Does the equation $\exp{X}=e$ have a discrete set of solutions?

Question

Let $G$ be a compact Lie group and $X\in\mathfrak{g}$ an element such that $\exp{X}=e$. Does this equation have a discrete set of solutions, in general? If not, are there any conditions on $G$ so that is does? If this is the case, please provide some references about this topic.

Answer 1

No. For example, the matrices $$X(\theta):=\begin{pmatrix}0 & 2\pi i e^{i\theta} \\ 2\pi ie^{-i\theta} & 0\end{pmatrix}\in\mathfrak{su}(2)$$ for $\theta\in\mathbb{R}$ have the property that $$\exp(X(\theta))=\begin{pmatrix}1&0\\0&1\end{pmatrix}\in\mathrm{SU}(2)$$ for all $\theta\in\mathbb{R}$.

In general, the equation $\exp(X)=z$ for $z\in Z_G$ is not discrete, since if there is one solution, say $X$, then $\mathrm{Ad}(g)X$ is another solution for all $g\in G$. Indeed, $$\exp(\mathrm{Ad}(g)X)=g\exp(X)g^{-1}=gzg^{-1}=zgg^{-1}=z$$ since $z$ commutes with everything. The set $\{\mathrm{Ad}(g)X:g\in G\}$ is called the adjoint orbit of $X$, and is discrete if and only if $X\in Z_{\mathfrak{g}}$ (so $X=0$ when $G$ is semisimple).