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
CN fixing 0 we may still seek analytic CN−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.
