Abstract
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 /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 spaceH p , the sequence Φ n /λn 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 p a .
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The author is partially supported by NSF Grant DMS 97-06408 and wishes to thank Professor P. Bourdon for sharing results and conjectures.
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Poggi-Corradini, P. Norm convergence of normalized iterates and the growth of Kœnigs maps. Ark. Mat. 37, 171–182 (1999). https://doi.org/10.1007/BF02384832
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DOI: https://doi.org/10.1007/BF02384832