Evolution of the Ergodic Theory
DOI:
https://doi.org/10.15407/sofs2019.04.102Keywords:
ergodic theorem, ergodic process, ergodic theory, dynamical system, invariant measureAbstract
The article provides a historical reconstruction of the origin, formation and development of the ergodic theory in the global context.
In 30s of the past century the applied tasks assisted formulation of the theory of nonlinear fluctuations, forming of bases of ergodic theory. G. Birkhoff laid the beginning to the concept of the dynamic system. In 1930s ideas of H. Poincaré laid the beginning of the ergodic theory (G. Birkhoff, John von Neumann, M. Krylov, M. Bogolubov).
The article describes the historical sequence of becoming and development of the ergodic theory in Ukraine (world context). In 1950–1970, the theory of the dynamical systems was rapidly developing. The ergodic theory develops as cleanly mathematical theory within the framework of the general theory of the dynamical systems and studies of transformation with an invariant measure.
The article shows ways of developing a set of concepts and ideas that resulted in creating of the ergodic theory (for example, in 1934 M. Bogolubov and M. Krylov introduced the concept of integral manifold, put beginning to asymptotic theory of nonlinear mechanics, in 1958–1959 А. Коlmogorov introduced two fundamental concepts – К-system and dynamical entropy, in 1959 Yakov Sinai developed a concept of entropy, in 1965 three American mathematicians put beginning to the concept of topological entropy).
In the article, opening is considered from the ergodic theory of the dynamic systems of А. Коlmogorov and his followers. Dynamical entropy is well-known as entropy of Коlmogorov – Sinai entropy (1959).
Numerous works are analyzed, published since 1979, in which the problem of classification of measures was examined on Cantor sets. Probability measures were studied in these works only, majority of results touched the case of Bernoulli measures.
Тhe article shows the contribution of Ukrainian scientists in formation of ergodic theory. Works of ergodic theory of M. Bogolubov and M. Krylov are analyzed. A review of works of the Kharkiv school, devoted to the ergodic theory, is made. Expansion of research directions, research of the dynamical systems operating in spaces of various nature are a modern tendency.
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