Complex Analysis and Dynamical Systems by Mark Agranovsky Anatoly Golberg Fiana Jacobzon David Shoikhet & Lawrence Zalcman

Author:Mark Agranovsky, Anatoly Golberg, Fiana Jacobzon, David Shoikhet & Lawrence Zalcman
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

2.1 Maps Fixing a Point in

As one knows, the disc automorphisms with a unique fixed point in are called elliptic. Denote by {φ [n]} the sequence of iterates of φ and recall the following [21]:

Theorem 1 (Denjoy–Wolff)

Let φ be an analytic selfmap of other than the identity or an elliptic disc automorphism. Then the sequence of iterates {φ [n]} converges uniformly on compacts to a point called the Denjoy–Wolff point of φ.

An immediate consequence is the fact that an analytic selfmap φ of , other than the identity, can have at most one fixed point in .

For any composition operator, the constant function 1 is obviously an eigenfunction associated to the eigenvalue 1, by the obvious equality 1 ∘ φ = 1. If φ, not the identity, fixes , then, by a simple computation, one establishes that the only other eigenvalues C φ can possibly have are (φ ′ (ω)) n , n = 1, 2, …. We call those complex numbers the Schröder eigenvalues of C φ , even when they are not eigenvalues of C φ (which can happen). The reason for this terminology is E. Schröder’s formulation of the eigenvalue functional equation for composition operators [63]. It is known that the Schröder eigenvalues always belong to the spectrum σ(C φ ) of C φ [21]. Denote by σ e (C φ ) the essential spectrum of C φ . Combining the above considerations with G. Koenig’s theorem on Schröder’s equation [37] (see also Theorem 7 in this paper), one easily gets:

Remark 1 ([49, Remark 1])

Let φ be a non-automorphic analytic selfmap of with a fixed point . If σ e (C φ ) is simply connected then

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