Flux in Tilted Potential Systems: Negative Resistance and Persistence

Abstract

Many real-world systems are well-modeled by Brownian particles subject to gradient dynamics plus noise arising, e.g., from the thermal fluctuations of a heat bath. Of central importance to many applications in physics and biology (e.g., molecular motors) is the net steady-state particle current or “flux” enabled by the noise and an additional driving force. However, this flux cannot usually be calculated analytically. Motivated by this, we investigate the steady-state flux generated by a nondegenerate diffusion process on a general compact manifold; such fluxes are essentially equivalent to the stochastic intersection numbers of Manabe (Osaka Math J 19(2):429–457, 1982). In the case that noise is small and the drift is “gradient-like” in an appropriate sense, we derive a graph-theoretic formula for the small-noise asymptotics of the flux using Freidlin–Wentzell theory. When additionally the drift is a local gradient sufficiently close to a generic global gradient, there is a natural flux equivalent to the entropy production rate—in this case our graph-theoretic formula becomes Morse-theoretic, and the result admits a description in terms of persistent homology. As an application, we provide a mathematically rigorous explanation of the paradoxical “negative resistance” phenomenon in Brownian transport discovered by Cecchi and Magnasco (Phys Rev Lett 76(11):1968, 1996).

Proofs of Lem. 6.6 and Prop. 6.5, 7.7

In this appendix we prove Lem. 6.6 and Prop. 6.5, 7.7; we also restate these results for convenience. We first prove Prop. 6.5 using the following Lem. A.1, which we prove using a technique from [FW12, p. 146, Lem. 1.5].

Lemma A.1

Let $\textbf{v}$ be a $C^1$ vector field on a closed Riemannian manifold M. Assume that $R(\textbf{v})$ is finite. Then for any $e\in \Pi (M)$, $Q_{\textbf{v}}(e) = 0$ if and only if e contains a piecewise $\textbf{v}$-integral curve.

Proof

Assume that e contains a piecewise integral curve in the sense of Def. 6.4. I.e., there is a finite sequence $(\gamma _j)_{j=1}^N$ of segments of $\textbf{v}$-integral curves having well-defined path homotopy classes $[\gamma _j]\in \Pi (M)$ satisfying $[\gamma _1][\gamma _2] \cdots [\gamma _N] = e$. By concatenating suitable restrictions $\tilde{\gamma }_j=\gamma _j|_{[a_j,b_j]}$ with short paths $c_j:[0,t_j]\rightarrow M$ having arbitrarily small actions $\mathcal {S}_{t_j}(c_j)$ [FW12, p. 143, Lem. 1.1] and using local simply connectedness of M, a path $\varphi \in C_e([0,T],M)$ with $\mathcal {S}_T(\varphi )<\varepsilon$ can be constructed for every $\varepsilon > 0$ (with $T\ge 0$ depending on $\varepsilon$), so $Q_{\textbf{v}}(e) = 0$.

To prove the converse, let $e\in \Pi (M)$ satisfy $Q_{\textbf{v}}(e) = 0$. Assume, to obtain a contradiction, that e does not contain a piecewise $\textbf{v}$-integral curve. Then e does not contain a constant curve since otherwise the trivial piecewise integral curve $c :\{0\}\rightarrow \{\mathfrak {s}(e)\}$ satisfies $[c]=e$. It follows that there exists a piecewise $\textbf{v}$-integral curve $c = (\gamma _1,\ldots , \gamma _N)$ such that

$$\begin{aligned} Q_{\textbf{v}}([c]^{-1}e) = 0, [c]\ne e,\text { and there is no forward extension of }c\text { satisfying this property}. \end{aligned}$$

(150)

Here $[c]^{-1}$ is the reversal (groupoid inverse) of the path homotopy class [c]. By a forward extension of c we mean (i) a forward extension of $\text {dom}(\gamma _N)$ if $\text {dom}(\gamma _N)$ is bounded above, or (ii) the addition to c of a new integral curve segment $\gamma _{N+1}$ satisfying $\omega ^*(\gamma _{N+1})= \omega (\gamma _N)$ if $\text {dom}(\gamma _N)$ is unbounded above.

