Hyperbolic Components

TL;DR

This paper studies the structure and classification of bounded hyperbolic components in polynomial map parameter spaces, showing they are topological cells with unique centers and classifying them via reduced mapping schemes.

Contribution

It establishes that each bounded hyperbolic component is a topological cell with a unique center, and classifies components by reduced mapping schemes, extending results to real and rational maps.

Findings

Bounded hyperbolic components are topological cells.

Each component contains a unique post-critically finite map.

Components with the same reduced scheme are biholomorphic.

Abstract

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. In the space of suitably normalized maps of this type, the hyperbolic maps form an open set called the hyperbolic locus. The various connected components of this hyperbolic locus are called hyperbolic components, and those hyperbolic components with compact closure (or equivalently those contained in the "connectedness locus") are called bounded hyperbolic components. It is shown that each bounded hyperbolic component is a topological cell containing a unique post-critically finite map called its center point. For each degree $d$, the bounded hyperbolic components can be separated into finitely many distinct types, each of which is characterized by a suitable reduced mapping scheme $\bar S_f$. Any two components with the same reduced mapping scheme are canonically biholomorphic to each other. There are similar statementsfor real polynomial maps, for polynomial maps with marked critical points, and for rational maps. Appendix A, by Alfredo Poirier, proves that every reduced mapping scheme can be represented by some classical hyperbolic component, made up of polynomial maps of $\C$. This paper is a revised version of [M2], which was circulated but not published in 1992.