A disconnected deformation space of rational maps

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.