Gluing hyperbolic plugs (Béguin–Bonatti–Yu, Prop. 1.1): the images of Λ_X and Λ_Y are hyperbolic sets of the glued field

Let $(U,X)$ and $(V,Y)$ be hyperbolic plugs: plugs whose maximal invariant sets $\Lambda_X$, $\Lambda_Y$ are hyperbolic sets with one-dimensional strong stable and strong unstable bundles. Let $T^{out}$ be a union of connected components of $\partial^{out}U$, let $T^{in}$ be a union of connected components of $\partial^{in}V$, and let $\varphi:U\to V$. 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 $Di_U(X)=Z\circ i_U$, $Di_V(Y)=Z\circ i_V$. Then the images of the two maximal invariant sets are hyperbolic sets of $Z$:

This is the part of the proof of Proposition 1.1 where the hyperbolic structures of $\Lambda_X$ and $\Lambda_Y$ are regarded as hyperbolic structures for $Z$ on $W$.

Formalization Note A hyperbolic set is defined as in the mission (IsHyperbolicSet): a continuous Riemannian metric, line fields $E^s$, $E^u$ with $E^s\oplus\mathbb R Z\oplus E^u=TW$ on the set, invariance under the derivatives of the time-$t$ maps of the flow, and exponential estimates with constants $C>0$, $\lambda>0$. $Z$ is not assumed to be C¹. The hypotheses on $T^{out}$ and $T^{in}$ are those of Proposition 1.1; the map $\varphi$ enters only through the gluing relation.