On the structure of iteration groups of homeomorphisms having fixed points (Q1390818)

Let \(I\) be a real interval and \(V\) a divisible subgroup of the additive group \((R,+)\). An iteration group on \(I\) over \(V\) is a family of homeomorphisms \(F(I,V):= \{f^t: I\to I\), \(t\in V\}\) such that \(f^t\circ f^s= f^{t+s}\), \(t,s\in V\). The problem of describing all iteration groups is connected with the investigation of transformations of systems of differential equations with several deviating arguments. [Cf. \textit{F. Neuman}, Czech. Math. J. 31(106), 87-96 (1981; Zbl 0463.34051).] In this paper, the author gives a characterization of the structure of all iteration groups of continuous functions on a real interval, without any additional assumptions, with respect to iterative parameter \(t\). He proves that for every iteration group \(F(I,V)\) there exists a family of pairwise disjoint open intervals \(I_\alpha\), \(\alpha\in M\) such that \(f^t[I_\alpha]= I_\alpha\) and \(f^t(x)= x\) for \(x\in I\setminus \bigcup_{\alpha\in M}I_\alpha\), \(t\in V\). Every iteration group \(F(J,V)\) where \(J\in \{I_\alpha, \alpha\in M\}\) satisfies one of the following conditions: (I) there exists \(t\in V\) such that \(f^t(x)\neq x\), \(x\in J\); (II) for every \(t\in V\), \(f^t\) has a fixed point in \(J\) and the family of functions \(F(J,V)\) has no common fixed point. Also, it is shown that one can build every group of type (I) by a special compilation of disjoint iteration groups i.e. iteration groups with fixed points defined on some subintervals of \(J\). And that every group of type (II) is built by means of a countable family of iteration groups of type (I).