Minimal-dilatation asymptotics for closed surfaces from the magic manifold

Let $\varSigma_g$ be a closed orientable surface of genus $g$. Write $\delta_g$ for the smallest dilatation of a pseudo-Anosov mapping class on $\varSigma_g$, and let $\delta_g^+$ denote the corresponding minimum among pseudo-Anosov mapping classes with orientable invariant foliations. Let $N(r)$ denote the Dehn filling of the magic 3-manifold along slope $r$ on one cusp.

Magic-manifold minimal-dilatation conjecture.

$$\lim_{g\to\infty}g\log\delta_g=\log\left(\frac{3+\sqrt{5}}{2}\right).$$

For large $g$, $\delta_g$ is achieved by the monodromy of a $\varSigma_g$-bundle over the circle obtained from either $N(\tfrac{3}{-2})$ or $N(\tfrac{1}{-2})$ by Dehn filling both cusps. Moreover,

$$\lim_{\substack{g\not\equiv 0\pmod{6}\g\to\infty}}g\log\delta_g^+=\log\left(\frac{3+\sqrt{5}}{2}\right).$$

For large $g$ such that $g\not\equiv0\pmod{6}$, $\delta_g^+$ is achieved by the monodromy of a $\varSigma_g$-bundle over the circle obtained from $N(\tfrac{3}{-2})$ or $N(\tfrac{1}{-2})$ by Dehn filling both cusps.

These conjectures describe the asymptotic smallest dilatations for pseudo-Anosov monodromies on closed surfaces and identify the relevant Dehn fillings of the magic manifold. The first half of the conjecture was also stated as a question by Hironaka; the proposed asymptotic and realization remain open in the source.