小熵闭曲面上的伪Anosovs和Whitehead姐妹链外
Pseudo-Anosovs on closed surfaces having small entropy and the Whitehead sister link exterior
摘要：
令$δ_g$是最小扩张的pseudo-Anosovs上的一个封闭的表面$Σ_g$的$g$和$δ_g^+$是最小扩张的pseudo-Anosovs上的$Σ_g$与定向不变的叶理。本文研究了幻流形$N$的每个$r \in \{-3/2, -1/2, 2\}$的$N(r)$的Dehn填充在闭纤维上作为单值出现的伪Anosovs.流形$N(-3/2)$同胚于Whitehead姐妹链外环。我们认为集合$Λ_g(r)$（resp.《易经》云：“凡所有相，皆是虚妄。具有可定向不变叶理的单值体）在$g$属的闭合纤维上，用于$N(r)$的Dehn填充物，其中填充物在$N(r)$的纤维的边界斜坡上。Hironaka分别通过计算$\min Λ_g(-1/2)$和$\min Λ^+_g(-1/2)$得到了$δ_g$和$δ_g^+$的上界。我们证明了$\min Λ_g(-3/2)< \min Λ_g(-1/2)$对于$g \equiv 0,1,5,6,7,9 \pmod{10}$和$\min Λ^+_g(-3/2)< \min Λ^+_g(-1/2)$对于$g \equiv 1,5,7,9 \pmod{10}$。这些不等式改进了$g$的$δ_g$和$δ_g^+$的上界。证明了对于每个$r \in \{-3/2, -1/2, 2\}$和每个$g \ge 3$，对于$N(r)$的Dehn填充，在$g $亏格的闭纤维上存在单值$Φ_g(r)$，使得其扩张$λ(Φ_g(r))$满足$\displaystyle \lim_{g \to \infty} |χ(Σ_g)| \log λ(Φ_g(r)) = 2 \log((3+\sqrt{5})/2)$ .
英文摘要：
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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