[Paper Review] David extension of circle homeomorphisms, welding, mating, and removability

Mikhail Lyubich, Sergei Merenkov+2 more|arXiv (Cornell University)|Oct 21, 2020

Mathematical Dynamics and Fractals4 citations

TL;DR

This paper establishes a David extension theorem for circle homeomorphisms conjugating expansive covering maps of the circle, allowing parabolic periodic points to map to parabolic points and hyperbolic points to either hyperbolic or parabolic points. The key contribution is a unified framework using David homeomorphisms to construct matings between anti-rational maps and reflection groups, prove conformal removability of Julia and limit sets, and reprove the existence of Suffridge polynomials.

ABSTRACT

We provide a David extension result for circle homeomorphisms conjugating two dynamical systems such that parabolic periodic points go to parabolic periodic points, but hyperbolic points can go to parabolics as well. We use this result, in particular, to prove the existence of a new class of welding homeomorphisms, to establish an explicit dynamical connection between critically fixed anti-rational maps and kissing reflection groups, to show conformal removability of the Julia sets of geometrically finite polynomials and of the limit sets of necklace reflection groups, to produce matings of anti-polynomials and necklace reflection groups, and to give a new proof of the existence of Suffridge polynomials (extremal points in certain spaces of univalent maps).

Motivation & Objective

• To develop a new David extension theorem for circle homeomorphisms that conjugate expansive covering maps of the circle, even when hyperbolic points map to parabolic points.
• To establish a general surgery technique using David integrability to transform hyperbolic dynamical systems into parabolic ones.
• To prove conformal removability of Julia sets of geometrically finite polynomials and limit sets of necklace reflection groups.
• To construct matings between anti-polynomials and reflection groups via David welding and topological conjugacies.
• To provide a new proof of the existence of Suffridge polynomials as extremal points in spaces of univalent maps.

Proposed method

• Use a David extension theorem based on results by Chen, Chen, He and Zakeri to extend circle homeomorphisms conjugating expansive covering maps to the unit disk.
• Apply the David Integrability Theorem to ensure the extended homeomorphisms are quasiconformal in a generalized sense with controlled distortion.
• Construct a topological conjugacy between the dynamics on the circle and the dynamics on the boundary of reflection group limit sets using Markov partitions and Böttcher coordinates.
• Utilize conformal conjugacies on the basin of infinity and extend them continuously to the Julia/limit sets via local connectivity and density of preimages.
• Leverage the fact that conformally removable sets with a topological conjugacy that is conformal outside the set must be Möbius conjugate, leading to affine conjugacies.
• Use the asymptotic behavior of maps at infinity and coefficient matching to show that the conjugating map is a rotation, establishing the desired conjugacy.

Experimental results

Research questions

• RQ1 Can a circle homeomorphism conjugating two expansive covering maps of the circle be extended as a David homeomorphism if parabolic points are preserved and hyperbolic points may map to parabolic points?
• RQ2 Can such a David extension be used to construct a welding homeomorphism between the Julia set of an anti-rational map and the limit set of a reflection group?
• RQ3 Are the Julia sets of geometrically finite polynomials and the limit sets of necklace reflection groups conformally removable?
• RQ4 Can anti-polynomials be mated with reflection groups via David homeomorphisms to produce new dynamical systems?
• RQ5 Can the existence of Suffridge polynomials be rederived using a David surgery framework based on extremal quasiconformal maps?

Key findings

• A general David extension theorem is established for circle homeomorphisms conjugating expansive covering maps, provided parabolic periodic points of the source map are mapped to parabolic periodic points of the target map.
• The existence of a new class of welding homeomorphisms is proven, linking the dynamics of anti-rational maps and reflection groups via David extensions.
• The Julia sets of geometrically finite polynomials and the limit sets of necklace reflection groups are shown to be conformally removable using the David extension framework.
• Mating constructions between anti-polynomials and reflection groups are established, yielding new dynamical systems with topological conjugacy across the boundary.
• A new proof of the existence of Suffridge polynomials is obtained by realizing them as extremal points in spaces of univalent maps via David surgery.
• The conjugacy between the dynamics of two reflection group actions on the Riemann sphere is shown to be affine, implying that the conjugating map is a rotation by a root of unity, thus establishing a precise geometric correspondence.