Simultaneous Abel equations

Abstract

Let ${\mathcal{S}}$ be a set of homeomorphisms of an open interval such that the group generated by ${\mathcal{S}}$ is disjoint, i.e., the graphs of any two distinct functions in it do not intersect. We give necessary and sufficient conditions for the system of Abel equations

$$\phi(f(x))=\phi(x)+\lambda(f),\quad f \in \mathcal{S}$$

to have a continuous solution, where ${{\lambda}:\mathcal{S}\to{\mathbb {R}}}$ is a given map. We describe this solution and show that there exists a specific map λ for which the above system always has a continuous solution. As an application we give a criterion for the embeddability of a noncyclic disjoint group of continuous functions in a continuous iteration group.