Inner Functions and Laminations

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 → zᵈ, we show that the number of pre-images of a point z ∈ D \ 0\ that lie in a ball of hyperbolic radius R centered at the origin satisfies N(z, R) \, ~ \, 1/2 log 1/|z| · 1∫∂ D log |F'| dm, as R → ∞. 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.