Small asymptotic translation lengths of pseudo-Anosov maps on the curve complex

Let $M$ be a hyperbolic fibered 3-manifold with $b_1(M)\ge 2$ and let $S$ be a fiber with pseudo-Anosov monodromy $\psi$. We show that there exists a sequence $(R_n,{\psi }_n)$ of fibers and monodromies contained in the fibered cone of $(S,\psi )$ such that the asymptotic translation length of ${\psi }_n$ on the curve complex $\mathcal{C}(R_n)$ behaves asymptotically like $1\mathrm{/}\mathrm{\mid }\chi (R_n){\mathrm{\mid }}^2$. As applications, we can reprove the previous result by Gadre–Tsai that the minimal asymptotic translation length of a closed surface of genus $g$ asymptotically behaves like $1\mathrm{/}{g}^2$. We also show that this also holds for the cases of hyperelliptic mapping class group and hyperelliptic handlebody group.