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| Format: | Preprint |
| Published: |
2022
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2211.04897 |
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Table of Contents:
- 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 $\inf q_n =0$.