I am reading my class notes from a Dynamical System course, I have a notation "define $φ(x_0, t)$ to be the solution at time $t$", and latter the notes introduce a symbol "$φ(·, t)$" when explaining the inverse of $φ(·, t)$ is $φ(·, -t)$.
It will be great helpful if anyone explain this symbol $φ(·, t)$ to me.
Thank you
The symbol $\varphi(\cdot, t)$, for fixed $t$, is a shorthand for the function that maps $x$ to $\varphi(x,t)$. Being pedantic, it would be $\varphi(\cdot,t)(x) \doteq \varphi(x,t)$.