Moduli space of cubic Newton maps

Abstract

In this article, we study the topology and bifurcations of the moduli space M3 of cubic Newton maps. It's a subspace of the moduli space of cubic rational maps, carrying the Riemann orbifold structure (Cˆ,(2,3,∞)). We prove two results:• The boundary of the unique unbounded hyperbolic component is a Jordan arc and the boundaries of all other hyperbolic components are Jordan curves;• 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 [37]. The second result confirms a conjecture of Tan Lei.