On the structure of iteration groups of homeomorphisms having fixed points

Abstract.

We give a description of all iteration groups of continuous functions $F (I, V) = \{ f^ {t} : I \mapsto I, t \in V \}$, where I is a real interval and V is a divisible subgroup of the additive group $(\Bbb {R}, +)$. We do not assume any regularity with respect to the iterative parameter t. 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. We show that one can build every group of type (I) by a special compilation of disjoint iteration groups i.e. iteration groups without fixed points defined on some subintervals of J. However, every group of type (II) is built by means of a countable family of iteration groups of type (I).