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

Abstract

For a generalized Cantor set E(ω) with respect to a sequence ω = {qn} ∞ n=1 ⊂ (0, 1), we consider Riemann surface X E(ω) := Ĉ \ E(ω) and metrics on Teichmüller space T (X E(ω) ) of X E(ω) . If E(ω) = C ( the middle one-third Cantor set), we find that on T (X 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 qn = 1. On the other hand, it is not easy to judge whether the metrics define the same topology or not if inf qn = 0. In this paper, we show that the two metrics define different topologies on T (X E(ω) ) for some ω = {qn} ∞ n=1 such that inf qn = 0.