Analysis of Boundary Behaviour of Quasidisks and Jordan Repellers

TL;DR

The paper addresses how boundary geometry influences harmonic measure and boundary rotation in simply connected domains, focusing on quasidisks and Jordan repellers. It develops a unified framework that couples geometric function theory with dynamical systems, centering on spectra that capture derivative growth and rotation, including the Minkowski distortion and dimension spectra and the new crosscut spectrum. A key contribution is proving the relation $d_ Omega(a,b)=(1-a)\,f_ Omegaig( frac{1}{1-a},- frac{b}{1-a}ig)$ for quasidisks and establishing equalities among various spectra for repellers via thermodynamic formalism and Carleson-type estimates, as well as introducing a propagation mechanism for spectra across scales. Together, these results rigorize folklore relations, provide tools to compare geometric and conformal spectra, and enable approximation of universal spectra through repeller-driven analyses with potential applications in complex dynamics and boundary behavior of conformal maps.

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.

Figures (3)

• Figure 1: This is an example showing how to disks of the same radius, centered at $x$ and $y$, could be close but have very different rotations. The same is true for two concentric disks (centered at $x$) with different radii. The dashed curves are curves connecting $z_0$ with disks centered at $x$. The red lines are curves connecting $z_0$ with larger disks, while the blue ones connect it to the smaller disks.
• Figure 2: The red curves are the curves $C_\nu$ which replace the parts of $C+L$ which is outside $\Omega$.
• Figure 3: This figure sketches the different sets and different conformal maps described above. Note that the arc $\tilde{\alpha}$ lies on the real line, where the harmonic measure of the half disk is linear.

Theorems & Definitions (62)

• Definition 2.1
• Definition 2.2
• Remark 2.3
• Theorem 2.4
• Definition 2.5
• Theorem 2.6: The Relation Theorem
• Definition 2.7
• Remark 2.8
• Definition 2.9
• Lemma 2.10: The Main Lemma