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 $x_0$. First of all, if the boundary of $V$, i.e., $\partial V$ is transversal to the vector field $f$, then $\partial V$ is a fundamental domain of $W^s(x_0)$. The intuition behind the fundamental domain has been clarified here.
In this question, I am thinking about another interesting concept, namely fundamental neighborhood which is tightly related to fundamental domain. By definition, any cross section of the vector field $f$ containing $\partial V$ and transversal to $W^s(x)$ is called the fundamental neighborhood $G(x)$ associated with $W^s(x)$.
Based on the concept of fundamental domain and fundamental neighborhood, we have the following results:
$W^s(x) = \cup_{t\in \mathbb{R}} \phi_t(\partial V) \cup {x}$,
$\cup_{t\ge 0} \phi_t(G(x)) \cup W^u(x)$ contains a neighborhood of $x$.
I saw the above results and definitions in Hirsch's paper (page 3, above Theorem 3-3), but there is no reference or clarification in the paper.
I greatly appreciate it if someone can clarify or provide a reference for the concept of fundamental neighborhood or the results 1 and 2.
Thank you!