Edit: I revise the question according to the comment of Gabe Conant.

What is an example of a non constant entire function $f:\mathbb{C}\to \mathbb{C}$ which satisfy the following?:

For every $z\in \mathbb{C}$, the sequence $z,f(z),f^2(z),\ldots,f^n(z),\ldots$ is a bounded sequence but $f$ is not in the form $f(z)=\lambda z,\; |\lambda|\leq 1$.

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    $\begingroup$ Do you also want $f$ to be non-constant? $\endgroup$ – Gabe Conant Jul 18 at 11:11
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    $\begingroup$ @GabeConant Yes. Thank you. i revise the question. $\endgroup$ – Ali Taghavi Jul 18 at 11:14

Given an entire function $f\colon\mathbb{C}\to\mathbb{C}$, the escaping set, $I(f)$, is the set of $z\in\mathbb{C}$ such that $f^n(z)\to\infty$. Per the Wikipedia article, the escaping set of a non-linear entire function is nonempty.

The reference for this is On the iteration of entire functions by Eremenko.

  • $\begingroup$ Thanks for your answer. $\endgroup$ – Ali Taghavi Jul 18 at 11:31
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    $\begingroup$ @AliTaghavi You're welcome! I assume you also didn't want a linear example like $f(z)=(z+1)/2$. $\endgroup$ – Gabe Conant Jul 18 at 11:35
  • $\begingroup$ Yes I should exclude the linear case too. but i did not pay attention (and i was not aware of ) to this linear case $(z+1)/2$ $\endgroup$ – Ali Taghavi Jul 18 at 11:41
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    $\begingroup$ Dynamically, it is the same as $z\mapsto z/2$ with translation of the fixed point. $\endgroup$ – Ali Taghavi Jul 18 at 12:09

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