Schröder equation and commuting functions on the circle

We show that if F:S1→S1 is a homeomorphism of the unit circle S1S1 and the rotation number α(F)α(F) of F is irrational, then the Schröder equationΦ(F(z))=e2πiα(F)Φ(z),z∈S1, has a unique (up to a multiplicative constant) continuous at a point of the limit set of F   solution. We apply this result to prove that if FF is a non-trivial continuous and disjoint iteration group or semigroup on S1S1 and a continuous at least at one point function G:S1→S1 commutes with a suitable element of FF, then G∈FG∈F.