Theorem. Let D be the open unit disk in C and let f be a holomorphic function mapping D into D which is not an automorphism of D. Then there is a unique point z in the closure of D such that the iterates of f tend to z uniformly on compact subsets of D. If z lies in D, it is the uniquefixed point of f. The mapping f leaves invariant hyperbolic disks centered on z, if z lies in D, and disks tangent to the unit circle at z, if z lies on the boundary of D. When the fixed point is at z = 0, the hyperbolic disks centred at z are just the Euclidean disks with centre 0. Otherwise f can be conjugated by a Möbius transformation so that the fixed point is zero. An elementary proof of the theorem is given below, taken from and. Two other short proofs can be found in.
If f has a fixed point z in D then, after conjugating by a Möbius transformation, it can be assumed that z = 0. Let M be the maximum modulus of f on |z| = r < 1. By the Schwarz lemma for |z| ≤ r, where It follows by iteration that for |z| ≤ r. These two inequalities imply the result in this case.
No fixed points
When f acts in D without fixed points, Wolff showed that there is a point z on the boundary such that the iterates of f leave invariant each disk tangent to the boundary at that point. Take a sequence increasing to 1 and set By applying Rouché's theorem to and, has exactly one zero in D. Passing to a subsequence if necessary, it can be assumed that The point z cannot lie in D, because, by passing to the limit, z would have to be a fixed point. The result for the case of fixed points implies that the maps leave invariant all Euclidean disks whose hyperbolic center is located at. Explicit computations show that, as k increases, one can choose such disks so that they tend to any given disk tangent to the boundary at z. By continuity, f leaves each such disk Δ invariant. To see that converges uniformly on compacta to the constantz, it is enough to show that the same is true for any subsequence, convergent in the same sense to g, say. Such limits exist by Montel's theorem, and if g is non-constant, it can also be assumed that has a limit, h say. But then for w in D. Since h is holomorphic and g open, for all w. Setting , it can also be assumed that is convergent to F say. But then f = w = f, contradicting the fact that f is not an automorphism. Hence every subsequence tends to some constant uniformly on compacta in D. The invariance of Δ implies each such constant lies in the closure of each disk Δ, and hence their intersection, the single point z. By Montel's theorem, it follows that converges uniformly on compacta to the constant z.
