Iteration groups, commuting functions and simultaneous systems of linear functional equations

Abstract

Let $( f^t )_{t \in \mathbb{R}}$ be a measurable iteration group on an open interval $I$. Under some conditions, we prove that the inequalies $g\circ f^a \leq f^a \circ g$ and $g\circ f^b \leq f^b\circ g$ for some $a,b \in \mathbb{R}$ imply that g must belong to the iteration group. Some weak conditions under which two iteration groups have to consist of the same elements are given. An extension theorem of a local solution of a simultaneous system of iterative linear functional equations is presented and applied to prove that, under some conditions, if a function $g$ commutes in a neighbourhood of $f$ with two suitably chosen elements $f^{a}$ and $f^{b}$ of an iteration group of $f$ then, in this neighbourhood, $g$ coincides with an element of the iteration group. Some weak conditions ensuring equality of iteration groups are considered.