Finite flow matching gives local charts at metric-end broken trajectories

At every intermediate critical point, use [F1] to write the local stable disk as a graph over its stable spectral space and the unstable disk as a graph over its unstable spectral space. The map sending the two spectral coordinates to the sum of their two graph parametrizations has derivative the identity at the critical point, so its inverse gives smooth coordinates in which the disks are the coordinate axes. Invariance makes the vector field $s^{\mathrm{\prime }}=A_{s}s+R_s(s,u)$, $u^{\mathrm{\prime }}=A_{u}u+R_u(s,u)$ with $R_s(0,u)=R_u(s,0)=0$ and $DR(0)=0$. Shrink the ball so that $⟨s,s^{\mathrm{\prime }}⟩<0$ and $⟨u,u^{\mathrm{\prime }}⟩>0$ for nonzero components, by the spectral gaps and the small diagonal derivatives of $R$. Fix entry $\mathrm{\mid }s\mathrm{\mid }=r$ and exit $\mathrm{\mid }u\mathrm{\mid }=r$. The boundary data of [F2] permit these sections because its boundary-data domain has radius $2r$. Every trajectory entering and leaving the small box crosses each of these sections exactly once.

At a negative break adjacent to the middle, choose a fixed time $-S^{\mathrm{\prime }}$ earlier than $-S$ such that $v(-S^{\mathrm{\prime }})$ lies in the critical chart. This anchor may be the critical point itself for a constant connector; no unstable exit crossing is required. Put $P={\mathrm{\Psi }}_{S^{\mathrm{\prime }},-S^{\mathrm{\prime }}}^{-1}{W}_{+}^s(q)$ at that fixed anchor. At the broken point, regularity says $W_{-}^u(a)⋔P$. In the axis coordinates $W_{-}^u(a)=\{s=0\}$, so projection of $P$ to the stable coordinates $s$ is a submersion. Choose coordinates ${\xi }_m$ on its fibre; the sheet has the form $u=B(s,{\xi }_m)$, and the unshifted middle solutions correspond to $s=0$. On the incoming section $\mathrm{\mid }s\mathrm{\mid }=r$, Morse--Smale transversality makes the unstable coordinate projection from the incoming $W_{-}^u(p)$ sheet a submersion. Its remaining coordinates ${\xi }_i$ parametrize the incoming orbit space, and it has the form $s=A(u,{\xi }_i)$ with $\mathrm{\mid }A\mathrm{\mid }=r$. The local equations are therefore $a=A({\alpha }_T(a,b),{\xi }_i)$ and $b=B({\zeta }_T(a,b),{\xi }_m)$, where $b$ lies in an open unstable ball about $v(-S^{\mathrm{\prime }})$ and may equal zero. They impose the original fixed-time continuation condition, without introducing or quotienting a middle phase. Positive adjacent breaking uses the time-reversed construction at a fixed $+S^{\mathrm{\prime }}$ anchor.

For an augmented family use the total endpoint sheet $P\subset (s,u,\lambda )$, rather than a vertical member. Transversality with $\{s=0\}\times I$ makes projection $P\to s$ a submersion. Thus local coordinates on $P$ are $(s,{\xi }_m)$ with $u=B(s,{\xi }_m)$ and $\lambda =L(s,{\xi }_m)$; ${\xi }_m$ is a chart on the full augmented broken-middle stratum. Recover the actual parameter by $\lambda =L({\zeta }_T(a,b),{\xi }_m)$ after matching; no vertical inverse is assumed. At a parameter corner write the parameter as $(\eta ,z)$ with $z\in [0,ϵ{)}^c$ the boundary coordinates. Facewise regularity makes the endpoint matching derivative onto using the flow variables and $\eta$ at fixed $z=0$. Choose an invertible normal minor from those variables and apply [F4], keeping $s$, $z$ and the complementary tangential variables free. This gives the same graph $(s,{\xi }_m)$ with $z$ among the free components of ${\xi }_m$ and the recovered parameter retaining exactly that $z$. Restricting $z$ to its orthant therefore preserves every parameter face after neck matching. For the two-parameter datum the stronger endpoint constancy permits $L=\lambda$ there. If two regular continuation pieces are separated by an autonomous plateau, retain the fixed outgoing anchor of the first and fixed incoming anchor of the second. The first transverse endpoint sheet is $s=A(u,{\xi }_1)$, the second is $u=B(s,{\xi }_2)$, and the equations are the same as in step 2.1, now with both $a$ and $b$ in open balls; neither endpoint needs to cross a sphere. The plateau length $T$ is the external datum parameter. For autonomous-to-autonomous junctions, use the two actual sections and the same graph normal variables from the two transverse orbit sheets, retaining their orbit-space coordinates. Every autonomous piece is nonconstant, so its section phase is fixed by a nonzero flow crossing; this phase convention is not imposed on the continuation middle.

