Commuting holomorphic maps and linear fractional models

Abstract

Let f be a holomorphic map of the open unit disc Δ of into itself, having no fixed points in Δ and Wolff point τε∂Δ. In the open case in which f ′(τ) = 1 we study the centralizer of f i.e., the family Gf of all holomorphic maps of Δ into itself which commute with f under composition. We prove that if the sequence of iterates {fn } converges to τ non tangentially, then Gf coincides with the set of all elements of the pseudo-iteration semigroup of f (in the sense of Cowen, see [5,6]) whose Wolff point is τ. In the same hypotheses we give a representation of the centralizer Gf in Aut(Δ) or Aut, study its main features and generalize a result due to Pranger ([15]).