Polynomial basins of infinity

TL;DR

This paper investigates the structure of polynomial basins of infinity, proving that the fibers of a certain projection are connected, which implies connectedness of various conjugacy classes, using analysis of model surfaces and maps.

Contribution

It establishes the connectedness of fibers of the projection from polynomial classes to basin classes, advancing understanding of polynomial dynamics and conjugacy classes.

Findings

All fibers of the projection are connected.

Quasiconformal and topological basin conjugacy classes are connected.

Analysis of model surfaces and maps is key to the proof.

Abstract

We study the projection $\pi: M_d \to B_d$ which sends an affine conjugacy class of polynomial $f: \mathbb{C}\to\mathbb{C}$ to the holomorphic conjugacy class of the restriction of $f$ to its basin of infinity. When $B_d$ is equipped with a dynamically natural Gromov-Hausdorff topology, the map $\pi$ becomes continuous and a homeomorphism on the shift locus. Our main result is that all fibers of $\pi$ are connected. Consequently, quasiconformal and topological basin-of-infinity conjugacy classes are also connected. The key ingredient in the proof is an analysis of model surfaces and model maps, branched covers between translation surfaces which model the local behavior of a polynomial.