MATHEMATICS

B. M. GUREVICH

ON A CERTAIN CLASS OF SPECIAL AUTOMORPHISMS AND SPECIAL FLOWS

(Presented by Academician A. N. Kolmogorov on 20 VI 1963)

The notion of a \(K\)-system was first introduced by A. N. Kolmogorov in [1]. In the present note we indicate some conditions under which special automorphisms and special flows (see below) are \(K\)-systems. Verification of these conditions in one concrete case (§ 3) makes it possible to obtain a number of examples of \(K\)-flows with finite entropy.

§ 1. Special automorphisms. Let \(T\) be an automorphism of a Lebesgue space \(X\) with measure \(\mu\), and let \(f(x)\), \(x \in X\), be a measurable positive function taking integer values \(a_1, a_2, \ldots\), with
\[ \int_X f(x)\,d\mu < \infty. \]
Denote by \(U\) the set of all integers, and assign to each point \(u \in U\) a measure equal to 1. In the direct product of the spaces with measure \(X \times U\), select the subspace \(\widetilde X\) consisting of those points \((x,u)\) for which \(0 \leq u \leq f(x)-1\). Dividing the measure in \(\widetilde X\) by the number
\[ \int_X f(x)\,d\mu, \]
we obtain a normalized measure. Denote it by \(\widetilde\mu\). Define a transformation \(\widetilde T\) of the space \(\widetilde X\) by the formula
\[ \widetilde T(x,u)= \begin{cases} (x,u+1), & \text{if } u<f(x)-1,\\ (Tx,0), & \text{if } u=f(x)-1. \end{cases} \]
It is not hard to verify that \(\widetilde T\) is an automorphism of the space \(\widetilde X\) with measure \(\widetilde\mu\). It is called a special automorphism, built over the automorphism \(T\) and the function \(f(x)\).

Assume now that \(T\) is a \(K\)-automorphism [1], i.e., there exists a measurable partition \(\xi_0\) of the space \(X\) with the following properties:
\[ 1)\quad T^n\xi_0=\xi_n \geq \xi_{n-1};\qquad 2)\quad \prod_{n=-\infty}^{\infty} T^n\xi_0=\varepsilon;\qquad 3)\quad \bigcap_{n=-\infty}^{\infty} T^n\xi_0=\nu, \]
where \(\varepsilon\) is the partition of the space \(X\) into individual points, and \(\nu\) is the trivial partition, whose only element is all of \(X\). A partition with properties 1)–3) is called a \(K\)-partition corresponding to the \(K\)-automorphism \(T\). Throughout the whole work we shall assume that the function \(f(x)\) is constant mod \(0\) on the elements of the partition \(\xi_0\).

Let \(\gamma_n\) be the partition of the space \(X\) into the preimages of the values of the function
\[ S_n(x)=f(x)+f(Tx)+\cdots+f(T^n x),\qquad n\geq 1. \]
Put \(\eta_n=\gamma_{n-1}\cdot \xi_{-n}\). It is easy to verify that \(\eta_n \leq \eta_{n-1}\). We shall call the natural numbers \(a_1,a_2,\ldots\) relatively prime if their greatest common divisor is equal to 1.

Theorem 1. If the function \(f(x)\) takes relatively prime values and the condition
\[ \bigcap_{n=1}^{\infty}\eta_n=\nu, \tag{*} \]
is fulfilled, then \(\widetilde T\) is a \(K\)-automorphism.

Proof. Let \(\widetilde{\xi}_0\) be the partition of the space \(\widetilde X\) obtained from the partition \(\xi_0\) of the space \(X\) by the following rule: the points \((x_1,u_1)\in \widetilde X\) and \((x_2,u_2)\in \widetilde X\) belong to the same element of the partition \(\widetilde{\xi}_0\) if the points \(x_1\in X\) and \(x_2\in X\) belong to the same element of the partition \(\xi_0\) and \(u_1=u_2\). The partition \(\widetilde{\xi}_0\) is, obviously, measurable. We shall show that, if the hypotheses of the theorem are satisfied, then \(\widetilde{\xi}_0\) is a \(K\)-partition corresponding to the automorphism \(\widetilde T\). Properties 1) and 2) follow at once from the corresponding properties of the partition \(\xi_0\) and the constancy of the function \(f(x)\) on the elements of this partition. Only property 3) requires verification.

