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

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