Broken continuation trajectories and geometric convergence

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)) for the smooth-bundle setup. Let $(f_s,g_s)$ be a regular continuation datum from $(f^{-},g^{-})$ to $(f^{+},g^{+})$ on a closed manifold $M$ (A regular continuation datum between Morse--Smale pairs), let $p\in \mathrm{Crit}(f^{-})$, $q\in \mathrm{Crit}(f^{+})$ (Morse--Smale pairs), and let $\mathcal{C}(p,q)$ be the continuation moduli spaces of the datum.

A broken continuation trajectory from $p$ to $q$ consists of nonconstant autonomous tail pieces and one continuation solution (which may be constant) $({\gamma }_1^{-},\dots ,{\gamma }_r^{-};v;{\gamma }_1^{+},\dots ,{\gamma }_s^{+}),r,s\ge 0,$ together with critical points $p=p_0^{-},p_1^{-},\dots ,p_r^{-}$ of $f^{-}$ and $q=p_0^{+},p_1^{+},\dots ,p_s^{+}$ of $f^{+}$ such that ${\gamma }_i^{-}\in {\mathcal{M}}^{-}(p_{i-1}^{-},p_i^{-}),v\in \mathcal{C}(p_r^{-},p_s^{+}),{\gamma }_j^{+}\in {\mathcal{M}}^{+}(p_j^{+},p_{j-1}^{+})$ for all $i,j$ (Unparametrized Morse trajectory moduli space, A Morse trajectory from one critical point to another). The pieces are read in the temporal order ${\gamma }_1^{-},\dots ,{\gamma }_r^{-},v,{\gamma }_s^{+},{\gamma }_{s-1}^{+},\dots ,{\gamma }_1^{+}$; the tuple displays the pieces of the two ends in increasing order of their index-drop chains. The space of broken continuation trajectories from $p$ to $q$ is denoted $\bar{\mathcal{C}}(p,q)$; the locus $r=s=0$ is $\mathcal{C}(p,q)$ itself.

The pieces carry the index drops $d_i:=\mathrm{ind}(p_{i-1}^{-})-\mathrm{ind}(p_i^{-})\ge 1,e_j:=\mathrm{ind}(p_j^{+})-\mathrm{ind}(p_{j-1}^{+})\ge 1,m:=\mathrm{ind}(p_r^{-})-\mathrm{ind}(p_s^{+})\ge 0,$ which are nonnegative by the emptiness of the corresponding spaces for nonpositive drop (No Morse--Smale trajectories for nonpositive index drop, A regular continuation datum between Morse--Smale pairs), and the index identity $\mathrm{ind}(p)-\mathrm{ind}(q)=r+s+m+{\sum }_{i=1}^r(d_i-1)+{\sum }_{j=1}^s(e_j-1)$ holds by telescoping the drops along the two chains (Nondegenerate critical points, nullity, index, and coindex). In particular $r+s\le \mathrm{ind}(p)-\mathrm{ind}(q)$, and equality holds precisely when every tail piece connects critical points of consecutive index and the middle piece has index difference $m=0$. The number of tail pieces is bounded by the index drop, exactly as for the broken Morse trajectories of a Morse--Smale pair (Breaking length is bounded by the index drop, Broken Morse trajectories).

A sequence $u_n\in \mathcal{C}(p,q)$ converges geometrically to such a configuration if $u_n\to v$ in $C^{\mathrm{\infty }}$ on compact time intervals without shifting the middle solution, and there are shifts $a_n^i\to -\mathrm{\infty }$ and $b_n^j\to +\mathrm{\infty }$ such that $u_n(\cdot +a_n^i)\to {\gamma }_i^{-},u_n(\cdot +b_n^j)\to {\gamma }_j^{+}$ in $C^{\mathrm{\infty }}$ on compact intervals, for chosen parametrized representatives of the autonomous orbit classes. Temporal order requires $a_n^{i+1}-a_n^i\to +\mathrm{\infty }$ and $b_n^{j-1}-b_n^j\to +\mathrm{\infty }$. For the middle and each shifted autonomous tail, $C^0$ convergence on compact intervals implies convergence of all derivatives by differentiating the corresponding smooth ODE; the shifted equation on every fixed compact tail interval is eventually the fixed autonomous end equation (The topology of compact convergence on $C(X,Y)$ for metric $X$ and $Y$: uniform convergence on each compact subset of $X$, Geometric convergence to a broken trajectory).

The topology on $\bar{\mathcal{C}}(p,q)$ uses compact-time tests on the unshifted middle solution and ordered regular-level transversal tests on the autonomous components. A neighbourhood specifies compact intervals and open neighbourhoods of the middle curve on them, and open transversal neighbourhoods along each tail; configurations may smooth some breaks but must pass these tests in the stated order. Shrinking the tests gives a neighbourhood basis, with the compact-convergence topology on the unbroken locus and the corresponding broken-end topology on each stratum. The middle solution's time coordinate is fixed: translating it generally changes the continuation equation. Every middle solution has critical limits by Continuation solutions have critical limits and exponential decay.