Inner Functions and Laminations

TL;DR

This paper investigates orbit counting for inner functions on Riemann surfaces, deriving asymptotic formulas for pre-image counts using geodesic flows, and extends results to parabolic inner functions with infinite height.

Contribution

It introduces new asymptotic formulas for pre-image counts of inner functions with finite Lyapunov exponent, including generalizations and analogues for parabolic cases.

Findings

Asymptotic formula for pre-image counts of finite Lyapunov exponent inner functions.

Extension of results to general inner functions via Cesàro averages.

Analogues established for parabolic inner functions of infinite height.

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 $$ \mathcal{N}(z, R) \, \sim \, \frac{1}{2} \log \frac{1}{|z|} \cdot \frac{1}{\int_{\partial \mathbb{D}} \log |F'| dm}, \quad \text{as }R \to \infty. $$ 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.