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MATHEMATICS
V. A. Rokhlin and Ya. G. Sinai
CONSTRUCTION AND PROPERTIES OF INVARIANT MEASURABLE PARTITIONS
(Presented by Academician A. N. Kolmogorov on 18 VII 1961)
1. Terminology and notation.
We shall use the terminology of the survey article (1), which describes the current state and problems of the theory of transformations with invariant measure.
\(M\) denotes a Lebesgue space; \(\varepsilon\) is the partition of the space \(M\) into individual points; \(\nu\) is the trivial partition with the single element \(M\); the notation \(\xi \leqslant \eta\), where \(\xi\) and \(\eta\) are partitions, means that \(\eta\) is a subpartition of \(\xi\); \(\prod\) and \(\bigcap\) denote the product and intersection of measurable partitions; \(H(\xi)\) is the entropy of the measurable partition \(\xi\); \(H(\xi \mid \eta)\) is the mean conditional entropy of the measurable partition \(\xi\) with respect to the measurable partition \(\eta\) (for the definition and properties of the function \(H(\xi \mid \eta)\), see (2)); \(Z\) is the set of partitions \(\xi\) with \(H(\xi) < \infty\); \(T\) is an automorphism of the space \(M\); \(\xi_T^n\) (briefly \(\xi^n\)), \(\xi_T^{-}\) (briefly \(\xi^{-}\)) and \(\xi_T\) are partitions defined from the partition \(\xi\) and the automorphism \(T\) by the formulas
\[ \xi_T^n=\prod_{k=0}^{n-1} T^k \xi,\qquad \xi_T^-=\prod_{k=1}^{\infty} T^{-k}\xi,\qquad \xi_T=\prod_{k=-\infty}^{\infty} T^k \xi; \]
\(h(T,\xi)\) and \(h(T)\) are functions defined by the formulas
\[ h(T,\xi)=H(\xi\mid \xi_T^-),\qquad h(T)=\sup h(T,\xi)\quad (\xi\in Z); \]
\(h(T)\) is the entropy of the automorphism \(T\).
M. S. Pinsker showed (3) that every automorphism has a largest factor-automorphism with zero entropy; in other words, for every automorphism \(T\) there exists a measurable partition \(\pi(T)\) such that, for \(\xi\in Z\), the condition \(\xi \leqslant \pi(T)\) is equivalent to the equality \(h(T,\xi)=0\). If \(\pi(T)=\nu\), then \(T\) is called an automorphism with completely positive entropy. \(T\) is called a \(K\)-automorphism, or a Kolmogorov automorphism, if there exists a measurable partition \(\zeta\) with three properties: \(T\zeta\geqslant \zeta\),
\[ \prod_n T^n \zeta=\varepsilon,\qquad \bigcap_n T^n \zeta=\nu. \]
2. Statement of results.
Theorem 1. For every automorphism \(T\) there exists a measurable partition \(\zeta\) with four properties: a) \(T\zeta\geqslant \zeta\); b) \(\prod_k T^k\zeta=\varepsilon\); c) \(\bigcap_k T^k\zeta=\pi(T)\); d) \(H(T\zeta/\zeta)=h(T)\).
Theorem 2. If a measurable partition \(\zeta\) has, with respect to an automorphism \(T\), properties a), b), then \(\bigcap_n T^n\zeta\geqslant \pi(T)\). If \(\zeta\) has properties a), d) and \(h(T)<\infty\), then \(\bigcap_n T^n\zeta\leqslant \pi(T)\). Thus, if \(h(T)<\infty\), then c) follows from a), b), d).
Corollary 1. The class of automorphisms with completely positive entropy coincides with the class of \(K\)-automorphisms.
Corollary 2. The factor automorphism of a \(K\)-automorphism is a \(K\)-automorphism.
Corollary 3. An automorphism inverse to a \(K\)-automorphism is a \(K\)-automorphism.
Corollary 4. An automorphism generated by a stationary Gaussian sequence with absolutely continuous spectrum is a \(K\)-automorphism.
