Hyperbolic components in spaces of polynomial maps

TL;DR

This paper studies the structure of hyperbolic components in spaces of polynomial maps, showing they are topological cells with unique centers, classified by a simplified invariant called the reduced mapping schema.

Contribution

It proves that each hyperbolic component is a topological cell with a unique center, and classifies components by a reduced mapping schema based on topological data.

Findings

Hyperbolic components are topological cells.

Each component contains a unique post-critically finite map.

Components are classified by a reduced mapping schema.

Abstract

We consider polynomial maps $f:\C\to\C$ of degree $d\ge 2$, or more generally polynomial maps from a finite union of copies of $\C$ to itself which have degree two or more on each copy. In any space $\p^{S}$ of suitably normalized maps of this type, the post-critically bounded maps form a compact subset $\cl^{S}$ called the connectedness locus, and the hyperbolic maps in $\cl^{S}$ form an open set $\hl^{S}$ called the hyperbolic connectedness locus. The various connected components $H_\alpha\subset \hl^{S}$ are called hyperbolic components. It is shown that each hyperbolic component is a topological cell, containing a unique post-critically finite map which is called its center point. These hyperbolic components can be separated into finitely many distinct ``types'', each of which is characterized by a suitable reduced mapping schema $\bar S(f)$. This is a rather crude invariant, which depends only on the topology of $f$ restricted to the complement of the Julia set. Any two components with the same reduced mapping schema are canonically biholomorphic to each other. There are similar statements for real polynomial maps, or for maps with marked critical points.