Global topology of hyperbolic components I: Cantor circle case

TL;DR

This paper investigates the topological structure of hyperbolic components in the moduli space of degree d rational maps, focusing on the Cantor circle locus and revealing their quotient space structure.

Contribution

It characterizes hyperbolic components in the Cantor circle locus as finite quotients of a product of Euclidean space and tori, extending understanding beyond the connectedness locus.

Findings

Hyperbolic components are finite quotients of R^{4d-4-n}×T^{n}.

The structure depends on the dynamics of the maps.

The proof uses Riemann surface theory, dynamical systems, and algebraic topology.

Abstract

The hyperbolic components in the moduli space ${M}_d$ of degree $d\geq2$ rational maps are mysterious and fundamental topological objects. For those in the connectedness locus, they are known to be the finite quotients of the Euclidean space $\mathbb{R}^{4d-4}$. In this paper, we study the hyperbolic components in the disconnectedness locus and with minimal complexity: those in the Cantor circle locus. We show that each of them is a finite quotient of the space $\mathbb{R}^{4d-4-n}\times\mathbb{T}^{n}$, where $n$ is determined by the dynamics. The proof relates Riemann surface theory (Abel's Theorem), dynamical system and algebraic topology.