The rational maps \(F_\lambda(z)= z^m+ \lambda/z^d\) have no Herman rings

DOI10.1007/s12044-010-0044-xzbMath1206.37025OpenAlexW2013637022MaRDI QIDQ611054

Wei-Yuan Qiu, Ying Qing Xiao

Publication date: 14 December 2010

Published in: Proceedings of the Indian Academy of Sciences. Mathematical Sciences (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1007/s12044-010-0044-x

zbMATH Keywords

Julia setrational mapHerman ring

Mathematics Subject Classification ID

Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable (30D05) Small divisors, rotation domains and linearization in holomorphic dynamics (37F50) Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets (37F10)

Related Items (8)

Singular perturbations with multiple poles of the simple polynomials ⋮ Dynamics of regularly ramified rational maps. I: Julia sets of maps in one-parameter families ⋮ Connectivity of the Julia sets of singularly perturbed rational maps ⋮ Rational maps without Herman rings ⋮ Singular perturbations of the unicritical polynomials with two parameters ⋮ No Herman rings for regularly ramified rational maps ⋮ Relating singularly perturbed rational maps to families of entire maps ⋮ Proof of a folklore Julia set connectedness theorem and connections with elliptic functions

Cites Work

• The escape trichotomy for singularly perturbed rational maps
• On the dynamics of the McMullen family 𝑅(𝑧)=𝑧^{𝑚}+𝜆/𝑧^{ℓ}
• On the quasiconformal surgery of rational functions
• The rational maps $z\mapsto 1+1/\omega z^d$ have no Herman rings
• The McMullen domain: Satellite Mandelbrot sets and Sierpinski holes
• Symbolic dynamics for a Sierpinski curve Julia set
• Structure of the McMullen domain in the parameter planes for rational maps
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