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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 Φ n /λ n  converges uniformly on compact subsets ofD to a function σ analytic inD which satisfiesσϕ=λσ. It is of interest in the study of composition operators to know if, whenever σ belongs to a Hardy spaceH  p , the sequence Φ n /λ n  converges to σ in the norm ofH  p . 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 anyH  p . 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 spaceL   a   p  .

Norm convergence of normalized iterates and the growth of Kœnigs maps