Articles written in Proceedings – Mathematical Sciences

    • Connectivity of the Julia sets of singularly perturbed rational maps


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      We consider a family of rational functions which is given by


      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.

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