Let $e_c :=[c]^{-1}e$, $x:=\mathfrak {s}(e_c) = \mathfrak {t}([c])$, and let $\varphi ^{(k)}\in C_{e_c}([0,T_k],M)$ be a sequence with $\mathcal {S}_{T_k}(\varphi ^{(k)})\rightarrow 0$. Since $\textbf{v}^{-1}(0)$ is finite, for sufficiently small $\varepsilon > 0$ the metric ball $B_{\varepsilon }(x)$ of radius $\varepsilon$ centered at $x = \mathfrak {t}([c])$ is simply connected, does not contain the entire image of any path representing $e_c$, and is disjoint from $\textbf{v}^{-1}(0){\setminus } \{x\}$. Let $S_\varepsilon (x):=\partial B_{\varepsilon }(x)$. Since by continuity each $\varphi ^{(k)}$ must pass through $S_\varepsilon (x)$, for each k

$$\begin{aligned} \tau _k:=\inf \{t> 0 :\varphi ^{(k)}([0,t])\not \subset B_{\varepsilon }(x)\} < \infty \end{aligned}$$

(151)

is well-defined and $q_k:=\varphi ^{(k)}(\tau _k)\in S_\varepsilon (x)$ by continuity.

First assume that $(\tau _k)$ is bounded. Then by passing to a subsequence we may assume that $\tau _k \rightarrow T \ge 0$. For each k define $\psi ^{(k)}:=\varphi ^{(k)}|_{[0,T]}$ if $\tau _k\ge T$ and otherwise define $\psi ^{(k)}$ to be the extension of $\varphi ^{(k)}|_{[0,\tau _k]}$ by the constant path $[\tau _k, T]\rightarrow \{\varphi ^{(k)}(\tau _k)\}$. Then $\mathcal {S}_T(\psi ^{(k)})\rightarrow 0$ as $k\rightarrow \infty$. Since $\mathcal {S}_T$-sublevel sets are compact in the compact-open topology on $C([0,T],M)$ by [FW12, p. 74; p. 135, Thm 3.2] it follows that a subsequence of $(\psi ^{(k)})$ converges uniformly to an absolutely continuous path $\gamma \in C([0,T],M)$ satisfying $\mathcal {S}_T(\gamma )= 0$. From this and (41) it follows that $\gamma$ is a $\textbf{v}$-integral curve segment such that $[c][\gamma ]\not = e$, and $Q_{\textbf{v}}(([c][\gamma ])^{-1}e) = Q_{\textbf{v}}([\gamma ]^{-1}e_c)=0$ by continuity of $Q_{\textbf{v}}$ [FW12, p. 143, Lem. 1.1]. This contradicts (150).

It remains only to consider the case that $(\tau _k)$ is unbounded. In this case, by passing to a subsequence we may assume that $\tau _k \ge k$ for all $k\in \mathbb {N}$. Hence for each $\ell \in \mathbb {N}$, $\psi ^{(k,\ell )}:[-\ell ,0]\rightarrow M$ given by $\psi ^{(k,\ell )}(t):=\varphi ^{(k)}(t+\tau _k)$ is well-defined for all $k\ge \ell$. By the compactness of sublevel sets of $\mathcal {S}_{-\ell ,0}$ and a diagonal argument, we can construct a single subsequence of the $\varphi ^{(k)}$ such that, after passing to this subsequence, for each $\ell$ the paths $\psi ^{(k,\ell )}_\ell$ converges uniformly as $k\rightarrow \infty$ to an absolutely continuous $\gamma ^{(\ell )}:[-\ell ,0]\rightarrow \infty$ satisfying $\gamma ^{(\ell )}(0)\in S_\varepsilon (x)$ and $\gamma ^{(\ell +1)}|_{[-\ell ,0]}=\gamma ^{(\ell )}$ for all $\ell$. Hence there is a nonconstant $\textbf{v}$-integral curve segment $\gamma :(-\infty ,0]\rightarrow M$ satisfying $\gamma |_{[-\ell ,0]}=\gamma ^{(\ell )}$ for all $\ell$, $\gamma (0)\in S_\varepsilon (x)$, and $\gamma ((-\infty ,0])\subset B_\varepsilon (x)$ since the image of each $\psi ^{k,\ell }$ is contained in $B_{\varepsilon }(x)$ by the definition of $\tau _k$. Since $R(\textbf{v})$ is finite, it follows that $\omega ^*(\gamma ) \subset B_\varepsilon (x) \cap \textbf{v}^{-1}(0)$. Since $B_\varepsilon (x)$ is disjoint from $\textbf{v}^{-1}(0){\setminus } \{x\}$ by construction, it follows that $x\in \textbf{v}^{-1}(0)$ and $\omega (\gamma _N) = \{x\}=\omega ^*(\gamma )$, so $(\gamma _1,\ldots ,\gamma _N,\gamma )$ is a piecewise $\textbf{v}$-integral curve. Moreover, $[c][\gamma ]\ne e$, and $Q_{\textbf{v}}(([c][\gamma ])^{-1}e)=Q_{\textbf{v}}([\gamma ]^{-1}e_c)=0$ by continuity. This contradicts (150) and completes the proof.

