A regular continuation datum between Morse--Smale pairs

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)) for the smooth-bundle setup. Let $M$ be a closed manifold and let $(f^{-},g^{-})$ and $(f^{+},g^{+})$ be Morse--Smale pairs in the metric sense (Morse--Smale pairs).

A continuation datum from $(f^{-},g^{-})$ to $(f^{+},g^{+})$ is a choice of $S>0$ together with smooth families $f_s:M\to \mathbb{R}$ and of Riemannian metrics $g_s$, $s\in \mathbb{R}$, such that $(f_s,g_s)=(f^{-},g^{-})(s\le -S),(f_s,g_s)=(f^{+},g^{+})(s\ge S)\mathrm{.}$ Its continuation equation is the non-autonomous first-order equation ${\mathrm{\partial }}_{s}u(s)=-{\mathrm{\nabla }}^{g_s}{f}_s(u(s)),s\in \mathbb{R},$ for smooth $u:\mathbb{R}\to M$, read as an equation for curves of the time-dependent field $s\mapsto -{\mathrm{\nabla }}^{g_s}{f}_s$ (Time-dependent vector fields and their evolution operators, The Riemannian gradient is the metric dual of the differential). Solutions are never quotiented by time translation. The right-hand side is generally not translation invariant; constant data are an autonomous exception, for which the same unquotiented convention applies. For $p\in \mathrm{Crit}(f^{-})$ and $q\in \mathrm{Crit}(f^{+})$ set $\mathcal{C}(p,q)=\{u\in C^{\mathrm{\infty }}(\mathbb{R},M):{\mathrm{\partial }}_{s}u=-{\mathrm{\nabla }}^{g_s}{f}_s(u),{\lim }_{s\to -\mathrm{\infty }}u(s)=p,{\lim }_{s\to +\mathrm{\infty }}u(s)=q\}\mathrm{.}$

Fix a smooth torsion-free background connection, for example the Levi--Civita connection of $g^{-}$. Along a solution, put $E_u=C_0^1(\mathbb{R},u^{\ast }TM)$ and $F_u=C_0^0(\mathbb{R},u^{\ast }TM)$, with supremum norms; the subscript means that the section (and its covariant first derivative in $E_u$) tends to zero at both ends. The linearization is $D_u\xi ={\mathrm{\nabla }}_s\xi +{\mathrm{\nabla }}_{\xi }({\mathrm{\nabla }}^{g_s}{f}_s)\mathrm{.}$ The datum is regular at $(p,q)$ if $D_u:E_u\to F_u$ is onto for every $u\in \mathcal{C}(p,q)$; it is regular if this holds for every critical pair. Empty solution spaces satisfy this condition vacuously.

Here is the finite-dimensional description and index calculation. Let ${\mathrm{\Psi }}_{S,-S}$ be evolution across the compact window. Local smooth evolution and compactness of $M$ extend it across every finite time interval: finitely many coordinate neighbourhoods give a common positive local existence time, which can be iterated; backward evolution is its inverse (Time-dependent vector fields have local smooth evolution operators). Evaluation at $-S$ identifies $\mathcal{C}(p,q)$ with the fibre product $\{(x,y)\in W_{-}^u(p)\times W_{+}^s(q):{\mathrm{\Psi }}_{S,-S}(x)=y\}\mathrm{.}$ The end stable/unstable disks and their transported tangent spaces have the expected dimensions and exponentially decaying variations (Stable and unstable manifolds for a Morse gradient with an arbitrary smooth metric). For smooth data the disks are smooth: differentiate the supplier's contraction fixed-point equation repeatedly in its finite-dimensional initial parameter; at every order the unknown derivative has the same linear contraction operator, while its forcing involves already obtained lower derivatives and bounded derivatives of the cut-off smooth vector field. Its Neumann series therefore gives a continuous derivative of each order. Finite-time smooth flow transport gives smooth global immersion charts. In a frame converging on both autonomous tails, $D_u$ becomes a first-order matrix operator with invertible self-adjoint limits. The whole-line operator lemma gives index $\mathrm{ind}(p)-\mathrm{ind}(q)$ and says that surjectivity is equivalent to the two transported decaying initial-value spaces spanning $T_{y}M$ (Fredholm index and range of an asymptotically hyperbolic first-order operator). These spaces are exactly $d{\mathrm{\Psi }}_{S,-S}(T_x{W}_{-}^u(p))$ and $T_y{W}_{+}^s(q)$. Thus regularity is transversality of this fibre product; it makes $\mathcal{C}(p,q)$ a smooth manifold of dimension $\mathrm{ind}(p)-\mathrm{ind}(q)$, empty for negative dimension (Transverse fibre products are embedded submanifolds). No translation quotient is taken, even for the constant datum, for which translations happen to be symmetries.

Existence and its qualification. Assume the Axiom of Choice (The Axiom of Choice) for the following genericity assertion. Allow both $f_s$ and $g_s$ to vary, with the two ends fixed. Regular data form a residual set in the smooth path space and can be obtained by arbitrarily small perturbations supported inside $(-S,S)\times M$. To see the required transversality, choose finitely many smooth functions ${\varphi }_j$ whose gradients span every tangent space (coordinate functions times cutoffs on a finite chart cover), and perturb $f_s$ by ${\sum }_j{a}_j\beta (s){\varphi }_j$, where $\beta$ is a nonnegative unit-integral bump supported very near some $s_0\in (-S,S)$. Differentiating evolution with respect to $a_j$ gives the integral of the transported vector $-\beta (s){\mathrm{\nabla }}^{g_s}{\varphi }_j$. As the support shrinks, these vectors converge uniformly in the initial point to $-d{\mathrm{\Psi }}_{S,s_0}{\mathrm{\nabla }}^{g_{s_0}}{\varphi }_j$; hence they span $T_{y}M$ for a sufficiently narrow bump. The universal endpoint map is therefore a submersion. Apply finite-dimensional parametric transversality to its fibre products with the stable/unstable immersion charts (Parametric transversality). Their countable chart covers and the finitely many critical pairs leave a null exceptional parameter set, so arbitrarily small good parameters exist. Transversality on each compact piece of a countable chart exhaustion is open and dense; intersecting these sets gives the residual assertion. This argument permits function variations, including along constant solutions.

For an arbitrary fixed function path, metric variations alone need not give regularity: a point critical for every $f_s$ remains a constant solution for every metric path. If its two end indices have negative difference, the linearization there has negative index and cannot be onto. The generic-metric statement in the autonomous distinct-end supplier The universal metric--trajectory projection is Fredholm does not cover this obstruction. Regularity of a specified datum is a hypothesis below, rather than a consequence of that supplier.