# An entire function all whose forward orbits are bounded

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

• Do you also want $f$ to be non-constant? – Gabe Conant Jul 18 '19 at 11:11
• @GabeConant Yes. Thank you. i revise the question. – 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.
• @AliTaghavi You're welcome! I assume you also didn't want a linear example like $f(z)=(z+1)/2$. – Gabe Conant Jul 18 '19 at 11:35
• Yes I should exclude the linear case too. but i did not pay attention (and i was not aware of ) to this linear case $(z+1)/2$ – Ali Taghavi Jul 18 '19 at 11:41
• Dynamically, it is the same as $z\mapsto z/2$ with translation of the fixed point. – Ali Taghavi Jul 18 '19 at 12:09