Corollary 5. The unitary operator \(U\) conjugate to an automorphism \(T\) has, in the orthogonal complement to the subspace of functions with integrable square of the modulus that are constant mod \(0\) on the elements of the partition \(\pi(T)\), a countably multiple Lebesgue spectrum.
Theorems 1 and 2 are proved below. Here we shall derive from them Corollaries 1–5. The fact that a \(K\)-automorphism has completely positive entropy was proved by M. S. Pinsker \((^3)\); this also follows from Theorem 2. The converse proposition follows from Theorem 1. Corollaries 2 and 3 follow in an obvious way from Corollary 1. Corollary 4 follows from Corollary 1 and from the fact that, in the case under consideration, the automorphism has completely positive entropy (see \((^3)\)). Corollary 5 follows from the fact that the orthogonal complement of a unitary subring in a unitary ring is countably infinite-dimensional (see \((^5)\)).
3. Three lemmas. We shall denote measurable partitions by \(\alpha,\beta,\gamma\).
Lemma 1. If \(\beta \preccurlyeq \alpha\) and \(H(\alpha\mid \beta^{-})<\infty\), or \(\alpha \preccurlyeq \beta\) and \(H(\beta\mid \alpha^{-})<\infty\), then
\[ \lim_{n\to\infty}\frac{1}{n}H(\alpha^n\mid \beta^{-})=H(\alpha\mid \alpha^{-}). \tag{1} \]
Proof. In the first case the sequence of partitions \(T^{-n}(\beta^{-}\alpha^n)\) converges, increasing, to \(\alpha^{-}\), so that \(H(\alpha\mid T^{-n}(\beta^{-}\alpha^n))\to H(\alpha\mid \alpha^{-})\). Applying the method of arithmetic means and using the relation
\[ H(\alpha^n\mid \beta^{-})=\sum_{k=1}^{n} H(\alpha\mid T^{-k}(\beta^{-}\alpha^n)), \]
we obtain (1). In the second case
\[ \lim_{n\to\infty}\frac{1}{n}H(\alpha^n\mid \beta^{-}) =\lim_{n\to\infty}\left[\frac{1}{n}H(\beta^n\mid \beta^{-})-\frac{1}{n}H(\beta^n\mid \alpha^n\beta^{-})\right]\ge \]
\[ \ge H(\beta\mid \beta^{-})-\lim_{n\to\infty}\frac{1}{n}H(\beta^n\mid \alpha^n\alpha^{-}) =\lim_{n\to\infty}\left[\frac{1}{n}H(\beta^n\mid \alpha^{-}\alpha^{-1})-\frac{1}{n}H(\beta^n\mid \alpha^n\alpha^{-})\right]= \]
\[ =\lim_{n\to\infty}\frac{1}{n}H(\alpha^n\mid \alpha^{-})=H(\alpha\mid \alpha^{-}), \]
and the reverse inequality is obvious.
Lemma 2. If \(\alpha\preccurlyeq \beta\) and \(H(\beta\gamma\mid \beta^{-})<\infty\), then
\[ \lim_{n\to\infty}H(\alpha\mid \beta^{-}T^{-n}\gamma^{-})=H(\alpha\mid \beta^{-}). \tag{2} \]
Proof. Since
\[ \sum_{k=0}^{n-1} H(\beta\mid \beta^{-}T^{-k}\gamma^{-})=H(\beta^n\mid \beta^{-}\gamma^{-}), \]
for \(\alpha=\beta\) formula (2) follows from Lemma 1 and the theorem on arithmetic means. In the general case,
\[ \lim_{n\to\infty}H(\alpha\mid \beta^{-}T^{-n}\gamma^{-}) =\lim_{n\to\infty}H(\beta\mid \beta^{-}T^{-n}\gamma^{-}) -\lim_{n\to\infty}H(\beta\mid \alpha\beta^{-}T^{-n}\gamma^{-})\ge \]
\[ \ge H(\beta\mid \beta^{-})-H(\beta\mid \alpha\beta^{-})=H(\alpha\mid \beta^{-}), \]
and the reverse inequality is obvious.
