Gluing plugs (Béguin–Bonatti–Yu, Prop. 1.1): Λ_X, Λ_Y and the connecting orbits lie in the maximal invariant set of the glued field

Let $X$ and $Y$ be vector fields on the compact 3-manifolds with boundary $U$ and $V$, let $T^{out}\subseteq U$, and let $\varphi:U\to V$. Write $\partial^{out}U$ for the set of boundary points of $U$ at which $X$ points strictly outward and $\partial^{in}V$ for the set of boundary points of $V$ at which $Y$ points strictly inward. Let $(W,Z)$ be a gluing of $(U,X)$ and $(V,Y)$ along $\varphi$: C¹ embeddings $i_U:U\to W$ and $i_V:V\to W$ with injective derivatives cover the compact 3-manifold $W$, identify exactly the points $x\in T^{out}$ with $\varphi(x)$, and satisfy $Di_U(X)=Z\circ i_U$, $Di_V(Y)=Z\circ i_V$. Write $\Lambda_X$, $\Lambda_Y$, $\Lambda_Z$ for the maximal invariant sets (points whose orbit is defined for all times), $L^u_X$ for the exit lamination of $(U,X)$ (points of $\partial^{out}U$ whose backward orbit is defined for all times) and $L^s_Y$ for the entrance lamination of $(V,Y)$ (points of $\partial^{in}V$ whose forward orbit is defined for all times). The connecting set $\mathcal C\subseteq W$ is the $Z$-orbit of $\varphi_*(L^u_X)\cap L^s_Y$: the set of points $w$ that lie on a complete integral curve $\gamma:\mathbb R\to W$ of $Z$ with $\gamma(0)=i_U(x)=i_V(\varphi(x))$ for some $x\in L^u_X\cap T^{out}$ with $\varphi(x)\in L^s_Y$. Then the three pieces lie in the maximal invariant set of $Z$:

This is the inclusion $\supseteq$ in the sentence "$\Lambda_Z$ is the union of $\Lambda_X$, $\Lambda_Y$ and the $Z$-orbit of $\varphi_*(L^u_X)\cap L^s_Y$" of the proof of Proposition 1.1.

Formalization Note No plug or hyperbolicity hypothesis is needed for this inclusion: only the gluing identities. $Z$ is not assumed to be C¹. The connecting set is phrased by the existence of a complete integral curve of $Z$ through the gluing point (see the companion statement plugGluing_maxInvSet_subset).