A characterization of the dynamics of Schröder’s method for polynomials with two roots

Both the paper and the model show that, after an affine–Möbius conjugacy, the Schröder map for f(z)=(z−a)^m(z−b)^n and the auxiliary map T_{m,n} reduce to the quadratic monomial R_{m,n}(u)=−(n/m)u^2. The paper implements this via A(z)=1+2(z−a)/(a−b) and M(z)=(z−1)/(z+1), proving M∘T_{m,n}∘M^{-1}(u)=−(n/m)u^2 and then reading off the Julia set as an Apollonius circle or a line (Theorems 1–2) . The model performs the same conjugacy in one step with w=(z−a)/(z−b)=M∘A, again yielding w↦−(n/m)w^2 and the same Julia sets, with the same center–radius formulas. Thus the arguments are essentially the same conjugacy-based proof, and the formulas coincide with those in the paper .