Characterizations of (H)PI extensions
Let X be a compact \(T_ 2\) space, T be a topological group and \(X=(T,X,\pi)\) be a topological transformation group (ttg) [\textit{W. A. Veech}, Bull. Am. Math. Soc. 83, 775-830 (1977; Zbl 0384.28018)]. A ttg X is called minimal (ergodic) iff \(X=\overline{Tx}\) for every \(x\in X\) \((X=\overline{TU}\) for every nonempty open \(U\subset X)\). Let \(\phi\) :X\(\to Y\equiv (T,Y,\rho)\) be a homomorphism (extension) and \(R^ n_{\phi}=\{(x_ 1,...,x_ n)\in X^ n:\phi (x_ 1)=...=\phi (x_ n)\}\). \(\phi\) :X\(\to Y\) is called an \(n-C^ 1\)-extension iff every ergodic subset \(A=\overline{TA}\) of \(R^ n_{\phi}\) with a dense set of almost periodic points is minimal. Theorem. Let \(\phi\) :X\(\to Y\) be a homomorphism of minimal ttg. Then the following statements are equivalent: (a) \(\phi\) is a PI extension; (b) \(\phi\) is an \(n-C^ 1\)-extension for some \(n\in {\mathbb{N}}\), \(n\geq 2\); (c) \(\phi\) is an \(n-C^ 1\)-extension for every \(n\in {\mathbb{N}}\), \(n\geq 2\). A characterization of HPI extensions of nonmetric minimal ttgs is also given.
minimal topological transformation group
scientific article; zbMATH DE number 3942486