On the convergence of Denjoy-Wolff points

If $\varphi$ is an analytic function from the unit disk $\mathbb{D}$ to itself, and $\varphi$ is not a conformal automorphism, we denote by $\lambda_{\varphi}$ its Denjoy-Wolff point, that is, the limit of the iterates $\varphi(\varphi(\cdots\varphi(0)\cdots))$. A result of Heins shows that, given a sequence $(\varphi_{n})_{n\in\mathbb{N}}$ of such analytic functions that convergence pointwise to $\varphi$, it follows that $\lim_{n\to\infty}\lambda_{\varphi_{n}}=\lambda_{\varphi}$. This allows us to improve results about the contnuous extensions of the subordination functions that arise in the study of free convolutions. We also offer an alternate proof of the result of Heins.Comment: The main technical tool in the first version is now appropriately credited to Maurice Hein