On the parabolic Fatou domains

2025/09/12 by Gao, Ning, Gao, Yan, Peng, Wenjuan
#37F10 #37F20 #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2509.09914

Abstract

Let f be a rational map with an infinitely-connected fixed parabolic Fatou domain U. We prove that there exists a rational map g with a completely invariant parabolic Fatou domain V, such that (f,U) and (g,V) are conformally conjugate, and each non-singleton Julia component of g is a Jordan curve which bounds a superattracting Fatou domain of g containing at most one postcritical point. Furthermore, we show that if the Julia set of f is a Cantor set, then the parabolic Fatou domain can be perturbed into an attracting one without affecting the topology of the Julia set.

Citations

• Hyperbolic-parabolic deformations of rational maps
• No invariant line fields on Cantor Julia sets
• Proof of the Branner-Hubbard conjucture on Cantor Julia sets
• Laminations in holomorphic dynamics

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