• Connectivity of the Julia sets of singularly perturbed rational maps

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      https://www.ias.ac.in/article/fulltext/pmsc/129/03/0032

    • Keywords

       

      Julia sets; connectivity; Herman ring; Cantor circles

    • Abstract

       

      We consider a family of rational functions which is given by

      $$f_{\lambda}(z)=\frac{z^n(z^{2n}-\lambda^{n+1})}{z^{2n}-\lambda^{3n-1}},$$

      where $n\geq 2$ and $\lambda\in\mathbb{C}^{*}-\{\lambda:\lambda^{2n-2}=1\}$.When $\lambda\neq 0$ is small, $f_{\lambda}$ can be seen as a perturbation of the unicritical polynomial $z\mapsto z^{n}$. It was known that in this case the Julia set $J(f_\lambda)$ of $f_\lambda$ is a Cantor set of circles on which the dynamics of $f_\lambda$ is not topologically conjugate to that of any McMullen maps. In this paper, we prove that this is the unique case such that $J(f_\lambda)$ is disconnected.

    • Author Affiliations

       

      JIANXUN FU1 YANHUA ZHANG2

      1. Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China
      2. Department of Mathematics, Qufu Normal University, Qufu 273165, People’s Republic of China
    • Dates

       
  • Proceedings – Mathematical Sciences | News

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