Furstenberg–Ellis–Namioka Structure Theorem on a CHART Group

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 \({\mathbb {T}}^3\).