A remark on generalized iterates (Q2266185)

Let f be an invertible function from R to R. Then the n-th iterate \(f^ n\) is defined for \(n\in Z\). Suppose that all iteration trajectories are non-periodic and let G be a group of order c \((=card R)\) which contains Z as a normal subgroup. Then there exist functions \(F^{\alpha}: R\to R\), \(\alpha\in G\) such that \(F^{\alpha}(F^{\beta})=F^{\alpha +\beta}\) for all \(\alpha\),\(\beta\) in G and \(F^ n=f^ n\) for \(n\in Z\). The results may also be stated in a more general setting.