A characterization of regularly almost periodic minimal flows
Let \((X,\mathbb{R},\pi)\) be a transformation group with compact metric space \(X\) and phase group \(\mathbb{R}\), the additive group of real numbers. Let \(\lambda\in\mathbb{R}\). We say that \(\lambda\) is an eigenvalue \((X,\mathbb{R},\pi)\) if there is a continuous mapping \(\chi_\lambda:X\to K=\{\xi\in C\), \(|\xi|=1\}\) such that \(\chi_\lambda(\pi(x,t))= \chi_\lambda(x) \exp(i\lambda t)\) for \(x\in X\) and \(t\in\mathbb{R}\). The set of eigenvalues of \(\pi\) is denoted by \(\Lambda(\pi)\), it is a countable subgroup of the additive group \(\mathbb{R}\). Theorem 1. \((X,\mathbb{R},\pi)\) is regularly almost periodic if and only if it is almost automorphic and \(\dim \Lambda(\pi)=1\). Here, the dimension is defined as the cardinal number of the largest rationally independent subset. Theorem 2. Assume \((X,\mathbb{R},\pi)\) to be a minimal flow. Then \(\pi\) is pointwise regularly almost periodic if and only if it is equicontinuous and \(\dim \Lambda(\pi)=1\).
scientific article; zbMATH DE number 897697
scientific article