Commuting analytic functions without fixed points

Let A be the set of nonidentity analytic functions which map the open unit disk into itself. Wolff has shown that the iterates of $f \in A$ converge uniformly on compact sets to a constant $T(f)$, unless f is an elliptic conformal automorphism of the disk. This paper presents a proof that if f and g are in A and commute under composition, and if f is not a hyperbolic conformal automorphism of the disk, then $T(f) = T(g)$. This extends, in a sense, a result of Shields. The proof involves the so-called angular derivative of a function in A at a boundary point of the disk.