Iteration groups, commuting functions and simultaneous systems of linear functional equations - Janusz Matkowski (Opuscula Mathematica, 2008)
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 \(g° f^a ≤ f^a ° g\) and \(g° f^b ≤ f^b° g\) 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.