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

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 $T(\Gamma )$.