Furstenberg–Ellis–Namioka Structure Theorem on a CHART Group

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Abstract

For \(E({\mathbb {T}})\) being the endomorphism group of the circle group \({\mathbb {T}}\), the Furstenberg–Ellis–Namioka Structure Theorem of the CHART group \(G=E({\mathbb {T}})\times {\mathbb {T}}\) with the product \((f,u)(g,v)=(fg,uvf\circ g(\mathrm{e}^{i}))\) is known to be equal to \(\{G,1_{\mathbb {T}}\times {\mathbb {T}},\{(1_{\mathbb {T}},1)\}\}\). A somewhat similar group structure is known to exist on \(E({\mathbb {T}})\times E({\mathbb {T}})\times {\mathbb {T}}\), studied by Milnes. We give an explicit characterization of the Furstenberg–Ellis–Namioka Structure Theorem for an admissible subgroup \(\Sigma \) of \(E({\mathbb {T}})\times E({\mathbb {T}})\times {\mathbb {T}}\), where \(\Sigma \) is the Ellis group of the Hahn-type skew product dynamical system on the 3-torus \({\mathbb {T}}^3\).

Keywords

Compact right topological group Ellis group Furstenberg–Ellis–Namioka structure theorem 

Mathematics Subject Classification

43A60 
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Notes

Acknowledgements

The authors would like to thank the anonymous referee for the kind suggestions. A support from Mahani Mathematical Research Center for the second author is gratefully acknowledged.

References

  1. 1.
    Ellis, R.: Locally compact transformation groups. Duke Math. J. 24, 119–125 (1957)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Ellis, R.: The Furstenberg structure theorem. Pac. J. Math. 76, 345–349 (1978)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Furstenberg, H.: The structure of distal flows. Am. J. Math. 83, 477–515 (1963)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Jabbari, A., Namioka, I.: Ellis group and the topological center of the flow generated by the map \(n\mapsto \lambda ^{{n}^{k}}\). Milan J. Math. 78, 503–522 (2010)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Jabbari, A., Vishki, H.R.E.: Skew product dynamical systems, Ellis groups and topological center. Bull. Aust. Math. Soc. 79, 129–145 (2009)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Lau, A.T.-M., Loy, R.J.: Banach algebras on compact right topological groups. J. Funct. Anal. 225, 263–300 (2005)MathSciNetCrossRefMATHGoogle Scholar
  7. 8.
    Moors, W.B.: Invariant means on CHART groups. Khayyam J. Math. 1, 36–44 (2015)MathSciNetMATHGoogle Scholar
  8. 9.
    Moors, W.B., Namioka, I.: Furstenberg’s structure theorem via CHART groups. Ergod. Theory Dyn. Syst. 33, 954–968 (2013)MathSciNetCrossRefMATHGoogle Scholar
  9. 10.
    Moran, A.: Minimal normal systems of compact right topological groups. Math. Proc. Camb. Phil. Soc. 123, 243–258 (1998)MathSciNetCrossRefMATHGoogle Scholar
  10. 11.
    Namioka, I.: Right topological groups, distal flows and a fixed-point theorem. Math. Syst. Theory 6, 193–209 (1972)MathSciNetCrossRefMATHGoogle Scholar
  11. 12.
    Rautio, J.: Enveloping semigroups and quasi-discrete spectrum. Ergod. Theory Dyn. Syst. 36, 2627–2660 (2016)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Iranian Mathematical Society 2018

Authors and Affiliations

  1. 1.Department of MathematicsKerman Branch, Islamic Azad UniversityKermanIran
  2. 2.Department of Pure Mathematics, Faculty of Mathematics and computerShahid Bahonar University of KermanKermanIran

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