Pseudo-Anosovs on closed surfaces having small entropy and the Whitehead sister link exterior

Let $δ_g$ be the minimal dilatation for pseudo-Anosovs on a closed surface $Σ_g$ of genus $g$ and let $δ_g^+$ be the minimal dilatation for pseudo-Anosovs on $Σ_g$ with orientable invariant foliations. This paper concerns the pseudo-Anosovs which occur as the monodromies on closed fibers for Dehn fillings of $N(r)$ for each $r \in \{-3/2, -1/2, 2\}$ of the magic manifold $N$. The manifold $N(-3/2)$ is homeomorphic to the Whitehead sister link exterior. We consider the set $Λ_g(r)$ (resp. $Λ_g^+(r)$) which consists of the dilatations of all monodromies (resp. monodromies having orientable invariant foliations) on a closed fiber of genus $g$ for Dehn fillings of $N(r)$, where the fillings are on the boundary slopes of fibers of $N(r)$. Hironaka obtained upper bounds of $δ_g$ and $δ_g^+$ by computing $\min Λ_g(-1/2)$ and $\min Λ^+_g(-1/2)$ respectively. We prove that $\min Λ_g(-3/2)< \min Λ_g(-1/2)$ for $g \equiv 0,1,5,6,7,9 \pmod{10}$ and $\min Λ^+_g(-3/2)< \min Λ^+_g(-1/2)$ for $g \equiv 1,5,7,9 \pmod{10}$. These inequalities improve the previous upper bounds of $δ_g$ and $δ_g^+$ for these $g$. We prove that for each $r \in \{-3/2, -1/2, 2\}$ and each $g \ge 3$, there exists a monodromy $Φ_g(r)$ on a closed fiber of genus $g$ for a Dehn filling of $N(r)$ such that its dilatation $λ(Φ_g(r))$ satisfies $\displaystyle \lim_{g \to \infty} |χ(Σ_g)| \log λ(Φ_g(r)) = 2 \log((3+\sqrt{5})/2)$.

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