Fixed point in the disk

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(r) be the maximum modulus of f on |z| = r < 1. By the Schwarz lemma^[4]

${\displaystyle |f(z)|\leq \delta (r)|z|,}$

for |z| ≤ r, where

${\displaystyle \delta (r)={M(r) \over r}<1.}$

It follows by iteration that

${\displaystyle |f^{n}(z)|\leq \delta (r)^{n}}$

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 ${\displaystyle r_{k}}$ increasing to 1 and set^[5][6]

${\displaystyle f_{k}(z)=r_{k}f(z).}$

By applying Rouché's theorem to ${\displaystyle f_{k}(z)-z}$ and ${\displaystyle g(z)=z}$, ${\displaystyle f_{k}}$ has exactly one zero ${\displaystyle z_{k}}$ in D. Passing to a subsequence if necessary, it can be assumed that ${\displaystyle z_{k}\rightarrow z.}$ 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 ${\displaystyle f_{k}}$ leave invariant all Euclidean disks whose hyperbolic center is located at ${\displaystyle z_{k}}$. 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 ${\displaystyle f^{n}}$ converges uniformly on compacta to the constant z, it is enough to show that the same is true for any subsequence ${\displaystyle f^{n_{k}}}$, 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 ${\displaystyle f^{n_{k+1}-n_{k}}}$ has a limit, h say. But then

${\displaystyle h(g(w))=g(w),}$

for w in D.

Since h is holomorphic and g(D) open,

${\displaystyle h(w)=w}$

for all w.

Setting ${\displaystyle m_{k}=n_{k+1}-n_{k}}$, it can also be assumed that ${\displaystyle f^{m_{k}-1}}$ is convergent to F say.

But then f(F(w)) = w = f(F(w)), 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 ${\displaystyle f^{n}}$ converges uniformly on compacta to the constant z.