Write the equations as $G_1=a-A({\alpha }_{1\mathrm{/}\rho }(a,b),{\xi }_i)=0$ and $G_2=b-B({\zeta }_{1\mathrm{/}\rho }(a,b),{\xi }_m)=0$. At $\rho =0$ the endpoint maps are zero, so the unique broken solution is $a=A(0,{\xi }_i)$, $b=B(0,{\xi }_m)$, and the derivative in the unknown $(a,b)$ is the identity. Work in an ambient open coordinate neighbourhood of $a$ even when its final value is on a sphere; the equation $a=A(\cdot )$ enforces that sphere constraint. By [F2] the endpoint substitutions are smooth and flat in $\rho$. Extend them by zero for negative $\rho$, then apply [F4] to obtain unique smooth $a(\rho ,\xi )$, $b(\rho ,\xi )$. The formulas construct exact solutions on the passage and exact exterior solutions, whose matching is exactly the fixed-anchor condition. For several necks, collect the small passage-end displacement coordinates into $e=({\alpha }_j,{\zeta }_j{)}_j$. At $e=0$ each exterior broken piece is independent. In its transverse fibre-product chart write its exterior equations as $H_k(e,{\xi }_k,{\eta }_k)=0$, where ${\xi }_k$ is its broken moduli coordinate and the chosen normal complement ${\eta }_k$ has invertible derivative, by step 3.1. The parameter-dependent implicit-function theorem solves ${\eta }_k={\eta }_k(e,{\xi }_k)$, so all section or anchor boundary data are smooth functions $a_j=A_j(e,\xi )$ and $b_j=B_j(e,\xi )$ of these displacements and the product broken coordinates. The augmented parameter is recovered in the same normal solution, with its full derivative used for the middle; autonomous exterior equations are independent of that parameter because the ends are fixed. Substitute the actual passage maps $e(\rho ,a,b)$ into these functions and impose $a_j-A_j(e(\rho ,a,b),\xi )=0$, $b_j-B_j(e(\rho ,a,b),\xi )=0$. At all zero necks, $e=0$ and $D_{(a,b)}e=0$ by flatness. Thus the unknown derivative of this entire joint system is the identity, regardless of cross-dependence of the exterior functions. The same finite implicit-function theorem solves all necks simultaneously. At partially zero necks it solves the same exterior equations with the corresponding endpoint displacements zero, so uniqueness identifies the face with the lower-neck construction.

The component estimates of [F2] imply that on a passage of length $T$, $\mathrm{\mid }s(t)\mathrm{\mid }\le C{e}^{-\beta t}$ and $\mathrm{\mid }u(t)\mathrm{\mid }\le C{e}^{-\beta (T-t)}$. Thus on fixed intervals after entry or before exit the passage converges to its stable or unstable axis orbit by finite-time uniqueness and continuous dependence; in the region far from both ends both components are small. The exterior matching variables converge to the broken variables of step 3.1. This proves geometric convergence as any neck tends to zero, with the continuation window unshifted. Conversely, a trajectory geometrically close to the chosen broken configuration meets the chosen autonomous entry and exit sections, or the section and fixed middle anchor, on the chosen immersion branches. Step 1.1 gives unique crossing times; adjacent to a middle the other time is the fixed anchor, so each $T_j$ is unique. In the two-window plateau case $T$ is the known external parameter. If a breaking sequence had bounded $T_j$, finite-time continuous dependence would connect a nonconstant stable entry to an unstable exit through the critical point in finite time, contradicting uniqueness. Hence $T_j$ tends to infinity at exactly the breaks. These crossings and the exterior representatives recover $\rho$ and $\xi$; uniqueness in step 4.1 recovers the same trajectory.

Shrink the geometric transversal neighbourhood so that all recovered exterior variables lie in the charts of step 3.1 and all passage times are above the threshold of step 4.1. Steps 4.1–5.1 show that this neighbourhood is exactly the image of the matching chart. The inverse crossing coordinates are continuous, including at a zero neck by the preceding divergence argument; the direct map is continuous by step 5.1. Thus the chart is a homeomorphism onto a neighbourhood and has the claimed stratumwise smoothness and inverse coverage. Partial broken configurations are included by setting the corresponding necks to zero; uniqueness of the same equations makes these face parametrizations compatible. All necks are positive exactly for unbroken solutions. Step 3.1 independently preserves the parameter boundary coordinates in $B$, so these unbroken solutions may lie on parameter faces; only $(0,\delta {)}^N\times \mathrm{int}B$ is the full chart interior.

Changing an autonomous section changes a passage time by finite exterior crossing times; changing a fixed anchor changes it by its fixed time difference. Changing the critical coordinates does not change the actual passage time between the same cuts. These are smooth functions of the section endpoints near the fixed stable and unstable crossings, since the crossings are transverse. Their limiting sum is a smooth function $c(\xi )$ of the broken variables; their difference from that limit is flat in the affected neck variable by [F2]. Therefore $T^{\mathrm{\prime }}=T+c(\xi )+e(\rho ,\xi )$ with $e$ smooth and flat at $\rho =0$, and ${\rho }^{\mathrm{\prime }}=\rho \mathrm{/}(1+c(\xi )\rho +e(\rho ,\xi )\rho )$. This is smooth with normal derivative $1$ at the face. Exterior stratum coordinates change smoothly by [F4] and the unique matching equations. Apply this argument separately to the finitely many necks; it proves compatibility at higher corners as well as at one break. The same exterior equation argument for the two-window plateau proves the final assertion of the statement.