Advertisement

Flows and Their Representations

Chapter
  • 955 Downloads
Part of the Texts and Readings in Mathematics book series (volume 6)

Abstract

Let (X, Ɓ) be a standard Borel space. A group τ t , t ∈ ℝ, of Borel automorphisms on (X, Ɓ) is called a jointly measurable flow, or simply a flow, if
  1. (i)

    the map (t, x) ↦ τ t x from ℝ × XX is measurable, where ℝ × X is given the usual product Borel structure,

     
  2. (ii)

    τ0x = x for all xX,

     
  3. (iii)

    τ0x = τt ο τ s (x) for all t,s ∈ ℝ and all x ∈ ℝ.

     

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. [1]
    W. Ambrose. Representation of Ergodic Flows, Annals of Mathematics, 42 (1941), 723–739.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    W. Ambrose and S. Kakutani. Structure and Continuity of Measurable Flows, Duke Math. J., 9 (1942), 25–42.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    G. D. Birkhoff. Proof of a Recurrence Theorem for Strongly Transitive Systems, Proc. Nat. Acad. Sci., U. S. A., 17 (1931), 650–655, Birkhoff: Collected Mathematical Papers, Vol. 2, 398–403, Dover, 1968.CrossRefzbMATHGoogle Scholar
  4. [4]
    S. G. Dani. Kolmogorov Automorphisms on Homogeneous Spaces, American J. Math., 98 (1976), 119–163.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    U. Krengel. Darstellungssatz für Strömungen und Halbströmungen. I II, Math. Ann., 176 (1968), 181–190, 182 (1969) 1–39.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    U. Krengel. On Rudolph’s Representation of Aperiodic Flows. Ann. Inst. Henri Poincaré, Vol XII (1976), 319–338.MathSciNetzbMATHGoogle Scholar
  7. [7]
    I. Kubo. Quasi-flows. Nagoya Math. Journal, 35 (1969), 1–30.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    J. Mathew and M. G. Nadkarni. Some Results on Cocycles and Spectra of Ergodic Flows, Statistics and Probability, Essays in Honour of C. R. Rao, edited by G. Kallianpur, P. Krishnaih, J. K. Ghosh, North-Holland, (1982) 493–504.Google Scholar
  9. [9]
    D. Rudolph. A Two Valued Step Coding of Ergodic Flows, Mathematische Zeitschrift (150), (3), (1976), 201–220. also see Ergodic Theory, Chapter 11, I. P. Kornfield, S. V. Fomin, Y. G. Sinai., Springer-Verlag, (1982).MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    V. M. Wagh. A Descriptive Version of Ambrose’ Representation Theorem for Flows, Proc. Ind. Acad. (Math. Sci.), 98 (2–3) (1988), 101–108.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Hindustan Book Agency 2013

Authors and Affiliations

  1. 1.University of MumbaiIndia

Personalised recommendations