Consider the ODE \begin{align} \dot x = f(x) \tag{1} \end{align} Let $x_0$ be a hyperbolic fixed point of $(1)$. Let $V$ be a neighborhood of $x_0$ in $W^s(x_0)$, where $W^s(x_0)$ is the stable manifold of $x_0$. I am having a hard time understanding the following statement:
"If the boundary of $V$, i.e., $\partial V$ is transversal to the vector field $f$, then $\partial V$ is a fundamental domain of $W^s(x_0)$."
I do not have a strong backgound in topology and geometry. I appreciate it if someone can clarify the above statement. what does fundamental domain mean? How is it related to the transversality of the vector field?
From Wikipedia:
In the case of a dynamical system $\dot{x} = f(x)$, the group action is the flow of this dynamical system. In particular, the 'orbits of the action' from the quote above are precisely the orbits in the dynamical systems sense.
The stable manifold of $x_0$ consists, by definition, of all orbits that flow to $x_0$ as $t \to \infty$. Thus, $W^s(x_0)$ is a collection of orbits. As orbits are unique as solutions to the ODE system $\dot{x}=f(x)$, every orbit can be uniquely represented by choosing a point on that orbit. So, we could try to represent the collection of orbits $W^s(x_0)$ by picking one point on every orbit in $W^s(x_0)$ in some way.
The 'fundamental domain' provides such a choice. That is, if you are given a neighbourhood $V$ of $x_0$ that lies completely within $W^s(x_0)$, then $V$ completely consists of orbit segments. Now, if the boundary of $V$ is transversal to $f$, it is transversal to the flow of the dynamical system. That means that all orbits in $V$ flow into $V$ ('into', since we consider $W^s$ and not $W^u$) through the boundary of $V$. Hence, we can represent every orbit in $W^s(x_0)$ by the point on $\partial V$ through which it flows into $V$.
The observation that all orbits flow into $V$ through the boundary of $V$ seems obvious, but if the boundary of $V$ would at some point $\hat{x}$ be tangent to the flow of the dynamical system, then the orbit through that point could touch $V$ without necessarily flowing into $V$. Since all of this happens inside $W^s(x_0)$, we know that this orbit will eventually flow to $x_0$ and therefore eventually enter $V$. Therefore, the tangency leads to the situation that this orbit is coincides with $\partial V$ more than once. This is contrary to the idea of a fundamental domain, where every point in that fundamental domain (in this case $\partial V$) corresponds to precisely one orbit in $V$.
Hence, the fundamental domain $\partial V$ completely characterises $W^s(x_0)$. This can be useful because the dimension of $\partial V$ is often lower than that of $W^s(x_0)$.