Moduli space of cubic Newton maps

TL;DR

This paper investigates the topology and bifurcation structure of the moduli space of cubic Newton maps, revealing boundary properties of hyperbolic components and characterizing the Head's angle map.

Contribution

It establishes the boundary structure of hyperbolic components and proves the surjectivity and monotonicity of the Head's angle map, confirming a conjecture of Tan Lei.

Findings

Boundary of the unbounded hyperbolic component is a Jordan arc.

Boundaries of all other hyperbolic components are Jordan curves.

The Head's angle map is surjective and monotone, with fibers fully characterized.

Abstract

In this article, we study the topology and bifurcations of the moduli space $\mathcal{M}_3$ of cubic Newton maps. It's a subspace of the moduli space of cubic rational maps, carrying the Riemann orbifold structure $(\mathbb{\widehat{C}}, (2,3,\infty))$. We prove two results: (1). The boundary of the unique unbounded hyperbolic component is a Jordan arc and the boundaries of all other hyperbolic components are Jordan curves. (2).The Head's angle map is surjective and monotone. The fibers of this map are characterized completely. The first result is a moduli space analogue of the first author's dynamical regularity theorem \cite{Ro08}. The second result confirms a conjecture of Tan Lei.