To the history of the appearance of the notion of $\epsilon$-entropy of an authomorphism of a~Lebesque space and $(\varepsilon,T)$-entropy of a~dynamical system with continuous time

D. Z. Arov

Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXV, Tome 436 (2015), pp. 76-100

Résumé

The article is devoted to the master thesis on “information theory” written by the author in 1956–57. The topic was suggested by his advisor A. A. Bobrov (a student of A. Ya. Khinchin and A. N. Kolmogorov), and the thesis was written under the influence of lectures by N. I. Gavrilov (a student of I. G. Petrovskii) on the qualitative theory of differential equations, which included the statement of Birkhoff's theorem for ergodic dynamical systems. In the thesis, the author used the concept of Shannon entropy in the study of ergodic dynamical systems $(f(p,t),p)$ in a separable compact metric space $R$ with an invariant measure $\mu$ (where $\mu(R)=1$) and introduced the concept of the $(\varepsilon,T)$-entropy of a system as a quantitative characteristics of the degree of mixing. In the work, not only partitions of $R$ were considered, but also partitions of the interval $(-\infty,\infty)$ into subintervals of length $T>0$. In particular, $f(p,T)$ was considered as an automorphism $S$ of $X=R$, and the $(\varepsilon,T)$-entropy is essentially the $\varepsilon$-entropy of $S$. But, despite some “oversights” in the definition of the $(\varepsilon,T)$-entropy and many years that have passed, the author decided to publish the corresponding chapter of the thesis in connection with the following: 1) There is a number of papers that refer to this work in the explanation of the history of the concept of Kolmogorov's entropy. 2) Recently, B. M. Gurevich obtained new results on the $\varepsilon$-entropy $h_\varepsilon(S)$, which show that for two ergodic automorphisms with equal finite entropies their $\varepsilon$-entropies also coincide for all $\varepsilon$, but, on the other hand, there are unexpected nonergodic automorphisms with equal finite entropies, but different $\varepsilon$-entropies for some $\varepsilon$. This shows that the concept of $\varepsilon$-entropy is of scientific value.