  In this paper, we study orbit counting problems for inner functions using
geodesic and horocyclic flows on Riemann surface laminations. For a one
component inner function of finite Lyapunov exponent with $F(0) = 0$, other
than $z \to z^d$, we show that the number of pre-images of a point $z \in
\mathbb{D} \setminus \{ 0\}$ that lie in a ball of hyperbolic radius $R$
centered at the origin satisfies $$ \mathcal{N}(z, R) \, \sim \, \frac{1}{2}
\log \frac{1}{|z|} \cdot \frac{1}{\int_{\partial \mathbb{D}} \log |F'| dm},
\quad \text{as }R \to \infty. $$ For a general inner function of finite
Lyapunov exponent, we show that the above formula holds up to a Ces\`aro
average. Our main insight is that iteration along almost every inverse orbit is
asymptotically linear. We also prove analogues of these results for parabolic
inner functions of infinite height.
