论文标题
在Riemann表面的长度光谱上
On the length spectrums of Riemann surfaces given by generalized Cantor sets
论文作者
论文摘要
对于序列相对于序列$ω= \ {q_n \} _ {n = 1}^{\ infty} \ subset（0,1）$，我们考虑riemann surface $ x_ {e（em x）在teichmüller空间上$ t（x_ {e（ω）}）$ $ x_ {e（ω）} $。如果$ e（ω）= \ MATHCAL {C} $（中间三分之一的Cantor集），我们发现在$ t（x _ {\ Mathcal {c}}）$上，TeichMüllerMetric$ d_t $定义了相同的拓扑，因为长度spectrum $ d_l $ d_l $。另外，我们可以轻松地检查$ d_t $不会定义与$ t（x_ {e（ω）}）$ $ d_l $的拓扑，如果$ \ sup q_n = 1 $。另一方面，如果$ \ inf q_n = 0 $，判断指标是否定义相同的拓扑并不容易。在本文中，我们表明，这两个指标为某些$ω= \ {q_n \} _ {n = 1}^{\ infty} $上的$ t（x_ {e（ω）}）$定义了不同的拓扑。
For a generalized Cantor set $E(ω)$ with respect to a sequence $ω=\{ q_n \}_{n=1}^{\infty} \subset (0,1)$, we consider Riemann surface $X_{E(ω)}:=\hat{\mathbb{C}} \setminus E(ω)$ and metrics on Teichmüller space $T(X_{E(ω)})$ of $X_{E(ω)}$. If $E(ω) = \mathcal{C}$ ( the middle one-third Cantor set), we find that on $T(X_{\mathcal{C}})$, Teichmüller metric $d_T$ defines the same topology as that of the length spectrum metric $d_L$. Also, we can easily check that $d_T$ does not define the same topology as that of $d_L$ on $T(X_{E(ω)})$ if $\sup q_n =1$. On the other hand, it is not easy to judge whether the metrics define the same topology or not if $\inf q_n =0$. In this paper, we show that the two metrics define different topologies on $T(X_{E(ω)})$ for some $ω=\{ q_n \}_{n=1}^{\infty}$ such that $\inf q_n =0$.