Lemma 3 \((^4)\). If \(H(\alpha\beta\mid \beta^{-})<\infty\), then
\[ H(\alpha\beta\mid \alpha^{-}\beta^{-})=H(\alpha\mid \alpha^{-}\beta_T)+H(\beta\mid \beta^{-}). \tag{3} \]
Proof. Since \(H(\alpha\mid \alpha^{-}\beta^{-}\beta^n)\to H(\alpha\mid \alpha^{-}\beta_T)\) and
\[ H(\alpha^n\mid \alpha^{-}\beta^{-}\beta^n) = \sum_{k=0}^{n-1} H(\alpha\mid \alpha^{-}\beta^{-}\beta^k), \]
we have
\[ \frac1n H(\alpha^n\mid \alpha^{-}\beta^{-}\beta^n)\to H(\alpha\mid \alpha^{-}\beta_T). \]
On the other hand, by Lemma 1,
\[ \begin{aligned} \lim_{n\to\infty}\frac1n H(\alpha^n\mid \alpha^{-}\beta^{-}\beta^n) &= \lim_{n\to\infty} \left[ \frac1n H(\alpha^n\beta^n\mid \alpha^{-}\beta^{-}) - \frac1n H(\beta^n\mid \alpha^{-}\beta^{-}) \right] \\ &= H(\alpha\beta\mid \alpha^{-}\beta^{-})-H(\beta\mid \beta^{-}). \end{aligned} \]
4. Proof of Theorem 1. Let \(\xi_1,\xi_2,\ldots\) be an increasing sequence of partitions from \(Z\), converging to \(\varepsilon\), and let \(n_1,n_2,\ldots\) be a sequence of integers. Put
\[ \eta_p=\prod_{k=1}^{p} T^{-n_k}\xi_k,\qquad \eta=\prod_{k=1}^{\infty} T^{-n_k}\xi_k,\qquad \zeta=\eta^{-}. \]
It is clear that \(\zeta\) has properties a) and b). We shall show that if the sequence \(n_1,n_2,\ldots\) increases sufficiently rapidly, then
\[ \lim_{p\to\infty}\left[H(\eta_p\mid \eta_p^{-})-H(\eta_p\mid \zeta)\right]=0, \tag{4} \]
and that, if this relation is satisfied, then \(\zeta\) has properties c) and d).
Relation (4) will certainly be satisfied if we subject the choice of the numbers \(n_1,n_2,\ldots\) to the conditions
\[ H(\eta_p\mid \eta_{q-1}^{-})-H(\eta_p\mid \eta_q^{-}) < \frac1p\,\frac1{2^{q-p}} \qquad (p<q). \tag{5} \]
Indeed, from (5) it follows that
\[ H(\eta_p\mid \eta_p^{-})-H(\eta_p\mid \eta^{-})<\frac1p,\qquad H(\eta_p\mid \eta_q^{-})-H(\eta_p\mid \zeta)\leq \frac1p. \]
The choice can be made inductively: if the numbers \(n_1,\ldots,n_{q-1}\) have been chosen, then \(n_q\) is chosen so large that the inequalities (5) are satisfied for \(p=1,\ldots,q-1\). This is possible by Lemma 2.
Since the sequence \(\xi_1,\xi_2,\ldots\) increases and converges to \(\varepsilon\), we have
\[ H(\eta_p\mid \eta_p^{-})=h(T,\eta_p)=h(T,\xi_p)\to h(T). \]
Since the sequence \(\eta_1,\eta_2,\ldots\) increases and converges to \(\eta\), we have
\[ H(\eta_p\mid \zeta)\to H(\eta\mid \zeta)=H(T\zeta\mid \zeta). \]
Therefore d) follows from (4).
