Let (ƒt)t∈R be a measurable iteration group on an open interval I. Under some conditions, we prove that the inequalies g o ƒa ≤ ƒa o g and g o ƒb ≤ ƒb o g for some a, b ∈ 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 ƒ with two suitably chosen elements ƒa and ƒb of an iteration group of ƒ then, in this neighbourhood, g coincides with an element of the iteration group. Some weak conditions ensuring equality of iteration groups are considered.
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