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 $f(z)=\lambda z,\; |\lambda|\leq 1$.

  • 2
    $\begingroup$ Do you also want $f$ to be non-constant? $\endgroup$ – Gabe Conant Jul 18 '19 at 11:11
  • 1
    $\begingroup$ @GabeConant Yes. Thank you. i revise the question. $\endgroup$ – Ali Taghavi Jul 18 '19 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.

| cite | improve this answer | |
  • $\begingroup$ Thanks for your answer. $\endgroup$ – Ali Taghavi Jul 18 '19 at 11:31
  • 1
    $\begingroup$ @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 '19 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 '19 at 11:41
  • 1
    $\begingroup$ Dynamically, it is the same as $z\mapsto z/2$ with translation of the fixed point. $\endgroup$ – Ali Taghavi Jul 18 '19 at 12:09

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.