Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-25T08:11:21.344Z Has data issue: false hasContentIssue false
  • English
  • Français

Quasi-Flows*

Published online by Cambridge University Press:  22 January 2016

Izumi Kubo
Izumi Kubo*
Affiliation:
Nagoya University
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The purpose of this paper is to investigate a quasi-flow which is a one-parameter group of non-singular measurable point transformations on a measure space. If, in particular, the transformations are all measure preserving (i.e. a flow is given), the ergodicity together with the mixing property, the spectral or metrical type, increasing partitions of the space and the entropy of the flow are our main interests. Those methods used in the study of a flow are frequently useful for our approach. For example, the concept of a special flow introduced by W. Ambrose [3] plays an important role and the representation of a given flow in terms of a special flow is a powerful tool in the study of flows. L.M. Abramov [1] calculates the entropy of a flow with the help of the representation. As another example we give attention to the work of G. Maruyama [10] and H. Totoki [15] where they discuss a general time-change of flows the basic idea of which was originated by E. Hopf [8]. They discuss the invariant measure of a general time-changed flow and prove that the ergodicity is inherited and the entropy is kept invariant by the time-change. In the study of quasi-flows, we shall use both the repesentation in terms of a special quasi-flow and a time-change. Besides a quasi-flow requires its own methods in the investigation and it gives us some further problems such as the existence of an invariant measure (cf. [4], [5], [6] and [7]) and related topics.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 35 , July 1969 , pp. 1 - 30
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1969

Footnotes

*

The results in this paper were presented at the Meeting of Mathematical Society of Japan on October 10, 1967.

References

[1] Abramov, L.M. On the entropy of a flow. Dokl. Akad. Nauk SSSR. 128 (1959), 873–875.Google Scholar
[2] Arnold, L.K. On σ-finite invariant measures. Doctorial Thesis Brown Univ.Google Scholar
[3] Ambrose, W. Representation of ergodic flows. Ann. of Math. 42 (1941), 723–729.CrossRefGoogle Scholar
[4] Ambrose, W. and Kakutani, S. Structure and continuity of measurable flow. Duke Math. J. 9 (1942), 25–42 CrossRefGoogle Scholar
[5] Dowker, Y.N. Finite and σ-finite invariant measures. Ann. of Math. 54 (1951), 595–608.CrossRefGoogle Scholar
[6] Halmos, P. Invariant measures. Ann. of Math. 48 (1947), 735–754.Google Scholar
[7] Hopf, E. Theory of measures and invariant integrales. Trans. AMS 34 (1932), 373–393.Google Scholar
[8] Hopf, E. Ergodentheorie. Leipzig (1937).CrossRefGoogle Scholar
[9] Krengel, U. Darstellungssätze für Strömungen und Halbströmungen II. (Preprint).Google Scholar
[10] Maruyama, G. Transformations of flows. J. Math. Soc. Japan, 18 (1966), 290–302.Google Scholar
[11] Rohlin, V.A. New progress in the theory of transformations with invariant measure. Russian Math. Surveys 15 No. 4 (1960), 1–22.Google Scholar
[12] Rohlin, V. A. On the fundamental ideas of measure theory. Amer. Math. Soc. Transl. Ser. 1, 10 (1961), 1–54.Google Scholar
[13] Sinai, Ya.G. Probabilistic ideas in ergodic theory. Amer. Math. Soc. Transl. Ser. 2, 31 (1963), 62–84.Google Scholar
[14] Sinai, Ya.G. II. Izv. Akad Nauk SSSR Ser. Mat. 30 No. 1 (1966), 15–68.Google Scholar
[15] Totoki, H. Time changes of flows. Mem. Fac. Sci. Kyushu Univ. 20 (1966), 27–55.Google Scholar
[16] Wiener, N. The ergodic theorem. Duke Math. J. 5 (1939), 1–18.CrossRefGoogle Scholar