Lie group homomorphism from $S^1$ to $S^1$

Question

What are the all the Lie group homomorphisms from $S^1$ to $S^1$?

I know that for each $n \in \mathbb{N}$, $z \mapsto z^n$ gives a Lie group homomorphism of $S^1$.

Thanks in advance!

Answer 1

Such an homomorphism induces a morphism of Lie algebra $f:\mathbb{R}\rightarrow \mathbb{R}$ which is a linear map $f(x)=ax$ such that $e^{ia(x+2\pi)}=e^{iax}$ we deduce that $e^{i2a\pi}=1$ and $a$ is an integer.