Gluing once-broken index-two trajectories: collar ends

Put $k=\lambda (r)$ and choose entry and exit levels $f(r)\pm \epsilon$. The incoming slice of $W^u(p)$ has dimension $k$ and is transverse to the stable sphere $S^{+}$ at its crossing $(0,z_{+})$. By [F5], a local sheet is a disk $D=\{(u,h(u))\}$ with $h(0)=z_{+}$ and $\mathrm{\mid }h(u){\mathrm{\mid }}^2=\epsilon +\mathrm{\mid }u{\mathrm{\mid }}^2$. The flow passage in [F3], with $g=h\mathrm{/}\mathrm{\mid }h\mathrm{\mid }$, extends in polar coordinates to $H(s,\theta )=(\sqrt{\epsilon +s^2}\theta ,sg(s\theta )),(s,\theta )\in [0,\delta )\times S^{k-1}\mathrm{.}$ This is a smooth embedded collar $Q$ of the unstable sphere $S^{-}$: its radial derivative at $s=0$ has nonzero stable component $g(0)$, the angular derivatives span the sphere tangent space, and $s=\mathrm{\mid }z\mathrm{\mid }$, $\theta =u\mathrm{/}\mathrm{\mid }u\mathrm{\mid }$ recover its parameters. For $s>0$, $Q$ is the passage image of $D\setminus \{(0,z_{+})\}$.

At the outgoing crossing $(u_{-},0)$, the stable slice of $W^s(q)$ has codimension $k-1$ in the exit level and is transverse to the $(k-1)$-sphere $S^{-}$ by [F2]. In local sphere coordinates its defining equations composed with $H$ thus have invertible angular derivative. Apply [F5], extending the formula for $H$ to negative $s$ for this local calculation, to solve uniquely $\theta =\theta (s)$ near $(0,u_{-}\mathrm{/}\sqrt{\epsilon })$. If $k=1$ there are no angular variables or equations and the assertion is immediate. The curve $\chi (s)=H(s,\theta (s))$ is a smooth embedded half-interval in $Q\cap W^s(q)$; for $s>0$ it lies in $W^u(p)$ as well. By [F4] it determines a smooth embedded family of ordinary orbit classes $\psi (s)$. The inverse passage gives entry points tending to $(0,z_{+})$ and exit points tending to $(u_{-},0)$, so [F1] makes the extension $\psi (0)=({\gamma }_1,{\gamma }_2)$ continuous.

Any trajectory sufficiently close to the broken point in the transversal neighbourhoods of [F1] has its entry point in $D$: closeness at the endpoint exit sphere of $p$ and smooth finite-time transport along the incoming component select precisely this local sheet of $W^u(p)$. Its exit point is therefore in $Q$, and closeness at the endpoint entry sphere of $q$ likewise selects the local sheet of $W^s(q)$. The uniqueness in step 2.1 forces that exit point to be $\chi (s)$. Shrinking the neighbourhood gives exactly one such half-interval; its inverse coordinate $s=\mathrm{\mid }z\mathrm{\mid }$ is continuous, including at the broken point. This proves the open-neighbourhood, homeomorphism and eventual-containment claims. Finally the passage time is $\frac{1}{2}\log (\sqrt{\epsilon +s^2}\mathrm{/}s)$ by [F3], which diverges as $s\to 0^{+}$.