Denote by \(\beta\) the partition of the space \(\widetilde X\) into sets \(B_i\) of the form
\[ B_i=\{(x,u): x\in X,\ u=i\}. \]
Since \(\widetilde\mu(B_0)>0\), for each \(n\) one can consider the partition \(\xi'_n\) of the set \(B_0\) formed by the intersections of the elements of the partition \(\widetilde{\xi}_n\) with \(B_0\). Define another sequence \(\{\eta'_n\}\), \(n>0\), of partitions of the set \(B_0\), assigning the points \((x_1,0)\in B_0\) and \((x_2,0)\in B_0\) to the same element of the partition \(\eta'_n\) if and only if \(x_1\) and \(x_2\) belong to the same element of the partition \(\eta_n\). All the constructed partitions are measurable and, for \(n>0\), satisfy the inequalities
\[ \xi'_{-n}\leq \xi'_{-n+1},\qquad \eta'_n\leq \eta'_{n-1}. \]
It is also clear that, by virtue of (*),
\[ \bigcap_{n>0}\eta'_n=\nu^*. \]

We shall prove that
\[ \bigcap_{n<0}\xi'_n=\nu . \tag{1} \]

If \(\sup_{x\in X} f(x)=M<\infty\), then (1) follows from the easily verified inequality
\[ \xi'_{-nM}\leq \eta'_n,\qquad n>0. \]

In the case of an unbounded function \(f(x)\), one can choose a sequence of integers \(k_n\to\infty\) and a sequence of sets \(A_n\subset B_0\) so that
\[ \widetilde\mu(A_n)\xrightarrow[n\to\infty]{}\widetilde\mu(B_0) \]
and on the set \(A_n\) the inequality \(\xi'_{-k_n}\leq \eta'_n\) holds; whence (1) again follows.

To complete the proof of the theorem, we derive from (1), using the mutual independence of the values of the function \(f(x)\), property 3) of the partition \(\widetilde{\xi}_0\). First note that the partition \(\beta\) is finite or countable. Then
\[ \bigcap_n \beta\cdot \widetilde{\xi}_n=\beta\cdot \bigcap_n \widetilde{\xi}_n . \]
Taking this into account, it is not difficult to obtain from (1) the inequality
\[ \bigcap_n \widetilde{\xi}_n\leq \beta . \]

Let \(B\) be the element of the partition \(\bigcap_n \widetilde{\xi}_n\) containing \(B_0\). The partition \(\bigcap_n \widetilde{\xi}_n\) is invariant with respect to the automorphism \(\widetilde T\). Hence, for any \(k\), the set \(\widetilde T^kB\) is an element of the partition \(\bigcap_n \widetilde{\xi}_n\). Since the automorphism \(\widetilde T\) is ergodic\({}^{**}\) and \(\widetilde\mu(B)>0\), we have
\[ d=\min_{k>0}(k:\widetilde T^kB=B)<\infty . \]
Then \(\widetilde T^kB=B \bmod 0\) for \(k=nd\), and
\[ \widetilde T^kB\cap B=\varnothing \bmod 0 \]
for \(k\ne nd,\ n=0,\pm1,\pm2,\ldots\). But

\[ {}^{*}\ \text{The set }B_0\text{ may be regarded as a subspace of the space }X;\ \nu\text{ denotes here the partition of }B_0\text{ whose only element is all of }B_0. \]

\[ {}^{**}\ \text{A special automorphism constructed from an ergodic automorphism and any function is, as is easy to see, ergodic.} \]

\(\mu(\widetilde T^{a_i} B_0 \cap B_0)>0\) for \(i=1,2,\ldots\); hence, \(\widetilde T^{a_i}B=B \bmod 0\), \(i=1,2,\ldots\). It is clear from this that \(d=1\), since the numbers \(a_i\) are relatively prime. Consequently, the set \(B\) is invariant with respect to \(\widetilde T\), and therefore \(\hat\mu(B)=1\), and \(B\) is the only \(\bmod\,0\) element of the partition \(\bigcap \xi_n\).

Let us give an example in which the function \(f(x)\) takes the values 1 and 2, but condition \((*)\) is not satisfied and \(\widetilde T\) is not a \(K\)-automorphism. Let \(T\) be a Bernoulli automorphism \((^3)\) with state space consisting of two points, 0 and 1, of equal measure, and

\[ f(x)= \begin{cases} 1, & \text{if } x_{-1}=1,\ x_0=0 \text{ or } x_0=1,\ x_1=0;\\ 2, & \text{if } x_{-1}=0,\ x_0=0 \text{ or } x_0=1,\ x_1=1. \end{cases} \]

As \(\xi_0\) one may take the partition generated by the values of \(x_i\) for all \(i\leqslant 1\). Setting \(A=\{x:x_0=1,\ x_1=0\}\), it is not difficult to verify that \(\lim_{n\to\infty}\mu(A/\eta_n)=0\) or 1. Hence condition \((*)\) is not satisfied for the function \(f(x)\). The automorphism \(\widetilde T\) is easily realized as a shift in the space of trajectories of a certain Markov chain with six states. This chain has two subclasses, whence it is clear that \(\widetilde T\) is not a \(K\)-automorphism.

