A GROUP STRUCTURE ON $\mathbb D$ AND ITS APPLICATION FOR COMPOSITION OPERATORS

International audience ; We present a group structure on $\mathbb D$ via the automorphisms which fix the point 1. Through the induced group action, each point of $\mathbb D$ produces an equivalence class which turns out to be a Blaschke sequence. We show that the corresponding Blaschke products are minimal/atomic solutions of the functional equation $\psi\circ\phi=\lambda\psi$ where $\lambda$ is a unimodular contant and $\phi$ is an automorphism of the unit disc. We also characterize all Blaschke products which satisfy this equation and study its application in the theory of composition operators on model spaces $K_\Theta$.