Schröder equation and commuting functions on the circle

We show that if F :S1→S1 is a homeomorphism of the unit circle S1 and the rotation number α(F) of F is irrational, then the Schröder equation Φ 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 F is a non-trivial continuous and disjoint iteration group or semigroup on S1 and a continuous at least at one point function G:S1→S1 commutes with a suitable element of F, then G ∈ F. © 2007 Elsevier Inc. All rights reserved.