We show that the Mandelbrot set for the family of renormalization transformations of 2-dimensional diamond-like hierachical Potts models in statistical mechanics is connected. We also give an upper bound for the Hausdorff dimension of Julia set when it is a quasi-circle.
WANG XiaoGuang 1,QIU WeiYuan 1 , YIN YongCheng 1 , QIAO JianYong 2 & GAO JunYang 2 1 School of Mathematical Sciences, Fudan University, Shanghai 200433, China
