Inner Functions and Laminations

Abstract

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 $$\mathcal{N}(z,R)\sim \frac{1}{2}\log \frac{1}{|z|}\cdot \frac{1}{{\int }_{\partial \mathbb{D}}\log |F^{\prime }|dm},\text{as }R\to \infty .$$ For a general inner function of finite Lyapunov exponent, we show that the above formula holds up to a Cesà ro 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.

75 pages, 2 figures