Geometric convergence to a broken trajectory

Let $(f,X)$ be a Morse--Smale pair with $X$ downward gradient-like in the normalized Morse-coordinate sense of Downward gradient-like vector fields for a Morse function, on a closed manifold $M$. Fix a compatible metric $d$ on $M$ (Metric space: $d(x,y)=0$ iff $x=y$, symmetry, and the triangle inequality; pseudometric and ultrametric, Topological manifolds are metrizable and paracompact). If $f(p)\le f(q)$, the broken-trajectory space is empty and has the empty topology. Otherwise each broken trajectory $v=(v_1,\dots ,v_r)$ from $p$ to $q$ has a height parametrization $h_v:[f(q),f(p)]⟶M\mathrm{.}$ On the height interval of a component, $h_v(a)$ is its unique point with $f$-value $a$; at an intermediate critical value it is that intermediate critical point, and at the endpoints it is $q$ and $p$. Strict descent, the component endpoint limits and Broken Morse trajectories make this a continuous map. Its noncritical pieces recover the orbit classes, so $v\mapsto h_v$ is injective.

The geometric-convergence topology is the topology transported to $\bar{\mathcal{M}}(p,q)$ from this image in $C^0([f(q),f(p)],M)$ with the topology of uniform convergence. Equivalently, it is induced by $d_H(v,w)={\max }_{a\in [f(q),f(p)]}d(h_v(a),h_w(a))\mathrm{.}$ The maximum exists because the domain is nonempty and compact. Uniform convergence on this domain is compact-open convergence (The compact-open topology on $C(X,Y)$ for arbitrary topological spaces, On a nonempty compact metric domain, the compact-open topology is the uniform topology). Different compatible metrics on the compact space $M$ give the same topology: the identity between the two compact metric spaces is uniformly continuous, as follows by taking a finite subcover of neighbourhoods on which its oscillation is prescribed.

For ordinary trajectories $v_n=[{\gamma }_n]$ and a broken trajectory $v=(v_1,\dots ,v_r)$, write $v_n⇝v_1\mathrm{\#}\cdots \mathrm{\#}{v}_r$ when, for parametrized representatives of the components, there are independent shifts $s_n^i$ such that ${\gamma }_n(\cdot +s_n^i)\to v_i$ in $C_{\mathrm{l}\mathrm{o}\mathrm{c}}^{\mathrm{\infty }}(\mathbb{R},M)$. Replacing representatives only changes the shifts. Smooth dependence for the flow (The fundamental theorem on flows) implies that $C^0$ convergence on compact time intervals forces all higher derivatives. The equivalence of this shift criterion with convergence in $d_H$ is proved in Compactness up to breaking of Morse trajectory spaces ↗.

Here is the neighbourhood description used for gluing (Audin–Damian §3.2.a, printed pp. 60–61). Write $p=p_0,\dots ,p_r=q$. Choose disjoint sufficiently small Morse neighbourhoods at all these critical points, including $p$ and $q$. For each $i=1,\dots ,r$, prescribe open level-set neighbourhoods $U_{i-1}^{-}$ of the exit point of $v_i$ from the chart at $p_{i-1}$ and $U_i^{+}$ of its entry point into the chart at $p_i$. A string $w$ belongs to $W(v,U^{-},U^{+})$ if its critical-point string is a subsequence $p_{i_0},\dots ,p_{i_k}$ with $0=i_0<\cdots <i_k=r$, and its $j$th component exits and enters each intervening chart through the prescribed neighbourhoods, in order. Thus it has at most $r$ components; for $r=1$ both endpoint transversals are still present.

These sets form a neighbourhood base for the height topology. Shrinking the prescribed transversals controls each compact noncritical segment by smooth finite-time flow dependence; inside a small Morse chart the equations $u(t)=e^{2t}u(0)$, $v(t)=e^{-2t}v(0)$ keep a crossing between its small entry and exit pieces in that chart. This gives uniform height control after subdividing into those finitely many segments and arbitrarily small critical neighbourhoods. Conversely, uniform height closeness forces passage through the prescribed level pieces and permits critical breaks only at points of the limiting string: the critical set is finite and all other critical points are separated from the limiting height graph. The level crossings vary continuously because $df(X)\mathrm{\ne }0$ there. This proves the two base containments. On the ordinary stratum the resulting topology is the quotient topology of the parametrized trajectory space, equivalently its regular-level slice topology (A regular level identifies unparametrized trajectories, The unparametrized trajectory space is a smooth manifold, The quotient topology of a surjection, quotient maps, saturated sets, and the quotient of a space by an equivalence relation with its canonical projection); the shift/height equivalence below also verifies this identification.