Dynamics of simply parabolic inner functions

We study the dynamics of Polya-Szeg\"o inner functions and discuss some of their basic properties such as equivalent conditions for simple and double parabolicity. We show that a simply parabolic Polya-Szeg\"o inner function admits forward and backward quotient half-cylinders, which allows one to enrich its dynamics with a Lavaurs map. To proceed, we restrict our attention to simply parabolic inner functions with finite Lyapunov exponent: $\int_{\mathbb{R}} \log |F'| d\ell < \infty$. We define a geodesic flow on the Riemann surface lamination associated to the Lavaurs semigroup and show that it is ergodic. As an application, we establish the Orbit Counting Theorem up to a Ces\`aro average for Lavaurs semigroups. If we additionally assume that $F$ is a parabolic one component inner function, then the geodesic flow is mixing and the full Orbit Counting Theorem holds.