Lemma A.1 now enables an easy proof of Prop. 6.5. For convenience we restate the proposition.

Proposition 6.5

Assume that the chain recurrent set $R(\textbf{v})$ is finite. Then for any $e\in \Pi (M)$,

$$\begin{aligned} Q_{\textbf{v}}(e) = 0 \quad \iff \quad \text {e contains a piecewise} \textbf{v}-\text {integral curve} \end{aligned}$$

and

$$\begin{aligned} Q_{\textbf{v}}(e) = \inf _{[\varphi ]=e}\left( -\int _{T_1}^{T_2}\langle \dot{\varphi }, \textbf{v}(\varphi ) \rangle dt\right) \quad \iff \quad \text {e contains a piecewise}\,\, (-\textbf{v})-\text {integral curve}, \end{aligned}$$

where the infimum is over absolutely continuous paths $\varphi$ of the form $\varphi :[T_1,T_2]\rightarrow M$ with square integrable derivative and satisfying $[\varphi ]=e$. In particular, if $\textbf{v}= \alpha ^\sharp$ is the metric dual of a closed one-form $\alpha$, then

$$\begin{aligned} Q_{\alpha ^\sharp }(e) = \int _{e}(-\alpha ) \quad \iff \quad \text {e contains a piecewise}\,\, (-\alpha ^\sharp )-\text {integral curve}. \end{aligned}$$

Remark A.2

If a continuous vector field $\textbf{v}$ on a closed manifold M has unique maximal integral curves, then there is a unique continuous flow $\Phi$ satisfying $\frac{d}{dt}\Phi ^t(x)|_{t=0}=\textbf{v}(x)$ [KK22, App. A.1]. In this case Def. 6.1, 6.3, and 6.4 and the definition $R(\textbf{v}):=R(\Phi )$ still make sense, and Lem. A.1 and Prop. 6.5 still hold for such a $\textbf{v}$ with all other hypotheses unchanged. The proofs are identical.

Proof

Fix $e\in \Pi (M)$. The first displayed statement is the content of Lem. A.1, and the third displayed statement is immediate from the second. To prove the second displayed statement we observe that Lem. 5.8 implies that, for any $\gamma \in C_e([T_1,T_2],M)$ satisfying $\mathcal {S}(\gamma )<+\infty$,

$$\begin{aligned} \mathcal {S}(\gamma ) \ge Q_{(-\textbf{v})}(e) + \inf _{[\varphi ]=e}\left( -\int _{T_1}^{T_2}\langle \dot{\varphi }, \textbf{v}(\varphi ) \rangle dt\right) . \end{aligned}$$

Taking the infimum over all such $\gamma$ yields

$$\begin{aligned} Q_{\textbf{v}}(e) \ge Q_{(-\textbf{v})}(e) + \inf _{[\varphi ]=e}\left( -\int _{T_1}^{T_2}\langle \dot{\varphi }, \textbf{v}(\varphi ) \rangle dt\right) , \end{aligned}$$

so the equality

$$\begin{aligned} Q_{\textbf{v}}(e)=\inf _{[\varphi ]=e}\left( -\int _{T_1}^{T_2}\langle \dot{\varphi }, \textbf{v}(\varphi ) \rangle dt\right) \end{aligned}$$

holds if and only if $Q_{(-\textbf{v})}(e) = 0$. Since $R(\textbf{v})$ is finite if and only if $R(-\textbf{v})$ is finite, the second statement of the proposition now follows from Lem. A.1 applied to the reversed vector field $(-\textbf{v})$. $\square$

We now prove Lem. 6.6. For convenience we restate the lemma.

