The Hausdorff dimension of the Julia sets concerning generated renormalization transformation

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, $\lambda \in \mathbb{C}$.