On one-parameter semigroups generated by commuting continuous injections

The problem of the embeddability of two commuting continuous injections f , g : I = ( 0 , b ] → I in Abelian semigroups is discussed. We consider the case when there is no iteration semigroup in which f and g can be embedded. Explaining this phenomenon we modify the definition of an iteration semigroup introducing a new notion – a T-iteration semigroup of f and g, that is a family { f t : I → I , t ∈ T } of continuous injections for which f u ∘ f v = f u + v , u , v ∈ T , such that f = f 1 and g = f s for an s ∈ T and s ∉ Q , where T ⊊ R + is a dense semigroup which can be extended to a group. We determine a maximal semigroup of indices Sem ( f , g ) ⊊ R + such that for every T-iteration semigroup T ⊂ Sem ( f , g ) . We give also a construction of maximal T-iteration semigroups of f and g that is such semigroups for which T = Sem ( f , g ) . We examine also some other Abelian semigroups of continuous functions containing f and g.