Analysis of Boundary Behaviour of Quasidisks and Jordan Repellers

TL;DR

This paper explores the boundary behavior of quasidisks and Jordan repellers through harmonic measure and boundary rotation, combining geometric function theory and dynamical systems to provide rigorous proofs and expand existing knowledge.

Contribution

It offers a rigorous, unified framework for analyzing boundary properties of quasidisks and Jordan repellers, filling gaps in the literature with a novel approach.

Findings

Detailed analysis of harmonic measure on quasidisks

New proofs for boundary rotation properties

Extension of classical results in geometric function theory

Abstract

We investigate the fine properties of harmonic measure and boundary rotation. By focusing on quasidisks and, in particular, on connected Jordan Repellers arising from conformal expanding dynamical systems, we explore those using the deep interplay between geometric function theory and dynamical systems as a unified framework. While some of the results presented here are regarded as 'folklore' among experts, they lack rigorous proofs in the existing literature. We fill this gap by providing a comprehensive, referable treatment using a novel approach that also expands existing results.