The Hausdorff dimension of the Julia sets concerning generated renormalization transformation

Abstract

Considering a family of rational map $ {U_{mn\lambda }} $ of the renormalization transformation of the generalized diamond hierarchical Potts model, we give the asymptotic formula of the Hausdorff dimension of the Julia sets of $ {U_{mn\lambda }} $ as the parameter $ \lambda $ tends to infinity, here $ {U_{mn\lambda }} = {\left[ {\frac{{{{\left( {z + \lambda - 1} \right)}^n} + \left( {\lambda - 1} \right){{\left( {z - 1} \right)}^n}}}{{{{\left( {z + \lambda - 1} \right)}^n} - {{\left( {z - 1} \right)}^n}}}} \right]^m}, $ where $ m \ge 2 $, $ n \ge 2 $ are two natural numbers, $ \lambda \in {{\mathbb{C}} } $.

Keywords

• renormlization transformation
• potts model
• asymptotic formula
• hausdorff dimension
• julia set