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Hausdorff dimensions of quasilines varying in the universal Teichmüller space
Author(s) -
Huo Shengjin,
Wu Shengjian
Publication year - 2013
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/bdt018
Subject(s) - mathematics , hausdorff dimension , hausdorff space , pure mathematics , space (punctuation) , dimension (graph theory) , hausdorff measure , set (abstract data type) , hausdorff distance , urysohn and completely hausdorff spaces , mathematical analysis , computer science , philosophy , linguistics , programming language
The aim of this paper is to study the relations between the Hausdorff dimensions of k ‐quasilines and the theory of extremal quasiconformal mappings. We show that there is an open and dense subset (Strebel points) of the universal Teichmüller space T (ℍ) such that, for every [ f ] in the set, the Hausdorff dimension of the k ‐quasiline determined by [ f ] is strictly less than 1 + k 2 . We also show that there are some points [ f ] ≠ [id] outside the open and dense set in the universal Teichmüller space such that the Hausdorff dimension of the quasiline determined by [ f ] is 1. Moreover, some results on the Hausdorff dimensions of the quasilines varying in the asymptotic Teichmüller space are also obtained.