Entire function invariant under different rotations is constant? How can we reason to show that an entire function that's invariant under two rotations of the plane, must be constant ? Assume the rotations are around different axes, and by rational multiples of $\pi$.
Concretely, let $f$ be an entire function. Let $\varphi(z)=e^{i\theta\pi}z$, $\theta\in \mathbb{Q}$, be a rotation around $0$, and let $\psi(z)=1+e^{i\eta\pi}(z-1)$, $\eta\in \mathbb{Q}$,  be a rotation around $1$. How to show that if $$f(\varphi(z))=f(z),\;\;\; f(\psi(z))=f(z)$$ for all $z\in \mathbb{C}$, then $f$ must be a constant function.
 A: Let $\mathscr{C} \subset \mathbb{C}$ be the unit circle.
For any $\omega \in \mathscr{C}$, define two maps:
$$
\phi_{\omega} : \mathbb{C} \ni z \mapsto \omega z \in \mathbb{C}\,\,\,\text{ and }\,\,\,
\psi_{\omega} : \mathbb{C} \ni z \mapsto 1 + \omega (z - 1) \in \mathbb{C}
$$
If $f$ is an entire function on $\mathbb{C}$ invariant under the coordinate transform $\phi_{\zeta}$ and $\psi_{\eta}$ for some $\zeta, \eta \in \mathscr{C}\setminus \{1\}$, i.e. 
$$ f\circ\phi_{\zeta} = f\circ\psi_{\eta} = f$$
$f$ will be invariant under their inverses $\phi_{\zeta}^{-1} = \phi_{\zeta^{-1}}$ and $\psi_{\eta}^{-1} = \psi_{\eta^{-1}}$ and hence under 
$\phi_{\zeta}\circ\psi_{\eta}\circ\phi_{\zeta^{-1}}\circ\psi_{\eta^{-1}}$, i.e.
$$f\circ\phi_{\zeta}\circ\psi_{\eta}\circ\phi_{\zeta^{-1}}\circ\psi_{\eta^{-1}} = f$$
But $\phi_{\zeta}\circ\psi_{\eta}\circ\phi_{\zeta^{-1}}\circ\psi_{\eta^{-1}}$ is the map
$$\mathbb{C} \ni z \mapsto z - (1-\zeta)(1-\eta) \in \mathbb{C}$$
which is a translation. 
If at least one of $\zeta, \eta \ne -1$, say $\eta \ne -1$, then by a similar argument on
$\zeta, \eta^{-1}$, we find $f$ is invariant under another translation: 
$$\mathbb{C} \ni z \mapsto z - (1-\zeta)(1-\eta^{-1}) \in \mathbb{C}$$
It is easy to check these two translations are linear independent from each other.
As a result, $f$ will be a doubly periodic entire function.
The fundamental domain of the lattice formed by the two translations is a compact
subset of $\mathbb{C}$. Since $f$ is continuous, $f$ is bounded on it. 
By double periodicity of $f$, $f$ is bounded over all $\mathbb{C}$. By Liouville's theorem, $f$ is a constant.
Conclusion: If an entire function $f$ is invariant under rotations around two points, some of which are rotations with angle differ from $180^{\circ}$, then $f$ is a constant.
A: If the orbit of some point is dense in the plane, then any continuous function invariant under the two rotations must be constant.
When this does not happen, I think the question is no longer really about complex analysis -- this is a question about group theory and plane geometry. Specifically, you want to know the subgroup of Euclidean motions of the plane generated by the two chosen rotations, and you want to understand how they can act on the plane.
