Norm convergence of normalized iterates and the growth of Kœnigs maps
Let ϕ be an analytic function defined on the unit diskD, with ϕ(D)⊂D, ϕ(0)=0, and ϕ′(0)=λ≠0. Then by a classical result of G. Kœnigs, the sequence of normalized iterates Φn/λn converges uniformly on compact subsets ofD to a function σ analytic inD which satisfiesσ°φ=λσ. It is of interest in the study of composition operators to know if, whenever σ belongs to a Hardy spaceHp, the sequence Φn/λn converges to σ in the norm ofHp. We show that this is indeed the case, generalizing a result of P. Bourdon obtained under the assumption that ϕ is univalent.When ϕ is inner, P. Bourdon and J. Shapiro have shown that σ does not belong to the Nevanlinna class, in particular it does not belong to anyHp. It is natural to ask, how bad can the growth of σ be in this case? As a partial answer we show that σ always belongs to some Bergman spaceLap.