On the dynamics of a family of singularly perturbed rational maps
The authors study the rational map \(z_{j+1}=f(z_j)\) with \(f(z)=z^n(z^{2n}-\lambda^{n+1})/(z^{2n}-\lambda^{3n-1})\), where \(n>1,\lambda\neq 0\) and \(\lambda^{2n-2}\neq 1\). This map can be thought of as a perturbation of the map \(z_{j+1}=z_j^n\) and in that sense is similar to the McMullen map \(f_M(z)=z^n+\lambda/z^d\). The main result is the determination of conditions under which the Julia set of \(f\) is (i) a quasicircle, (ii) a Cantor set of circles, (iii) a Sierpinski carpet, (iv) a degenerated Sierpinski carpet.
scientific article; zbMATH DE number 6381769
scientific article