Critical heights on the moduli space of polynomials

TL;DR

This paper studies the structure of the moduli space of complex polynomials through critical heights, analyzing quotient spaces and their topological properties, revealing a compact, contractible space with a natural simplicial structure.

Contribution

It introduces the space ^* and its projectivization, showing their topological properties and linking simplices to conjugacy classes of stable polynomials.

Findings

^* is compact and contractible.

The shift locus has a natural simplicial structure.

Top simplices correspond to conjugacy classes of stable 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 $\mathcal{T}_d^*$ obtained by collapsing each connected component of a fiber of $G$ to a point. The space $\mathcal{T}_d^*$ 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 $\mathcal{T}_d^*$ to the space of trees $\mathcal{T}_d$. We show that the projectivization $\mathbb{P}\mathcal{T}_d^*$ is compact and contractible; further, the shift locus in $\mathbb{P}\mathcal{T}_d^*$ 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.