EVOLUTION OF THE ERGODIC THEORY
1 Т.V. KILOCHYTSKA
1 T.H. Shevchenko National University “Chernihiv Collegium”
Nauka naukozn. 2019, 4(106): 102-115
Section: Science and technology history
Abstract: 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.
Keywords: ergodic theorem, ergodic process, ergodic theory, dynamical system, invariant measure.
- Bogoliouboff, N. (1931). Sur l’approximation trigonometriques des fonctions dans l’intervalle infini. Proceedings of the USSR Academy of Sciences, 1(2), 23–54 [in Russian].
- Fermi, E. (1923). Beweis dass ein Mechnisches Normalsystem in Allgemeinen Quasi-ergodisch ist. Phys. Zs., 24, 261–265.
- Krylov, N.M. & Bogolyubov N.N. (1934). Applications of methods of non-linear mechanics to the theory of stationary vibrations. Kyiv: All-Ukrainian Academy of Sciences, 108 [in Russian].
- Kryloff, N. & Bogoliouboff, N. (1937). La théorie générale de la mesure dans son applications a l’étude des système dynamiques de la mécanique non linéaire. Ann. Math., 38, 65–113. https://doi.org/10.2307/1968511
- Krylov, M.M., Boholiubov, M.M. (1937). The general theory of measure in non-linear mechanics. Collection of works on non-linear mechanics. Kyiv: the USSR Academy of Sciences, 55–112 [in Ukrainian].
- Kolmogorov, A.N. (1938). A simplified proof of the ergodic Birgof – Klinchin theorem. Advances of mathematical sciences, 5, 52–56 [in Russian].
- Kolmogorov, A.N. (1958). A new metric invariant of transit dynamic systems and automorphisms of the Lebed space. Reports of the USSR Academy of Sciences, 119(5), 861–864 [in Russian].
- Kolmogorov, A.N. (1959). Entropy per time unit: a metric invariant of automorphisms. Reports of the USSR Academy of Sciences, 124(4), 754–755 [in Russian].
- Sinay, Ya. G. (1959). The notion of the dynamic system’s entropy. Reports of the USSR Academy of Sciences, 124/4, 768–771 [in Russian]
- Abramov, L.M. & Sinay, Ya.G. (1959). A seminar devoted to the metric theory of dynamic system of Moscow State University, supervised by V.A. Rokhlin. Advances of mathematical sciences, 14/6(90), 223–225 [in Russian].
- Rokhlin, V.A. (2010). Selected works. Supplements to the biography. MTsNMO [in Russian].
- Sinay, Ya.G. (1963). Justification of the ergodic hypothesis for one dynamic system of the statistical mechanics. Reports of the USSR Academy of Sciences, 153(6), 1261–1264 [in Russian].
- Sinay, Ya.G. (1966). Classical dynamic systems with the even Lebedev spectrum. II. Proceedings of the USSR Academy of Sciences. Series: Mathematics, 30(1), 1568 [in Russian].
- Sinay, Ya.G. (1970). Dynamic systems with elastic reflections. Advances of mathematical sciences, 25(4), 141–192 [in Russian].
- Ornsteyn, D. (1978). The ergodic theory, randomness and dynamic systems. Moscow: Mir. [in Russian].
- Adler, R.L., Konheim, A.G. & Andrew, Мс. (1965). Topological entropy. Мс. Andrew – Trans. AMS., 114, 309–319. https://doi.org/10.1090/S0002-9947-1965-0175106-9
- Riecan, В. (1974). Abstract entropy. Acta F.R.N. Univ. Comen. – Mat., 55–67.
- Otokar, Grošek. (1979). Entropy on algebraic structures. Mathematica Slovaca. 29 (4), 411–424 [in Russian].
- Bratteli, O. (1972). Inductive limits of finite-dimensional C*-algebras. Trans. Am. Math. Soc., 171, 195–234. https://doi.org/10.2307/1996380
- Vershik, A. M. (1982). The theorem on Markov periodic approximation in the egrodic theory. Proceedings of scientific seminars of Leningrad Optical Mechanical Institute, 115, 72–82 [in Russian].
- Herman, R.H., Putnam, I. & Skau, C. (1992). Ordered Bratteli diagrams, dimension groups, and topological dynamics. Int. J. Math., 3, 827–864. https://doi.org/10.1142/S0129167X92000382
- Medynets, K. (2006). Cantor aperiodic systems and Bratteli diagrams. Comptes Rendus Mathematique, 342, issue 1, 43–46. https://doi.org/10.1016/j.crma.2005.10.024
- Oxtoby, J.C. & Ulam, S.M. (1941). Measure preserving homeomorphisms and metrical transitivity. Ann. Math. (2), 42, 874–920. https://doi.org/10.2307/1968772
- Alpern, S. & Prasad, V.S. (2000). Typical Dynamics of Volume Preserving Homeomorphisms. Cambridge: Cambridge University Press, 240. https://doi.org/10.1017/CBO9780511543180
- Navarro-Bermudez, F.J. (1979). Topologically equivalent measures in the Cantor space. Proc. Am. Math. Soc., 77, 229–236. https://doi.org/10.2307/2042644
- Akin, E., Dougherty, R., Mauldin, R.D. & Yingst, A. (2008). Which Bernoulli measures are good measures? Colloq. Math., 110, 243–291. https://doi.org/10.4064/cm110-2-2
- Austin, T.D. (2007). A pair of non-homeomorphic product measures on the Cantor set. Mathematical Proceedings of the Cambridge Philosophical Society, 142, 103–110. https://doi.org/10.1017/S0305004106009741
- Giordano, T., Putnam, I. & Skau, C. (1995). Topological orbit equivalence and C*-crossed products. Journal für die reine und angewandte Mathematik, 469, 51–112. https://doi.org/10.1515/crll.1995.469.51
- Durand, F., Host, B. & Skau, C. (1999). Substitutional dynamical systems. Bratteli diagrams and dimension groups. Ergodic Theory and Dynamical Systems, 19, 953–993. https://doi.org/10.1017/S0143385799133947
- Bezuglyi, S., Kwiatkowski, J. & Medynets, K. (2009). Aperiodic substitution systems and their Bratteli diagrams. Ergodic Theory and Dynamical Systems, 29(1), 37–72. https://doi.org/10.1017/S0143385708000230
- Akin, E. (2005). Good Measures on Cantor space. Transactions of the American Mathematical Society, 357(7), 2681–2722. https://doi.org/10.1090/S0002-9947-04-03524-X
- Karpel, O. (2012). Infinite measures on Cantor spaces. Journal of Difference Equations and Applications, 18(4), 703–720. https://doi.org/10.1080/10236198.2011.620955
- Karpel, O. (2012). Good measures on locally compact Cantor sets. J. Math. Phys. Anal. Geom., 8(3), 260–279.
- Bezuglyi, S. & Karpel, O. (2014). Orbit Equivalent Substitution Dynamical Systems and Complexity. Proceedings of the American Mathematical Society, 142, 4155–4169. https://doi.org/10.1090/S0002-9939-2014-12139-3
- Bezuglyi, S., Karpel, O. & Kwiatkowski, J. (2015). Subdiagrams of Bratteli diagrams supporting finite invariant measures. J. Math. Phys. Anal. Geom., 11(1), 3–17. https://doi.org/10.15407/mag11.01.003