March 2024 On the length spectrums of Riemann surfaces given by generalized Cantor sets
Erina Kinjo
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Kodai Math. J. 47(1): 34-51 (March 2024). DOI: 10.2996/kmj47103

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 $X_{E(\omega)}:=\hat{\mathbf{C}} \setminus E(\omega)$ 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 $\omega=\{q_n\}_{n=1}^{\infty}$ such that $\inf q_n =0$.

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Erina Kinjo. "On the length spectrums of Riemann surfaces given by generalized Cantor sets." Kodai Math. J. 47 (1) 34 - 51, March 2024. https://doi.org/10.2996/kmj47103

Information

Received: 11 November 2022; Revised: 26 May 2023; Published: March 2024
First available in Project Euclid: 13 March 2024

MathSciNet: MR4736283
Digital Object Identifier: 10.2996/kmj47103

Subjects:
Primary: 30F60
Secondary: 32G15

Keywords: generalized Cantor set , length spectrum , Riemann surface of infinite type , Teichmüller space

Rights: Copyright © 2024 Tokyo Institute of Technology, Department of Mathematics

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Vol.47 • No. 1 • March 2024
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