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

TL;DR

This paper investigates the relationship between Teichmüller and length spectrum metrics on Riemann surfaces derived from generalized Cantor sets, revealing conditions under which these metrics induce different topologies.

Contribution

It demonstrates that for certain generalized Cantor sets, the Teichmüller and length spectrum metrics define different topologies on the associated Teichmüller space.

Findings

For the middle-third Cantor set, metrics induce the same topology.

If  q_n = 1, metrics induce different topologies.

For some  q_n o 0, metrics induce different topologies.

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 $\inf q_n =0$.