On the set of normalized dilatations of fully-punctured pseudo-Anosov maps

Abstract

We improve the bound on the number of tetrahedra in the veering triangulation of a fully-punctured pseudo-Anosov mapping torus in terms of the normalized dilatation. When the mapping torus has only one boundary component, we can improve the bound further. Together with the author's work with Hironaka in the case when the mapping torus has at least two boundary components, this allows us to understand small elements of the set $\mathcal{D}$ of normalized dilatations of fully-punctured pseudo-Anosov maps using computational means. In particular, we certify that the minimum element of $\mathcal{D}$ is $\mu^2$ and the minimum accumulation point of $\mathcal{D}$ is $\mu^4$, where $\mu$ is the golden ratio.