Limits of sequences of pseudo-Anosov maps and of hyperbolic 3-manifolds

There are two objects naturally associated with a braid $\beta \in B_n$ of pseudo-Anosov type: a (relative) pseudo-Anosov homeomorphism ${\phi }_{\beta }\text{\hspace{0.17em}⁣}:S^2\to S^2$; and the finite volume complete hyperbolic structure on the 3-manifold $M_{\beta }$ obtained by excising the braid closure of $\beta$, together with its braid axis, from $S^3$. We show the disconnect between these objects, by exhibiting a family of braids $\{{\beta }_q:q\in \mathbb{Q}\cap (0,1\mathrm{/}3]\}$ with the properties that: on the one hand, there is a fixed homeomorphism ${\phi }_0\text{\hspace{0.17em}⁣}:S^2\to S^2$ to which the (suitably normalized) homeomorphisms ${\phi }_{{\beta }_q}$ converge as $q\to 0$; while on the other hand, there are infinitely many distinct hyperbolic 3-manifolds which arise as geometric limits of the form $\lim_{k\to\infty} M_{\beta_{q_k}}$,forsequences$q_k\to 0$.