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

DOI DOI 10 References Open Access
  • Kinjo Erina
    Department of Mechanical Engineering, Ehime University

Description

<p>For a generalized Cantor set E(ω) with respect to a sequence <img align="middle" src="./Graphics/abst-1.jpg"/>, we consider Riemann surface <img align="middle" src="./Graphics/abst-2.jpg"/> and metrics on Teichmüller space T(XE(ω)) of XE(ω). If E(ω) = <img align="middle" src="./Graphics/abst-3.jpg"/> (the middle one-third Cantor set), we find that on <img align="middle" src="./Graphics/abst-4.jpg"/>, Teichmüller metric dT defines the same topology as that of the length spectrum metric dL. Also, we can easily check that dT does not define the same topology as that of dL on T(XE(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(XE(ω)) for some <img align="middle" src="./Graphics/abst-5.jpg"/> such that inf qn = 0.</p>

Journal

  • Kodai Mathematical Journal

    Kodai Mathematical Journal 47 (1), 34-51, 2024-03-11

    Department of Mathematics, Tokyo Institute of Technology

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