Let us show that c) follows from (4). Let \(\alpha\in Z\) and \(\alpha\leq \bigcap_n T^n\zeta\). Then \(\eta_p^{-}\alpha_T^{-}\leq \zeta\). Decomposing \(H(\eta_p\alpha\mid \eta_p^{-}\alpha^{-})\) in two ways by formula (3), we obtain
\[ h(T,\alpha)=H(\alpha\mid \alpha^{-}) = H(\alpha\mid \alpha^{-}(\eta_p)_T) + H(\eta_p\mid \eta_p^{-}) - H(\eta_p\mid \eta_p^{-}\alpha_T). \tag{6} \]
Since the sequence \((\eta_1)_T,(\eta_2)_T,\ldots\) increases and converges to \(\varepsilon\), we have
\[ H(\alpha\mid \alpha^{-}(\eta_p)_T)\to 0, \]
while the difference
\[ H(\eta_p\mid \eta_p^{-})-H(\eta_p\mid \eta_p^{-}\alpha_T) \]
does not exceed the difference (4) and therefore also tends to zero. Consequently,
\[ h(T,\alpha)=0,\qquad \alpha\leq \pi(T) \]
and
\[ \bigcap_n T^n\zeta\leq \pi(T). \]
The reverse inequality follows from Theorem 2.
5. Proof of Theorem 2. Put
\[ \zeta_0=\bigcap_n T^n\zeta, \]
and let \(\eta\in Z\), \(\eta\leq \pi(T)\). Further, let \(\xi\) be a partition from \(Z\) satisfying, for some \(m\), the inequality \(\xi\leq T^m\zeta\). We shall show that
\[ H(\xi\mid \zeta_0\eta_T)=H(\xi\mid \zeta_0). \]
By condition b), this will prove that \(\zeta_0\eta_T=\zeta_0\), i.e. that \(\eta\leq \zeta_0\), and hence it will be proved that
\[ \pi(T)\leq \zeta_0. \]
For any natural \(p\),
\[ H(\xi \mid \zeta_0) \geq H(\xi \mid \zeta_0\eta_T) \geq H(\xi \mid \xi_{\tau^p}^{-}\zeta_0\eta_T) \tag{7} \]
(where \(\xi_{\tau^p}^{-}=\prod_{k=1}^{\infty} T^{-pk}\xi\)). It is clear that \(T\xi_0=\xi_0\), and since \(\eta \ll \pi(T)\), we have \(\eta_T=\eta^{-}=T^{pn}\eta^{-}\) (for any \(n\)). This makes it possible to apply Lemma 2 to the right-hand side of formula (7), with \(T^p\) in the role of \(T\) (one must put \(\alpha=\xi,\ \beta=\xi\xi_0,\ \gamma=\prod_{k=1}^{p}T^{-kn}\eta\)). The result is:
\[ H(\xi \mid \xi_{\tau^p}^{-}\zeta_0\eta_T)=H(\xi \mid \xi_{\tau^p}^{-}\zeta_0). \]
Since the sequence of partitions \(\xi_{\tau^p}^{-}\zeta_0\) decreases and converges to \(\zeta_0\) as \(p\to\infty\), it follows that \(H(\xi\mid \xi_{\tau^p}^{-}\zeta_0)\to H(\xi\mid \zeta_0)\). Consequently,
\[ H(\xi\mid \zeta_0\eta_T)=H(\xi\mid \zeta_0). \]
The proof of the second part of the theorem is close to the arguments of the preceding paragraph. Let \(\alpha\in Z\) and \(\alpha \ll \bigcap_n T^n\xi\). Choose in \(Z\) an increasing sequence \(\eta_1,\eta_2,\ldots\) converging to \(\xi\), and again write formula (6). Since \(\alpha\ll \xi\), we have \(H(\alpha\mid \eta_p)\to0\), and since \(H(\alpha\mid \alpha^{-}(\eta_p)_T)\leq H(\alpha\mid \eta_p)\), we also have \(H(\alpha\mid \alpha^{-}(\eta_p)_T)\to0\). The difference \(H(\eta_p\mid \eta_p^{-})-H(\eta_p\mid \eta_p^{-}\alpha_T)\) does not exceed the difference \(h(T)-H(\eta_p\mid \xi^{-})\), as \(p\to\infty\), which converges to
\[ h(T)-H(\xi\mid \xi^{-})=h(T)-H(T\xi\mid \xi)=0. \]
Therefore, \(h(T,\alpha)=0,\ \alpha\ll \pi(T)\), and \(\bigcap_n T^n\xi\ll \pi(T)\).
Moscow State University
named after M. V. Lomonosov
Received
18 VII 1961
REFERENCES
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