Continuity of Julia set and its Hausdorff dimension of Yang-Lee zeros
In this paper, the authors studied the Julia set of the rational functions \[ S_{\lambda}(z)=\left(\frac{z^3+3(\lambda-1)z+(\lambda-1)(\lambda-2)}{3z^2+ 3(\lambda-2)z+\lambda^2-3\lambda+3}\right)^2, \] where \(\lambda\) is a real number. In fact, they proved that the Julia set \(J(S_{\lambda})\) of \(S_{\lambda}\) is discontinuous if and only if \(\lambda \in \{\alpha_0,0,\beta_0\},\) and the Hausdorff dimension \(HD(J(S_{\lambda}))\) of \(J(S_{\lambda})\) is continuous at \(\lambda\) if \(\lambda\in \mathbb{R}\setminus\{\alpha_0,0,\beta_0\},\) where \[ \alpha_0=\min_{1<t<2}\frac{(t^2-1)(\sqrt{4t+1}-3)}2,\quad \beta_0=\max_{0<t<1}\frac{(1-t^2)(\sqrt{4t+1}+3)}2. \]
scientific article; zbMATH DE number 5871212