Norm convergence of normalized iterates and the growth of Kœnigs maps

Abstract

Let ϕ be an analytic function defined on the unit disk D, 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 of D to a function σ analytic in D which satisfies σ°φ=λσ. It is of interest in the study of composition operators to know if, whenever σ belongs to a Hardy space H_p, the sequence Φ_n/λⁿ converges to σ in the norm of H_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 any H_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 space L ${}_{a}^{p}$ .