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This question may be trivial but still I want to know the answer.

Question: Is there any necessary condition (except boundedness of the Fatou component) for the existence of a multiply connected Fatou component of a transcendental entire function?

Purpose: I was trying to show that a particular type of transcendental entire function has a Baker wandering domain but I was stuck to show that the function has a multiply connected Fatou component. In this regard I need to know the answer of my question.

Definition: A Fatou component $W$ is called wandering if $W_m\neq W_n$ for $m\neq n$ where $m$ and $n$ are integers. A Baker wandering domain is a wandering component $W$ of the Fatou set $\mathcal{F}(f)$ of $f$ such that, for $n$ large enough, $W_n$ is bounded, multiply connected and surrounds $0$, and $f^n(z)\to\infty$ as $n\to\infty$ for $z\in W$, where $W_n$ is the Fatou component containing $f^n{(W)}$.

Any suggestion or direction will be highly appreciated. Thank you.

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  • $\begingroup$ In your question, what "except boundedness" means? Boundedness of what? $\endgroup$
    – Alexandre Eremenko
    Commented Feb 7 at 13:15
  • $\begingroup$ I mean to say bounded Fatou components $\endgroup$
    – Factorial_zero
    Commented Feb 7 at 13:19
  • $\begingroup$ One such necessary condition is that the set of critical and asymptotic values is unbounded. $\endgroup$
    – Alexandre Eremenko
    Commented Feb 7 at 13:27
  • $\begingroup$ Thank you Sir @Alexandre Eremenko $\endgroup$
    – Factorial_zero
    Commented Feb 7 at 13:29

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