Number of fixed points of a meromorphic function

Question

I would like to know whether a meromorphic function on the whole complex plane with at most one pole can have infinitely many fixed points or not. Many thanks in advance.

Answer 1

An answer to your question may be that the two situations can occur.

Consider the meromorphic function: $$f(z)=\frac{\cos(z)}{z}+z.$$

Note that $z_{0}=0$ is a pole of degree 1 of $f$, indeed: $$f(z)=\frac{\cos(z)}{z}+z=\frac{1}{z}\sum_{n=0}^{\infty}(-1)^{n}\frac{z^{2n}}{n!}+z=\frac{1}{z}+z+\sum_{n=1}^{\infty}(-1)^{n}\frac{z^{2n}}{n!}.$$

Also, note that $f$ has infinitely many fixed points because, in particular, the points of the form: $$z=\pi n -\frac{\pi}{2} \qquad \mbox{ for each }n\in \mathbb{Z}$$ are fixed points of $f$. So, $f$ is a function that satisfies your requirements and has infinitely many fixed points.

For other hand, consider $g(z)=\frac{1}{z}$, clearly $g$ has a pole in $z_{0}=0$ of degree 1, but $g$ have not infinitely many fixed points.