Let $A$ be a diagonal matrix with real entries. I want to find out whether the exponential map $t \mapsto e^{itA}$ is open, where $t \in \mathbb{R}$. My observation: the map is injective since fixing $t, t' \in \mathbb{R}\setminus \{0\}$, we have $e^{itA} \neq e^{it'A}$ for all diagonal $A$. Also, the map is continuous since the exponential map is continuous. So the closest thing I can think of is the invariance of domain. However, I realized that the invariance of domain requires that both the source and the target of the map be $\mathbb{R}^n$, which is not true in this case where the source is $\mathbb{R}$ and the target is $GL(\mathbb{C})$.
Are my observations correct? Can someone point out what I need to check to determine whether the map is open? Thanks in advance!
EDIT: Thanks to the comment by @peek-a-boo and @copper.hat, I realized that my previous observations are wrong. In particular, $e^{itA}$ is not injective for all real diagonal $A$. To see this, simply take $A$ to be a real number, and note that $e^{2\pi i} = e^{4\pi i} = 1$. It occurs to me that the map cannot be injective if $A$ is just a real number. But if $A$ is a $2 \times 2$ real diagonal matrix, then the map can be made injective if at least one of the diagonal entries is not rational. In this case, can $t \mapsto e^{itA}$ be open even if $\| e^{itA}\| = 1$? Can the image of some open set in $\mathbb{R}$ under the map still cover the entire unit sphere (which is a closed set) if it is injective? I got lost in the topological notions. Can someone share some insights? Thanks in advance!