Loosely speaking, an inner function $F$ is a holomorphic self-map of the unit disk $\mathbb{D}$ which extends to a measure-theoretic dynamical system on the unit circle. Let $F$ be an inner function with derivative in the Nevanlinna class and an attracting fixed point at the origin. For a point $z \in \mathbb{D} \setminus \{0\}$, let
$$
\mathcal N(z, R) = \# \left \{ w \in B_{\mathrm{hyp}}(0,R) : F^{\circ n}(w) = z \text{ for some } n \ge 0 \right \}
$$
denote the number of repeated pre-images of $z$ inside a ball of hyperbolic radius $R$. In this talk, I will show an averaged form of the asymptotic formula
$$
\mathcal N(z, R) \, \sim \, \frac{1}{2} \log \frac{1}{|z|} \cdot \frac{e^R}{\int \log |F'| dm}, \qquad \text{as }R \to \infty.
$$
The proof follows an argument of McMullen from 2008 which involves geodesics flows on Riemann surface laminations. (According to Sullivan’s dictionary, these are analogous to unit tangent bundles of Riemann surfaces.) Our key insight is that backward iteration with respect to an inner function is essentially linear along almost every inverse orbit.
(This is joint work with Mariusz Urbański.)