# Distortion and Distribution of Sets under Inner Functions

## TL;DR

This paper investigates how inner functions with boundary fixed points distort and distribute sets on the unit circle, providing new insights into preimage sizes, distribution, and boundary behavior, with applications to irregular points and interpretations in the upper half-plane.

## Contribution

It extends classical measure invariance results to inner functions with boundary fixed points, analyzing set distortion and distribution, and offers applications to irregular points and upper half-plane interpretations.

## Key findings

- Lebesgue measure invariance under boundary fixed inner functions
- Distribution estimates of preimages of sets under such functions
- Size bounds for irregular points omitting large sets

## 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.