On conditions guaranteeing that mappings are elements of iteration groups

In this note, we prove that if F = { f v : I ⟶ I , v ∈ V } is a suitable iteration group on an open interval I and a function g : I ⟶ I is continuous at least at one point and commutes with two mappings f a , f b ∈ F such that b a is irrational, then g ∈ F . If, moreover, f a < id < f b , then the commutativity of g with f a and f b can be replaced by the inequalities g ∘ f a ≤ f a ∘ g and g ∘ f b ≤ f b ∘ g . As an application of the obtained results, we give a contribution to Schwaiger’s problem concerning iteration groups which are commuting in pairs.