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 $${\mathbb {T}}^3$$ .