Critical heights on the moduli space of polynomials

Abstract

Let $M_d$ be the moduli space of one-dimensional complex polynomial dynamical systems. The escape rates of the critical points determine a critical heights map $G:M_d\to {\mathbb{R}}^{d-1}$. For generic values of $G$, each connected component of a fiber of $G$ is the deformation space for twist deformations on the basin of infinity. We analyze the quotient space ${\mathcal{T}}_d^{\ast }$ obtained by collapsing each connected component of a fiber of $G$ to a point. The space ${\mathcal{T}}_d^{\ast }$ is a parameter-space analog of the polynomial tree $T(f)$ associated to a polynomial $f:\mathbb{C}\to \mathbb{C}$, studied by DeMarco and McMullen, and there is a natural projection from ${\mathcal{T}}_d^{\ast }$ to the space of trees ${\mathcal{T}}_d$. We show that the projectivization $\mathbb{P}{\mathcal{T}}_d^{\ast }$ is compact and contractible; further, the shift locus in $\mathbb{P}{\mathcal{T}}_d^{\ast }$ has a canonical locally finite simplicial structure. The top-dimensional simplices are in one-to-one corespondence with topological conjugacy classes of structurally stable polynomials in the shift locus.

Cited by in corpus (1)

• Enumerating the basins of infinity of cubic polynomials