Assume \(f(z)\) is analytic on the unit disk \(\mathbb{D} = \{z \in \mathbb{C} \mid |z| < 1\}\), maps \(\mathbb{D} \to \mathbb{D}\), satisfies \(f(0) = 0\), and is not a rotation (i.e. \(f(z) \ne e^{i\theta} z\)). Show that the series of iterates: \[\sum_{n=1}^\infty f^n(z)\] converges uniformly on compact subsets of \(\mathbb{D}\) to an analytic function, where \(f^1(z) = f(z)\) and \(f^{n+1}(z) = f(f^n(z))\).
Solution
By Schwarz’s lemma, \[ |f(z)|\le|z| \qquad(z\in\mathbb D). \] Since \(f\) is not a rotation, equality cannot occur at any nonzero point, and \[ |f'(0)|<1. \] Define \[ g(z)= \begin{cases} f(z)/z,&z\ne0,\\ f'(0),&z=0. \end{cases} \] Then \(g\) is holomorphic on \(\mathbb D\) and \[ |g(z)|<1 \qquad(z\in\mathbb D). \]
1. Let \(K\subset\mathbb D\) be compact. Choose \(r<1\) with\[ K\subseteq\overline{D_r}. \]By compactness,\[ c:=\max_{|z|\le r}|g(z)|<1. \]Hence\[ |f(z)|\le c|z| \qquad(|z|\le r). \]
2. Since \(f(\overline{D_r})\subseteq\overline{D_r}\), induction gives\[ |f^n(z)|\le c^n|z|\le rc^n \qquad(z\in K). \]
3. The geometric series\[ \sum_{n=1}^\infty rc^n \]converges. The Weierstrass \(M\)-test therefore implies that\[ \sum_{n=1}^\infty f^n(z) \]converges uniformly on \(K\). Since \(K\) was arbitrary, the convergence is locally uniform on \(\mathbb D\).
4. Every iterate \(f^n\) is holomorphic, so the locally uniform limit theorem implies that the sum is holomorphic on \(\mathbb D\).