Dynamics of inner functions revisited

TL;DR

This paper investigates circle restrictions of inner functions on the unit disk, establishing that local invertibility of a restriction is independent of the singularity set and linking this to a local analytic characterization of conditional expectations. It develops a spectral framework for transfer operators acting on Hardy and weighted Hilbert spaces, proving quasicompactness and central limit-type behavior for stochastic processes driven by inner-function restrictions under perturbations. Key tools include Clark measures, Denjoy–Wolff theory, and arc-map/differentiability structure that connect boundary regularity to dynamical properties inside the disk. The results provide a rigorous bridge between the ergodic theory of boundary restrictions and the spectral theory of transfer operators, with implications for limit theorems and perturbative analysis in complex-analytic dynamical systems.

Abstract

We study the circle restrictions of inner functions of the unit disc showing that the local invertibility of a restriction is independent of its singularity set and proving a local characterization of analytic conditional expectations. We establish central limit properties for some stochastic processes driven by probability preserving restrictions via spectral analysis of their perturbed transfer operators.