At the same time, the automorphism \(\widetilde T_1\), constructed from \(T\) and the function \(f_1(x)=3-f(x)\), is a \(K\)-automorphism, since it is isomorphic to a shift in the space of realizations of a Markov chain with six states having only one subclass. For each \(n>0\), the partition \(\gamma_n^1\) of the space \(X\) into preimages of the values of the function \(S_n^1(x)=f_1(x)+f_1(Tx)+\cdots+f_1(T^n x)\) coincides with \(\gamma_n\), and consequently condition \((*)\) is also not satisfied for the function \(f_1(x)\). This example shows that condition \((*)\) is not necessary in order that \(\widetilde T\) be a \(K\)-automorphism.

§ 2. Special flows. Keeping the assumptions and notation of § 1 concerning the space \(X\) and its automorphism \(T\), consider the special flow \(\{S_t\}\) \((^3)\), constructed from \(T\) and an integrable function \(f(x)\), \(x\in X\), bounded below by a positive constant. The space in which the flow \(\{S_t\}\) acts consists of those points \((x,u)\) of the direct product \(X\times(u)\) of the space \(X\) with the numerical line \((u)\), with the usual Lebesgue measure, for which \(0\leqslant u<f(x)\). By definition, for \(t<\inf_{x\in X} f(x)\),

\[ S_t(x,u)= \begin{cases} (x,u+t), & \text{if } t<f(x)-u,\\ Tx,\ t+u-f(x), & \text{if } t\geqslant f(x)-u. \end{cases} \]

For the remaining \(t\), the automorphism \(S_t\) can be defined from the fact that \(\{S_t\}\) is a group.

We require, as in § 1, that the function \(f(x)\) take a finite or countable number of values and be constant \(\bmod\,0\) on the elements of the partition \(\xi_0\). The notation \(\eta_n\) will then have the same meaning as in § 1.

Positive numbers \(\alpha_1,\alpha_2,\ldots\) will be called incommensurable if the totality of all possible finite sums of the form \(\sum_i n_i\alpha_i\), with integer coefficients \(n_i\), is everywhere dense on the numerical line. For incommensurability it is sufficient, for example, that among the \(\alpha_i\) there are two numbers whose ratio is irrational, or that the set \(\{\alpha_i\}\) contain an infinite bounded subset.

Theorem 2. If the values of the function \(f(x)\) are incommensurable and condition \((*)\) is fulfilled (see § 1), then \(\{S_t\}\) is a \(K\)-flow.

The proof of this theorem is similar to the proof of Theorem 1.

§ 3. Examples. In this section, by \(T\) we shall mean the shift in the space \(X\) of trajectories of a homogeneous ergodic Markov chain
\(Z=\{\ldots,x_{-n},\ldots,x_0,\ldots,x_n,\ldots\}\) with a finite number of states \(\omega_1,\omega_2,\ldots,\omega_s\). Assuming the initial distribution to be stationary, it is easy to show that \(T\) is a \(K\)-automorphism, while the partition \(\xi_0\), generated by the random variables \(x_i,\ i\leq 0\), is a \(K\)-partition. Denote by \(F_0\) the collection of positive functions \(f(x)\), \(x\in X\), measurable with respect to the random variable \(x_0\).

Theorem 3. In order that the special automorphism constructed from \(T\) and any function \(f\in F_0\) with mutually coprime values be a \(K\)-automorphism, it is necessary and sufficient that the Markov chain \(Z\) satisfy the local limit theorem in the Kolmogorov form \((^4)\).

The proof of this theorem is not difficult and rests mainly on the following known facts:

1) The special automorphism constructed from \(T\) and \(f\in F_0\) is the shift in the space of realizations of some stationary Markov chain with a finite number of states.

2) Ergodicity of a Markov chain with a finite number of states consists in the existence of a single essential class of states containing no subclasses.

3) An ergodic Markov chain with a finite number of states satisfies the local limit theorem \((^4)\) if and only if its rank \(((^5), p. 87)\) is equal to the number of states.

Theorem 4. In order that the automorphism \(T\) and any function \(f\in F_0\) satisfy condition \((*)\), it is necessary and sufficient that the Markov chain \(Z\) satisfy the local limit theorem in the Kolmogorov form \((^4)\).

The necessity follows immediately from Theorems 1 and 3. In proving sufficiency, a number of asymptotic formulas derived in \((^4)\) are used.

Comparison of Theorems 2 and 4 leads to the following result.

Theorem 5. If the Markov chain \(Z\) satisfies the local limit theorem in the Kolmogorov form \((^4)\) and the function \(f(x)\in F_0\) takes incommensurable values, then the special flow constructed from \(T\) and \(f\) is a \(K\)-flow.

In conclusion, I take this opportunity to express my gratitude to Ya. G. Sinai, under whose supervision this work was carried out.

Moscow State University
named after M. V. Lomonosov

Received
14 VI 1963

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