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: (relative) homeomorphism $\varphi_\beta\colon S^2\to S^2$; and the finite volume complete hyperbolic structure on 3-manifold $M_\beta$ obtained by excising closure $\beta$, together its axis, from $S^3$. We show disconnect between these objects, exhibiting family braids $\{\beta_q:q\in{\mathbb{Q}}\cap(0,1/3]\}$ properties that: one hand, there is fixed $\varphi_0\colon S^2$ to which (suitably normalized) homeomorphisms $\varphi_{\beta_{q}}$ converge as $q\to 0$; while other infinitely many distinct 3-manifolds arise geometric limits form $\lim_{k\to\infty} M_{\beta_{q_k}}$, for sequences $q_k\to 0$.