Schroder equation and commuting functions on the circle

We show that if F: S-1 -> S-1 is a homeomorphism of the unit circle S-1 and the rotation number alpha(F) of F is irrational, then the Schroder equation Phi(F(z)) = e(2 pi i alpha(F)) Phi(z), z is an element of S-1, 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 S-1 and a continuous at least at one point function G: S-1 -> S-1 commutes with a suitable element of F, then G is an element of F. (C) 2007 Elsevier Inc. All rights reserved.