Abstract:We study the dynamics of Polya-Szegö 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-Szegö 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 Cesàro 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.
From: Oleg Ivrii [view email]
[v1]
Wed, 17 Jun 2026 16:37:42 UTC (75 KB)
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