Furstenberg–Ellis–Namioka Structure Theorem on a CHART Group
- 15 Downloads
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 theoremMathematics Subject Classification
43A60Notes
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.Ellis, R.: Locally compact transformation groups. Duke Math. J. 24, 119–125 (1957)MathSciNetCrossRefMATHGoogle Scholar
- 2.Ellis, R.: The Furstenberg structure theorem. Pac. J. Math. 76, 345–349 (1978)MathSciNetCrossRefMATHGoogle Scholar
- 3.Furstenberg, H.: The structure of distal flows. Am. J. Math. 83, 477–515 (1963)MathSciNetCrossRefMATHGoogle Scholar
- 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.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.Lau, A.T.-M., Loy, R.J.: Banach algebras on compact right topological groups. J. Funct. Anal. 225, 263–300 (2005)MathSciNetCrossRefMATHGoogle Scholar
- 8.Moors, W.B.: Invariant means on CHART groups. Khayyam J. Math. 1, 36–44 (2015)MathSciNetMATHGoogle Scholar
- 9.Moors, W.B., Namioka, I.: Furstenberg’s structure theorem via CHART groups. Ergod. Theory Dyn. Syst. 33, 954–968 (2013)MathSciNetCrossRefMATHGoogle Scholar
- 10.Moran, A.: Minimal normal systems of compact right topological groups. Math. Proc. Camb. Phil. Soc. 123, 243–258 (1998)MathSciNetCrossRefMATHGoogle Scholar
- 11.Namioka, I.: Right topological groups, distal flows and a fixed-point theorem. Math. Syst. Theory 6, 193–209 (1972)MathSciNetCrossRefMATHGoogle Scholar
- 12.Rautio, J.: Enveloping semigroups and quasi-discrete spectrum. Ergod. Theory Dyn. Syst. 36, 2627–2660 (2016)MathSciNetCrossRefMATHGoogle Scholar