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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) 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.

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.

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

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

Purpose: I was trying to show that a particular type 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.

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

Question: Is there any necessary condition (except boundedness) 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.