Distortion and Distribution of Sets under Inner Functions
Complex Variables
2020-10-28 v1 Classical Analysis and ODEs
Abstract
It is a classical result that Lebesgue measure on the unit circle is invariant under inner functions fixing the origin. In this setting, the distortion of Hausdorff contents has also been studied. We present here similar results focusing on inner functions with fixed points on the unit circle. In particular, our results yield information not only on the size of preimages of sets under inner functions, but also on their distribution with respect to a given boundary point. As an application, we use them to estimate the size of irregular points of inner functions omitting large sets. Finally, we also present a natural interpretation of the results in the upper half plane.
Cite
@article{arxiv.1812.01510,
title = {Distortion and Distribution of Sets under Inner Functions},
author = {Matteo Levi and Artur Nicolau and Odí Soler i Gibert},
journal= {arXiv preprint arXiv:1812.01510},
year = {2020}
}
Comments
11 pages, no figures, comments welcome