Exponents and almost periodic orbits (Q2701844)

Let \(X\) be a metric space and \(f:\mathbb R \to X\) a continuous mapping. A sequence \(\{t_i\}, t_i\in \mathbb R\), is called an \(f\)-sequence if the corresponding sequence \(\{f(t_i)\}\) converges in \(X\). The author defines the group of exponents, denoted by \({\mathcal E}_f\), to be \(\{\alpha \in \mathbb R\mid \{\pi(\alpha t_i)\}\) converges in \(S^1\) for all \(f\)-sequences \(\{t_i\}\}\) where \(\pi :\mathbb R \to \mathbb R/\mathbb Z=S^1\) is the standard quotient map. It is shown that \({\mathcal E}_f\) generalizes the subgroup generated by the Fourier-Bohr exponents of an almost periodic orbit. Furthermore, any minimal almost periodic flow in a complete metric space is shown to be determined by \({\mathcal E}_f\), up to topological equivalence. Results and connections to the rotation number are also presented.