On the length spectrums of Riemann surfaces given by generalized Cantor sets

Abstract

For a generalized Cantor set $E(\omega)$ with respect to a sequence $\omega=\{ q_n \}_{n=1}^{\infty} \subset (0,1)$, we consider Riemann surface $X_{E(\omega)}:=\hat{\mathbb{C}} \setminus E(\omega)$ and metrics on Teichm\"uller space $T(X_{E(\omega)})$ of $X_{E(\omega)}$. If $E(\omega) = \mathcal{C}$ ( the middle one-third Cantor set), we find that on $T(X_{\mathcal{C}})$, Teichm\"uller 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 $T(X_{E(\omega)})$ 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 $T(X_{E(\omega)})$ for some $\omega=\{ q_n \}_{n=1}^{\infty}$ such that $\inf q_n =0$.