Take the stereotypical ODE $\dot x = f(x)$ where $x \in R^n$.
Say now I take a compact, $n$-dimensional subset $P$ of my phase space in $R^n$. Can I always find a $(n-1)$-dimensional submanifold S $\subset P$ such that all of $P$ can be reached from initial conditions in $S$? In other words, can I always foliate $P$ with orbits emerging from some submanifold $S$ with codimension 1?
If the answer is positive, what other assumptions would I need to guarantee that $S$ is connected, smooth and so on?
Thank you all in advance!