关于Riemann surfaces的某些伪Anosov映射的穿孔表示
On representations of certain pseudo-Anosov maps of Riemann surfaces with punctures
摘要：
设$S$是一个具有$3p+n>4$和$n\geq 1$的$(p,n)$型黎曼曲面.设$α_1,α_2\subset S$是两个简单的闭合测地线，使得$\{α_1, α_2\}$填充$S$。Thurston指出，通过沿$α_1$和$α_2$的Dehn扭曲获得的大多数映射都是伪Anosov映射。让$a$成为一个穿刺。本文研究了$S$上的伪Anosov映射族$\mathcal{F}(S,a)$，当$a$被填充时，该映射族投射到平凡映射上，并证明了在$\mathcal{F}(S,a)$中有无穷多个元素不能由沿两条填充测地线的Dehn扭得到.我们进一步刻画了$\mathcal{F}(S,a)$中所有可以由两条填充测地线构造的元素。最后，对任意点$b\in S$，我们得到了$S\backslash \{b\}$上的一类伪Anosov映射$\mathcal{H}$，这类映射不是由Thurston构造得到的，并且当$b$被填充时，它们投影到元素$χ\in \mathcal{F}(S,a)$上，并讨论了$\mathcal{H}$中元素的一些性质.
英文摘要：
Let $S$ be a Riemann surface of type $(p,n)$ with $3p+n>4$ and $n\geq 1$. Let $α_1,α_2\subset S$ be two simple closed geodesics such that $\{α_1, α_2\}$ fills $S$. It was shown by Thurston that most maps obtained through Dehn twists along $α_1$ and $α_2$ are pseudo-Anosov. Let $a$ be a puncture. In this paper, we study the family $\mathcal{F}(S,a)$ of pseudo-Anosov maps on $S$ that projects to the trivial map as $a$ is filled in, and show that there are infinitely many elements in $\mathcal{F}(S,a)$ that cannot be obtained from Dehn twists along two filling geodesics. We further characterize all elements in $\mathcal{F}(S,a)$ that can be constructed by two filling geodesics. Finally, for any point $b\in S$, we obtain a family $\mathcal{H}$ of pseudo-Anosov maps on $S\backslash \{b\}$ that is not obtained from Thurston's construction and projects to an element $χ\in \mathcal{F}(S,a)$ as $b$ is filled in, some properties of elements in $\mathcal{H}$ are also discussed.
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