Abstract

Under broad conditions, two analytic self-maps of the disk fixing 0 commute under composition precisely when they have the same Schroeder map, where the Schroeder map for an analytic ϕ : D → D with ϕ(0) = 0 is the unique analytic function σ on D solving Schroeder’s equation σ ∘ ϕ = ϕ′(0)σ and satisfying σ′(0) = 1. For analytic self-maps of the ball in C^N fixing 0 we may still seek analytic C^N−valued solutions σ to Schroeder’s equation with σ′(0) = I, but considerable complications for existence and uniqueness of σ may ensue. Nevertheless, we show that there are reasonably general hypotheses under which it will still be the case that two analytic self-maps of the ball fixing 0 commute if and only if they share a common Schroeder map σ with σ′(0) = I.

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