Collet-Eckmann type conditions and conformal welding of unicritical quadratic laminations

TL;DR

This paper introduces a Collet-Eckmann type condition for unicritical laminations, showing it guarantees a H"older continuous conformal welding that yields Julia sets, providing new insights into quadratic polynomial dynamics.

Contribution

It establishes a novel Collet-Eckmann condition for unicritical laminations and links it to conformal welding and Julia set formation, offering a new proof for properties of quadratic polynomials.

Findings

Collet-Eckmann condition implies H"older continuous conformal welding.

Almost all angles produce quadratic polynomials with H"older Fatou components.

New proof avoiding Beurling's theorem for quadratic polynomial properties.

Abstract

In this paper, we introduce a Collet-Eckmann type condition for the unicritical laminations on the unit circle. We prove that this condition implies the lamination admits a H\"older continuous conformal welding which produces a Julia set for some unicritical polynomial. In consequence, we present a new proof that almost all angles on the unit circle produce quadratic polynomials with H\"older Fatou components, without the use of Beurling's theorem.