On a metric view of the polynomial shift locus

We relate generic points in the shift locus ${\mathcal{S}}_D$ of degree $D\ge 2$ polynomials to metric graphs. Using thermodynamic metrics on the space of metric graphs, we obtain a distance function ${\rho }_D$ on ${\mathcal{S}}_D$. We study the (in)completeness of the metric space $({\mathcal{S}}_D,{\rho }_D)$. We prove that when $D\ge 3$, the space $({\mathcal{S}}_D,{\rho }_D)$ is incomplete and its metric completion contains a subset homeomorphic to the space $\mathbb{P}{\mathcal{S}\mathcal{T}}_D^{\ast }$ introduced by DeMarco and Pilgrim. This provides a new way to understand the space $\mathbb{P}{\mathcal{S}\mathcal{T}}_D^{\ast }$.