varepsilon_{n+1}}^{\,2^n-1}h_n = \underbrace{0\ldots 0}_{n+1}\cup \underbrace{1\ldots 1}_{n+1}, \]
and on the intersection of the supports the functions \(h_n\) and
\[ U_{\varepsilon_{n+1}}^{\,2^n-1}h_n \]
coincide. If, however,
\[ 1<k<2^n-1, \]
then
\[ \operatorname{supp}h_n\cap \operatorname{supp}U_{\varepsilon_{n+1}}^{\,k}h_n=\varnothing. \]

Since \(T_{\varepsilon_{n+1}}\) is an automorphism, we have

\[ \operatorname{supp} U_{\varepsilon_{n+1}}^{k} h_n \cap \operatorname{supp} U_{\varepsilon_{n+1}}^{k+1} h_n = T_{\varepsilon_{n+1}}^{-k}\underbrace{0\ldots 01}_{n+1} \cup T_{\varepsilon_{n+1}}^{-k}\underbrace{1\ldots 10}_{n+1}, \]

and on this intersection \(U_{\varepsilon_{n+1}}^{k}h_n\) and \(U_{\varepsilon_{n+1}}^{k+1}h_n\) differ in sign, while all other shifts \(U_{\varepsilon_{n+1}}^{l}h_n\) are zero on this set. Hence it follows that, if

\[ \sum_{k=0}^{2^n-1} c_k U_{\varepsilon_{n+1}}^{k} h_n=0, \]

then

\[ c_0=c_1=c_2=\cdots=c_{2^n-2}=c_{2^n-1}=-c_0, \]

and therefore \(c_k=0\). Thus the function \(h_{n+1}\) is a linear combination of shifts \(U_{\varepsilon_{n+1}}^{k}h_n\). It remains to note that linear combinations of the functions \(U_{\varepsilon_n}^{k}h_n\) are everywhere dense in \(H_{-1}\).

Put \(\sigma_{-1}(\Delta)=(E(\Delta)h_1,h_1)\), \(\{\lambda_n\}=\{\chi:\chi(\varepsilon_n)=\lambda_n\}\). Simple calculations give

\[ \sigma_{-1}(\{\lambda_n\})=|(\varphi_{\lambda_n},h_1)|^2. \]

From the properties of the functions \(a(y,\varepsilon_k)\) it follows that

\[ (\varphi_{\lambda_n},h_1)=\frac12(1-\lambda_n)(\varphi_{\lambda_n}^{2},h_1). \]

Finally we obtain

\[ \sigma_{-1}(\{\lambda_n\}) = \frac{1}{4^n}\prod_{k=1}^{n}|1-\lambda_k|^2. \]

The sets \(\{\lambda_n\}\), taken together, form a net \(S\) (for the definition of a net see, for example, (4)). The derivative \(d\sigma_{-1}/d\chi\) of the measure \(\sigma_{-1}\) with respect to the net \(S\) is represented in the form

\[ \lim_{n\to\infty}\frac{1}{2^n}\prod_{k=1}^{n}|1-\lambda_n|^2. \]

It follows from this representation that \(d\sigma_{-1}/d\chi=0\) almost everywhere, and hence \(\sigma_{-1}\) is singular.

Thus, the spectral characteristic of the representation \(g\to T_g\) is as follows: the spectrum is simple; the spectral measure \(\sigma\) is represented in the form \(\sigma_1+\sigma_{-1}\); \(\sigma_1\) is a discrete measure concentrated on the subgroup of characters \(\chi\) satisfying the conditions \(\chi(\varepsilon_n)\to 1\), \(n\to\infty\), \(\chi(\varepsilon_1)=1\); \(\sigma_{-1}\) is a continuous singular measure vanishing on the subgroup \(\widehat{G}_1\) of characters for which \(\chi(\varepsilon_1)=1\).

Moreover, \(\sigma_{-1}*\sigma_{-1}\) is a continuous measure concentrated on the subgroup \(\widehat{G}_1\), and, consequently, \(\sigma\) does not dominate \(\sigma*\sigma\).

It follows from this (Theorem 4) that the vectors of the class \(F^{2,2}\) do not form an everywhere dense set in \(L^2(X,\mu)\). Moreover, if \(f\in F^{2,2}\), then (Theorem 3) \(f\) necessarily belongs to \(H_1\).

In conclusion, I express my sincere gratitude to F. A. Berezin, A. A. Kirillov, and Ya. G. Sinai for their attention to this work.

Moscow State University
named after M. V. Lomonosov

Received
23 XI 1965

REFERENCES

1. Ya. G. Sinai, UMN, 18, no. 5 (1963).
2. Ya. G. Sinai, DAN, 150, no. 6, 1235 (1963).
3. Ya. G. Sinai, Theory of Probability and Its Applications, 8, no. 4 (1963).
4. G. E. Shilov, B. L. Gurevich, Integral, Measure and Derivative, Moscow, 1964.