Every broken trajectory is a limit of ordinary trajectories

First let the string be $a\to c\to b$, with $k=\lambda (c)$. On $f=f(c)+\epsilon$, write the incoming crossing as $(0,z_{+})$. The slice of $W^u(a)$ is transverse to the stable sphere $S^{+}=\{u=0,\mathrm{\mid }z{\mathrm{\mid }}^2=\epsilon \}$ by [F4]. Its projection to $u\in {\mathbb{R}}^k$ is therefore a submersion at $z_{+}$. By [F5], fixing its surplus local coordinates gives a $k$-disk $D$ of the form $(u,h(u))$, where $h(0)=z_{+}$ and $\mathrm{\mid }h(u){\mathrm{\mid }}^2=\epsilon +\mathrm{\mid }u{\mathrm{\mid }}^2$. Put $g(u)=h(u)\mathrm{/}\mathrm{\mid }h(u)\mathrm{\mid }$. The crossing map to $f=f(c)-\epsilon$, obtained by solving the flow in [F3], sends $(u,h(u))$ to $(\mathrm{\mid }h(u)\mathrm{\mid }u\mathrm{/}\mathrm{\mid }u\mathrm{\mid },\mathrm{\mid }u\mathrm{\mid }g(u))$ for $u\mathrm{\ne }0$. In polar coordinates it extends smoothly to $H(\rho ,\theta )=(\sqrt{\epsilon +{\rho }^2}\theta ,\rho g(\rho \theta )),\rho \ge 0,\theta \in S^{k-1}\mathrm{.}$ At $\rho =0$ its radial derivative has nonzero stable part $g(0)$, independent of the angular derivatives; it is an embedding near each boundary point, with $\rho =\mathrm{\mid }z\mathrm{\mid }$ and $\theta =u\mathrm{/}\mathrm{\mid }u\mathrm{\mid }$ recovering its parameters. Its boundary is the unstable sphere $S^{-}$.

At the outgoing crossing $(u_{-},0)$ of $c\to b$, the slice of $W^s(b)$ has codimension $\lambda (b)$ in the exit level and is transverse to $S^{-}$ by [F4]. Local defining equations composed with $H$ thus have a surjective derivative in the angular variables at $(0,u_{-}\mathrm{/}\sqrt{\epsilon })$. Choose an invertible block of $\lambda (b)$ angular coordinates and fix the others. By [F5] they can be solved as smooth functions of $\rho$ for all sufficiently small $\rho \ge 0$, giving points $H(\rho ,\theta (\rho ))\in W^s(b)$. If $\lambda (b)=0$ there are no equations to solve. For $\rho >0$ these points come from $D\subset W^u(a)$ and hence give ordinary trajectories from $a$ to $b$. Their entry and exit points tend to $(0,z_{+})$ and $(u_{-},0)$; smooth finite-time flow dependence controls all other prescribed transversals. They therefore approach the given once-broken trajectory in [F2]. This proves once-broken gluing existence without asserting invertibility of a sliced Fredholm operator.

Induct on the number $r$ of components. For $r=1$, $v$ itself is ordinary. For $r>1$, approximate its first $r-1$ components by an ordinary trajectory $u$ so closely that $u\mathrm{\#}{v}_r$ lies in a prescribed open neighbourhood $W$ of $v$; this is possible by the induction hypothesis and the finite transversal description in [F2]. Since $W$ is open and contains $u\mathrm{\#}{v}_r$, step 2.1 gives an ordinary trajectory in $W$. Hence every neighbourhood of every finite string meets the ordinary stratum, proving density.