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The failure of analyticity of Hausdorff dimensions of quasi-circles of Fuchsian groups of the second kind


Authors: Shengjin Huo and Shengjian Wu
Journal: Proc. Amer. Math. Soc. 143 (2015), 1101-1108
MSC (2010): Primary 30C62, 30F35, 30F60
DOI: https://doi.org/10.1090/S0002-9939-2014-12205-2
Published electronically: November 4, 2014
MathSciNet review: 3293725
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Abstract: Let $ \Gamma $ be a Fuchsian group. Any $ [\mu ]$ in the Teichmüller space $ T(\Gamma )$ determines a quasi-circle $ f_{\mu }(\partial \mathbb{D}).$ In this paper, we prove that, for any Fuchsian group $ \Gamma $ of the second kind, the Hausdorff dimension $ \delta ([\mu ])=dim f_{\mu }(\partial \mathbb{D})$ is not a real analytic function in $ T(\Gamma )$.


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Additional Information

Shengjin Huo
Affiliation: LMAM and School of Mathematical Sciences, Peking University, Beijing, 100871, People’s Republic of China
Address at time of publication: Department of Mathematics, Tianjin Polytechnic University, Tianjin, 300387, People’s Republic of China
Email: sjhuo@pku.edu.cn

Shengjian Wu
Affiliation: LMAM and School of Mathematical Sciences, Peking University, Beijing, 100871, People’s Republic of China
Email: wusj@math.pku.edu.cn

DOI: https://doi.org/10.1090/S0002-9939-2014-12205-2
Keywords: Fuchsian group, Carleson measure, Hausdorff dimension, Teichm\"uller space
Received by editor(s): April 20, 2012
Received by editor(s) in revised form: March 11, 2013, and March 13, 2013
Published electronically: November 4, 2014
Additional Notes: The authors were supported by the National Natural Science Foundation of China (Grant No. 11371035 and Grant No. 11401432)
Communicated by: Jeremy Tyson
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.