A disconnected deformation space of rational maps

TL;DR

This paper investigates the topology of deformation spaces of rational maps, revealing that certain quadratic rational maps have deformation spaces with infinitely many disconnected components.

Contribution

It demonstrates that some deformation spaces of rational maps are disconnected, providing explicit examples with infinitely many components, which was previously unknown.

Findings

Deformation spaces can be disconnected.

Some quadratic rational maps have infinitely many components.

Explicit examples of disconnected deformation spaces are constructed.

Abstract

Let $f:(\mathbb{P}^1,P)\to(\mathbb{P}^1,P)$ be a postcritically finite rational map with postcritical set $P$. William Thurston showed that $f$ induces a holomorphic pullback map $\sigma_f:\mathcal{T}_P\to\mathcal{T}_P$ on the Teichm\"uller space ${\mathcal T}_P:=\mathrm{Teich}(\mathbb{P}^1,P)$. If $f$ is not a flexible Latt\`es map, Thurston proved that $\sigma_f$ has a unique fixed point. In his PhD thesis, Adam Epstein generalized Thurston's ideas and defined a deformation space associated to a rational map $f:(\mathbb{P}^1,A)\to (\mathbb{P}^1,B)$ where $A \subseteq B$, allowing for maps $f$ which are not necessarily postcritically finite. By definition, the deformation space $\mathrm{Def}_B^A(f)\subseteq \mathcal{T}_B$ is the locus where the pullback map $\sigma_f:\mathcal{T}_B\to\mathcal{T}_A$ and the forgetful map $\sigma_A^B:\mathcal{T}_B\to\mathcal{T}_A$ agree. Using purely local arguments, Epstein showed that $\mathrm{Def}_B^A(f)$ is a smooth analytic submanifold of $\mathcal{T}_B$ of dimension $|B-A|$. In this article, we investigate the question of whether $\mathrm{Def}_B^A(f)$ is connected. We exhibit a family of quadratic rational maps for which the associated deformation spaces are disconnected; in fact, each has infinitely many components.