On the dynamics of a family of singularly perturbed rational maps

In this paper, we study the dynamical behavior of the family of complex rational maps which is given byfλ(z)=zn(z2n−λn+1)z2n−λ3n−1, where n≥2n≥2 and λ∈C⁎−{λ:λ2n−2=1}λ∈C⁎−{λ:λ2n−2=1}. This family of rational maps can be seen as a perturbation of the unicritical polynomial z↦znz↦zn if λ   is small. We prove that the Julia set J(fλ)J(fλ) of fλfλ is either a quasicircle, a Cantor set of circles, a Sierpiński carpet or a degenerated Sierpiński carpet provided one of the free critical points of fλfλ is escaping to the origin or to the infinity. In particular, we prove that there exists suitable λ   such that the Julia set J(fλ)J(fλ) is a Cantor set of circles, but fλfλ is not topologically conjugate to any McMullen map on their corresponding Julia sets.