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Minimal extensions of flows with amenable acting groups

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Abstract

We construct minimal extensions of flows with locally compact amenable acting groups Γ in the form of fibre-preserving flows on fibre bundles and affine flows on group bundles. We use these constructions to show that for a large family of groups Γ, the class of the compact second countable spaces, on which Γ acts in a (point-free and) minimal way, is closed with respect to countable products.

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References

  1. J. Auslander, Minimal Flows and their Extensions, North-Holland Mathematics Studies, Vol. 153. Notas de Matemática Mathematical Notes], Vol. 122, North-Holland, Amsterdam, 1988.

    MATH  Google Scholar 

  2. A. Blokh, L. Oversteegen and E. D. Tymchatyn, On minimal maps of 2-manifolds, Ergodic Theory and Dynamical Systems 25 (2005), 41–57.

    Article  MATH  MathSciNet  Google Scholar 

  3. P. Le Calvez and J. C. Yoccoz, FrUn théorème d’indice pour les homéomorphismes du plan au voisinage d’un point fixe, Annals of Mathematics 146 (1997), 241–293.

    Article  MATH  MathSciNet  Google Scholar 

  4. H. Chu and M. A. Geraghty, The fundamental group and the first cohomology group of a minimal set, Bulletin of the American Mathematical Society 69 (1963), 377–381.

    Article  MATH  MathSciNet  Google Scholar 

  5. J. de Vries, Elements of Topological Dynamics, Mathematics and its Applications, Vol. 257, Kluwer Academic Publishers Group, Dordrecht, 1993.

    Book  MATH  Google Scholar 

  6. M. Dirbák and P. Maličký, On the construction of non-invertible minimal skew products, Journal of Mathematical Analysis and Applications 375 (2011), 436–442.

    Article  MATH  MathSciNet  Google Scholar 

  7. R. Ellis, The construction of minimal discrete flows, American Journal of Mathematics 87 (1965), 564–574.

    Article  MATH  MathSciNet  Google Scholar 

  8. A. Fathi and M. R. Herman, Existence de difféomorphismes minimaux, Astérisque 49 (1977), 37–59.

    MathSciNet  Google Scholar 

  9. B. R. Fayad, Topologically mixing and minimal but not ergodic, analytic transformation on T5, Boletim da Sociedade Brasileira de Matématica 31 (2000), 277–285.

    Article  MATH  MathSciNet  Google Scholar 

  10. E. Glasner, Ergodic Theory via Joinings, Mathematical Surveys and Monographs, Vol. 101, American Mathematical Society, Providence, RI, 2003.

    MATH  Google Scholar 

  11. S. Glasner and B. Weiss, On the construction of minimal skew products, Israel Journal of Mathematics 34 (1979), 321–336.

    Article  MATH  MathSciNet  Google Scholar 

  12. S. Glasner and B. Weiss, Processes disjoint from weak mixing, Tranasctions of the American Mathematical Society 316 (1989), 689–703.

    Article  MATH  MathSciNet  Google Scholar 

  13. W. H. Gottschalk and G. A. Hedlund, Topological Dynamics, American Mathematical Society Colloquium Publications, Vol. 36, American Mathematical Society, Providence, RI, 1955.

    MATH  Google Scholar 

  14. O. Hájek, Non-minimality of 3-manifolds, in Recent advances in topological dynamics (Proc. Conf., Yale Univ., New Haven, Conn., 1972; in honor of Gustav Arnold Hedlund), Lecture Notes in Mathematics, Vol. 318, Springer, Berlin, 1973, pp. 140–142.

    Google Scholar 

  15. K. H. Hofmann and S. A. Morris, The Structure of Compact Groups. A Primer for the Student—a Handbook for the Expert, de Gruyter Studies in Mathematics, Vol. 25, Walter de Gruyter & Co., Berlin, 2006.

    MATH  Google Scholar 

  16. A. B. Katok, Minimal diffeomorphisms on principal S 1-fiber spaces, in Proceedings of the VI Soviet Topological Conference, Tbilisi, Metzniereba, Tbilisi, 1972, pp. xxx–xxx.

    Google Scholar 

  17. H. Kneser, Reguläre Kurvenscharen auf den Ringflächen, Mathematische Annaalen 91 (1924), 135–154.

    Article  MATH  MathSciNet  Google Scholar 

  18. S. Kolyada, L’. Snoha and S. Trofimchuk, Proper minimal sets on compact connected 2-manifolds are nowhere dense, Ergodic Theory and Dynamical Systems 28 (2008), 863–876.

    MATH  MathSciNet  Google Scholar 

  19. M. Lemańczyk and E. Lesigne, Ergodicity of Rokhlin cocycles, Journal d’Analyse Mathématique 85 (2001), 43–86.

    Article  MATH  Google Scholar 

  20. M. Lemańczyk and M. K. Mentzen, Topological ergodicity of real cocycles over minimal rotations, Monatshefte für Mathematik 134 (2002), 227–246.

    Article  MATH  Google Scholar 

  21. M. Lemańczyk and F. Parreau, Lifting mixing properties by Rokhlin cocycles, Ergodic Theory and Dynamical Systems 32 (2012), 763–784.

    Article  MATH  MathSciNet  Google Scholar 

  22. N. G. Markley, The Poincaré-Bendixson theorem for the Klein bottle, Transactions of the American Mathematical Society 135 (1969), 159–165.

    MATH  MathSciNet  Google Scholar 

  23. M. K. Mentzen and A. Siemaszko, Cylinder cocycle extensions of minimal rotations on monothetic groups, Colloquium Mathematicum 101 (2004), 75–88.

    Article  MATH  MathSciNet  Google Scholar 

  24. S. A. Morris, Pontryagin Duality and the Structure of Locally Compact Abelian Groups, London Mathematical Society Lecture Note Series, Vol. 29, Cambridge University Press, Cambridge-New York-Melbourne, 1977.

    Book  MATH  Google Scholar 

  25. W. Parry, A note on cocycles in ergodic theory, Compositio Mathematica 28 (1974), 343–350.

    MATH  MathSciNet  Google Scholar 

  26. J. P. Pier, Amenable Locally Compact Groups, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1984.

    Google Scholar 

  27. D. J. Rudolph, k-fold mixing lifts to weakly mixing isometric extensions, Ergodic Theory and Dynamical Systems 5 (1985), 445–447.

    Article  MATH  MathSciNet  Google Scholar 

  28. D. J. Rudolph, Z n and R n cocycle extensions and complementary algebras, Ergodic Theory and Dynamical Systems 6 (1986), 583–599.

    Article  MATH  MathSciNet  Google Scholar 

  29. K. Schmidt, Cocycles on Ergodic Transformation Groups, Macmillan Lectures in Mathematics, Vol. 1, Macmillan Company of India, Ltd., Delhi, 1977.

    MATH  Google Scholar 

  30. S. Smale, Mathematical problems for the next century, in Mathematics: Frontiers and Perspectives, American Mathematical Society, Providence, RI, 2000, pp. 271–294.

    Google Scholar 

  31. R. J. Zimmer, Extensions of ergodic group actions, Illinois Journal of Mathematics 20 (1976), 373–409.

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Faculty of Natural Sciences, Matej Bel University, Tajovského 40, 974 01, Banská Bystrica, Slovakia

    Matúš Dirbák

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  1. Matúš Dirbák
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Dirbák, M. Minimal extensions of flows with amenable acting groups. Isr. J. Math. 207, 581–615 (2015). https://doi.org/10.1007/s11856-015-1184-6

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  • DOI: https://doi.org/10.1007/s11856-015-1184-6

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