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Measure-theoretic problems in topological dynamics

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Abstract

This paper, motivated by a conjecture raised by Choksi in 1984 about homogeneous spaces, investigates the topological connexions between transformation groups and product spaces; our approach, based on the Furstenberg structure theorem, provides a unified treatment for (Baire) measures on any minimal distal flow and for measures on a product of compact metric spaces of the same topological weight.

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References

  1. G. Birkhoff,Dynamical Systems, Amer. Math. Soc. Colloq. Publ., Vol. 9, Amer. Math. Soc., Providence, R.I., 1927.

    MATH  Google Scholar 

  2. N. Bourbaki,Intégration, Chap. 8, Hermann, Paris, 1959–1967.

    Google Scholar 

  3. G. E. Bredon,Introduction to Compact Transformation Groups, Academic Press, New York, 1972.

    MATH  Google Scholar 

  4. J. R. Choksi,Automorphisms of Baire measures on generalized cubes I, Z. Wahr. Verw. Gebiete22 (1972), 195–204;23 (1972), 97–102.

    Article  MathSciNet  Google Scholar 

  5. J. R. Choksi,Measurable transformations on compact groups, Trans. Amer. Math. Soc.184 (1973), 101–124.

    Article  MathSciNet  Google Scholar 

  6. J. R. Choksi,Recent developments arising out of Kakutani's work on completion regularity of measures, Contemporary Math.26 (1984), 81–94.

    MATH  MathSciNet  Google Scholar 

  7. J. R. Choksi, S. T. Eigen, J. C. Oxtoby and V. S. Prasad,The work of Dorothy Maharam on measure theory, ergodic theory and category algebras, Contemporary Math.94 (1989), 57–71.

    MathSciNet  Google Scholar 

  8. J. R. Choksi and D. H. Fremlin,Completion regular measures on product spaces, Math. Ann.241 (1979), 113–128.

    Article  MATH  MathSciNet  Google Scholar 

  9. J. R. Choksi and R. R. Simha,Measurable transformations on homogeneous spaces, inStudies in Probability and Ergodic Theory, Adv. in Math. Suppl. Stud.2 (1978), 269–286.

  10. R. Ellis,Lectures on Topological Dynamics, W. A. Benjamin, Inc., New York, 1969.

    MATH  Google Scholar 

  11. R. Ellis,The Furstenberg structure theorem, Pacific J. Math.76 (1978), 345–349.

    MATH  MathSciNet  Google Scholar 

  12. R. Ellis, S. Glasner and L. Shapiro, Preprint, Univ. of Minesota.

  13. D. H. Fremlin,Measure algebras, inHandbook of Boolean Algebra (ed. J. D. Monk), North-Holland, Amsterdam, 1989.

    Google Scholar 

  14. H. Furstenberg,The structure of distal flows, Amer. J. Math.85 (1963), 477–515.

    Article  MATH  MathSciNet  Google Scholar 

  15. S. Graf,Realizing automorphisms of quotients of product σ-fields, Pacific J. Math.99 (1982), 19–30.

    MATH  MathSciNet  Google Scholar 

  16. S. Grekas,Isomorphic measures on compact groups, Math. Proc. Camb. Phil. Soc.112 (1992), 349–360 and a corrigendum to appear in Math. Proc. Camb. Phil. Soc.

    MATH  MathSciNet  Google Scholar 

  17. S. Grekas,Structural properties of compact groups with measure-theoretic applications, Isr. J. Math.87 (1994), 89–95.

    Article  MATH  MathSciNet  Google Scholar 

  18. R. A. Johnson,Disintegrating measures on compact group extensions, Z. Wahr. Verw. Gebiete53 (1980), 271–281.

    Article  MATH  Google Scholar 

  19. R. A. Johnson,Strong liftings commuting with minimal distal flows, Pacific J. Math.90, No. 1 (1980), 77–85.

    MATH  MathSciNet  Google Scholar 

  20. D. Maharam,On homogeneous measure algebras, Proc. Natl. Acad. Sci. Washington28 (1942), 108–111.

    Article  MATH  MathSciNet  Google Scholar 

  21. D. Maharam,Automorphisms of products of measure spaces, Proc. Amer. Math. Soc.9 (1958), 702–707.

    Article  MATH  MathSciNet  Google Scholar 

  22. D. Maharam,Realizing automorphisms of category algebras, General Topology and its Applications10 (1979), 161–174.

    Article  MathSciNet  Google Scholar 

  23. D. Maharam,Weak mixing and a note on a structure theorem for minimal transformation groups, Illinois J. Math.20 (1976), 186–197.

    MathSciNet  Google Scholar 

  24. D. McMahon and T. S. Wu,On proximal and distal extensions of minimal sets, Bull. Inst. Math. Acad. Sinica2 (1974), No. 1, 93–107.

    MATH  MathSciNet  Google Scholar 

  25. D. Montgomery and L. Zippin,Topological Transformation Groups, Interscience New York, 1955.

    MATH  Google Scholar 

  26. J. von Neumann,Einige Sättze über die messbare Abbildungen, Ann. of Math. (2)33 (1932), 574–586.

    Article  MathSciNet  Google Scholar 

  27. W. A. Veech,Point-distal flows, Amer. J. Math.92 (1970), 205–242.

    Article  MATH  MathSciNet  Google Scholar 

  28. W. A. Veech,Topological dynamics, Bull. Amer. Math. Soc.83 (1977), 775–830.

    Article  MATH  MathSciNet  Google Scholar 

  29. J. de Vries,Abstract topological dynamics, inRecent Progress in General Topology (eds. M. Husek and J. van Mill), Elsevier, Amsterdam, 1992.

    Google Scholar 

  30. A. Weil,L'intégration dans les groupes topologiques et ses applications, 2nd edition, Hermann, Paris, 1951.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Athens, Athens, Greece

    S. Grekas

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  1. S. Grekas
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In memory of my sister

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Grekas, S. Measure-theoretic problems in topological dynamics. J. Anal. Math. 65, 207–220 (1995). https://doi.org/10.1007/BF02788772

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  • DOI: https://doi.org/10.1007/BF02788772

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