Suppose $f$ is a nonconstant analytic function, a point $L$ is said to be a fixed point if $f(L)=L$, we call the value $\lambda=f'(L)$ "multiplier". (By minimum modulus principle applied to $f(z)-z$, $L$ must exist.)
If the limit $\lim_{z\to{e^{i\beta}}\infty}f'(z)$(in a specific direction) exists and $\lim_{z\to{e^{i\beta}}\infty}f(z)-z=0$ we call the point at infinity also a fixed point of f. For instance, $f(z)=z+e^z$ has a fixed point at $-\infty$.
It's essential in dynamic systems whether $$\frac{\ln(\lambda)}{2\pi{i}}\notin\mathbb{Q}\setminus\mathbb{Z}.$$ If this holds, the dynamic systems have a continuous flow, otherwise it's ultra-difficult to gnerate such flow, even can be impossible. (Such fixed point is called constructable.)
So it's crucial to ask whether any analytic function has at least one fixed point $L$ satisfying $\frac{\ln(f'(L))}{2\pi{i}}\notin\mathbb{Q}\setminus\mathbb{Z}$, allowing anyone to build up a complex continuous flow?
The linear case $f(z)=az+b$ is not in discussion.
Any functions $g$ conjugate to the linear case $f$ (for a function $g$ there exists an analytic(can be multivalued) function $\phi$ that $\phi\circ{g}\circ\phi^{-1}=f$) are not in discussion. For instance, functions $2z\sqrt{1-z^2},\frac{az+b}{cz+d},az^r$ are conjugate to a linear case, hence they have closed-form flows, so we're not interested in these cases.
What we already know:
- A quadratic function $f(z)=az^2+bz+c$ always have at least one constructable fixed point. This can be proved simply through Vieta's theorem.
- Any polynomial functions having no multiple fixed points (or all fixed points are distinct) always have at least one constructable fixed point, proved by contradiction if assumed it has no such fixed points.