I don't understand we the map $\Phi(\cdot ,x):I_x \rightarrow M$ from the excerpt from below of the lecture notes of my professor has to be injectiv.
(Here $M$ denotes the domain of the function on the RHS of the autonomous ODE, $f$ and $\Phi$ denotes, as usual, the flow of the autonomous ordinary differential equation, defined on the union $\bigcup_{x\in M} I_x \times \{x\}$, where $I_x$ is the maximal open interval of the solution with inital value $x$.)
Here's the excerpt:
Since $\Phi(\cdot ,x)$ (if I didn't understood it wrong) is just another notation for the solution with inital condition $x$, I would be somewhat astounded, if there weren't injective solutions for autonoums ODEs. (Already the definition of a periodic orbit suggests that $\Phi(\cdot ,x)$ isn't injective). So does the error lie on my side or is the text faulty ?