On the convergence of Denjoy-Wolff points

2022/03/31 by Belinschi, Serban, Bercovici, Hari, Ho, Ching Wei
#30D05 #46L54 #Dynamical Systems (math.DS) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2203.16728

Abstract

If φ is an analytic function from the unit disk \mathbbD to itself, and φ is not a conformal automorphism, we denote by λ_φ its Denjoy-Wolff point, that is, the limit of the iterates φ(φ(⋯φ(0)⋯)). A result of Heins shows that, given a sequence (φ_n)_n∈ℕ of such analytic functions that convergence pointwise to φ, it follows that lim_n→∞λ_φ_n=λ_φ. This allows us to improve results about the contnuous extensions of the subordination functions that arise in the study of free convolutions. We also offer an alternate proof of the result of Heins.

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