Mathematics

M. S. Pinsker

DYNAMICAL SYSTEMS WITH COMPLETELY POSITIVE AND ZERO ENTROPY*

(Presented by Academician A. N. Kolmogorov on 6 IV 1960)

Let in a Lebesgue space \(M\) with Boolean algebra \(\alpha\) of measurable subsets of \(M\) and measure \(\mu(\cdot)\), defined on \(\alpha\), there be given a dynamical system \(\{S_t\}\), i.e., a one-parameter group of automorphisms of the space \(M\) (see (1)). If \(\xi\) is a measurable partition of \(M\), invariant with respect to \(\{S_t\}\), then in the factor space \(M|\xi\) there is induced a dynamical system \(\{S'_t\}\), called a factor-system of the system \(\{S_t\}\).

Definition 1. A dynamical system \(\{S_t\}\) is called a system with completely positive entropy if every nontrivial factor-system has positive entropy (for the entropy of dynamical systems see (2–4)).

Definition 2. A dynamical system \(\{S_t\}\) is called regular in the sense of A. N. Kolmogorov (2), if there exists a closed subalgebra \(\alpha_0\) of the algebra \(\alpha\) whose shifts \(\alpha_t=S_t\alpha_0\) have the following properties:

\[ \alpha_t \subseteq \alpha_{t'} \quad \text{for } t \leq t'; \tag{1} \]

\[ \bigvee_t \alpha_t = \alpha; \tag{2} \]

\[ \bigwedge_t \alpha_t = \mathfrak{N}; \tag{3} \]

\(\mathfrak{N}\) is the trivial subalgebra of the Boolean algebra \(\alpha\).

Definition 3. A dynamical system \(\{S_t\}\) is called singular in the sense of Kolmogorov if every closed subalgebra \(\alpha_0\) satisfying conditions (1) and (2) coincides with \(\alpha\).

Definition 4. A closed subalgebra \(\alpha_0\) of the algebra \(\alpha\) is called a generating one for the dynamical system \(\{S_t\}\) if

\[ \bigvee_t \alpha_t=\alpha. \]

Definition 5. Factor-systems \(\{S'_t\}\) and \(\{S''_t\}\) of the dynamical system \(\{S_t\}\), defined on the factor spaces \(M|\xi'\) and \(M|\xi''\), respectively, are called mutually independent if, for any \(K'\in\xi'\), \(K''\in\xi''\),** one has
\[ \mu(K'\cdot K'')=\mu(K')\mu(K''). \]

We introduce notation. For finite subalgebras \(\beta_0\) of the algebra \(\alpha\) and \(h>0\), set
\[ \beta^{(h)}=\bigvee_n S_{nh}\beta_0,\qquad \bar{\beta}^{(h)}=\bigwedge_n \bigvee_{m<n} S_{mh}\beta_0 \]
and
\[ \bar{\alpha}=\bigvee_{\substack{h>0\\ \beta_0\subseteq\alpha}} \bar{\beta}^{(h)}, \]
and let \(\{\bar{S}_t\}\) be the factor-system of the system \(\{S_t\}\) acting in the factor space \(M|\bar{\alpha}\); \(\{S_n(\beta^{(h)})\}\) and \(\{S_n(\bar{\beta}^{(h)})\}\) are factor-systems of the system \(\{S_{nh}\}\), generated by auto-

* The terminology used in the present note was proposed by A. N. Kolmogorov, V. A. Rokhlin, and Ya. G. Sinai.
** We denote a partition of the space \(M\) and the \(\sigma\)-algebra generated by this partition by the same Greek letter.

by the morphism \(S_h\), acting in the factor spaces \(M \mid \beta^{(h)}\) and \(M \mid \bar{\beta}^{(h)}\), respectively.

Theorem 1. Any factor system of a dynamical system \(\{S_t\}\) with completely positive or zero entropy is a system with completely positive or zero entropy, respectively.

The proof of this theorem follows directly from the corresponding definitions.

Theorem 2. In order that \(\{S_t\}\) be a dynamical system with completely positive or zero entropy, the following condition is necessary and sufficient.

For any finite subalgebra \(\beta_0\) and any admissible \(h>0\), the system \(\{S_n(\beta^{(h)})\}\) is, respectively, regular or singular in the sense of Kolmogorov.

Theorem 3. A dynamical system regular in the sense of Kolmogorov has completely positive entropy; a dynamical system with zero entropy is singular in the sense of Kolmogorov.

Corollary. A dynamical system \(\{S_t\}\) with a finite generator is regular or singular in the sense of Kolmogorov if and only if \(\{S_t\}\) is a system with completely positive or zero entropy, respectively.

Theorem 4. The factor system \(\{\bar S_t\}\) has zero entropy. Any factor system with zero entropy is a factor system of \(\{\bar S_t\}\).

Theorem 5. The factor systems \(\{S'_t\}\) and \(\{\bar S_t\}\), with completely positive and zero entropy, respectively, are mutually independent.

Here the following questions arise: do maximal factor systems with completely positive entropy exist for every dynamical system, and can every ergodic dynamical system be decomposed into a direct product of independent factor systems with completely positive or zero entropy?

Theorem 6. A dynamical system generated by a multidimensional stationary Gaussian random process with absolutely continuous spectral functions is a system with completely positive entropy. A dynamical system generated by a multidimensional stationary random process with singular spectral functions has zero entropy.

Received
7 IV 1960

CITED LITERATURE

1. V. A. Rokhlin, UMN, 4, No. 2 (30), 57 (1949).
2. A. N. Kolmogorov, DAN, 119, No. 5, 861 (1958).
3. Ya. G. Sinai, DAN, 124, No. 4, 768 (1959).
4. A. M. Abramov, DAN, 128, No. 5, 873 (1959).