Lemma 6.6

Assume that $R(\textbf{v})$ consists of a finite number of hyperbolic zeros. Then there exists $C>0$ such that, if $\gamma$ is any integral curve of $\textbf{v}$,

$$\begin{aligned} \text {length}(\gamma )<C. \end{aligned}$$

(56)

Proof

We begin by bounding the length of trajectory segments near a zero $z\in \textbf{v}^{-1}(0)$. Fix $z\in \textbf{v}^{-1}(0)$ and any $\varepsilon > 0$. By the hyperbolicity of z, there exists $K > 0$ and a smooth local chart $\psi :U \rightarrow \mathbb {R}^n = \mathbb {R}^{n_x+n_y}$ in which any integral curve of $\textbf{v}$ satisfies the ODE

$$\begin{aligned} \begin{aligned} \dot{x}&= Ax + R(x,y)x\\ \dot{y}&= By + Q(x,y)y \end{aligned} \end{aligned}$$

(152)

on $\mathbb {R}^{n_x+n_y}$ with all eigenvalues of A having negative real part, all eigenvalues of B having positive real part, and with $\Vert R\Vert _{a_x}, \Vert Q\Vert _{a_y} < \varepsilon$ on the product $B_{K}^{n_x}\times B_K^{n_y}$ of balls of radius K centered at the origins of $\mathbb {R}^{n_x}$ and $\mathbb {R}^{n_y}$. Here $\Vert \,\cdot \,\Vert _{a_x}, \Vert \,\cdot \,\Vert _{a_y}$ are the norms induced by an adapted inner products $\langle \,\cdot \,, \,\cdot \, \rangle _{a_x}$, $\langle \,\cdot \,, \,\cdot \, \rangle _{a_y}$ on $\mathbb {R}^{n_x},\mathbb {R}^{n_y}$ chosen so that $\langle x, Ax \rangle _{a_x} < -k_0 \Vert x\Vert ^2_{a_x}$ and $\langle y, By \rangle _{a_y}>k_0\Vert y\Vert _{a_y}^2$ for some $k_0 > 0$ [HS74, pp. 279–280]. We may and do assume that the balls $B_{K}^{n_x}$, $B_{K}^{n_y}$ are defined with respect to these adapted norm. We compute

$$\begin{aligned} \begin{aligned} \frac{d}{dt}\Vert x\Vert ^2_{a_x}&= \langle x, Ax+R(x,y)x \rangle _{a_x} < -(k_0-\varepsilon ) \Vert x\Vert ^2_{a_x}\\ \frac{d}{dt}\Vert y\Vert ^2_{a_y}&= \langle y, By + Q(x,y)y \rangle _{a_y}>(k_0-\varepsilon )\Vert y\Vert ^2_{a_y}, \end{aligned} \end{aligned}$$

and by taking $\varepsilon$ smaller if necessary we may assume that $(k_0-\varepsilon ) > 0$. Define $k:=(k_0-\varepsilon )/2>0$. Grönwall’s inequality [Kva18, App. E] implies that any integral curve segment (x(t), y(t)) with image contained in $B_K^{n_x}\times B_K^{n_y}$ satisfies

$$\begin{aligned} \begin{aligned} \forall t\ge 0&:\Vert x(t)\Vert _{a_x}\le e^{-kt}\Vert x(0)\Vert _{a_x} \quad \text {and} \quad \Vert y(t)\Vert _{a_y}\ge e^{kt}\Vert y(0)\Vert _{a_y}\\ \forall t\le 0&:\Vert x(t)\Vert _{a_x}\ge e^{-kt}\Vert x(0)\Vert _{a_x} \quad \text {and} \quad \Vert y(t)\Vert _{a_y}\le e^{kt}\Vert y(0)\Vert _{a_y}. \end{aligned} \end{aligned}$$

(153)

The second and third inequalities imply that the smallest nonnegative time $T^+\in [0,+\infty )$ an integral curve with initial condition in the region $B_K^{n_x}\times B_K^{n_y}$ exits this region through $B_{K}^{n_x}\times \partial B_K^{n_y}$ and the largest nonpositive exit time $T^{-}\in (-\infty ,0]$ through $\partial B_{K}^{n_x}\times B_{K}^{n_y}$ are well-defined. Define $\Vert (x,y)\Vert _a:=\Vert x\Vert _{a_x} + \Vert y\Vert _{a_y}$. If (x, y) is an integral curve with initial condition $(x(0),y(0))\in B_{K}^{n_x}\times B_{K}^{n_y}$, then the first and fourth inequalities imply that the connected component of (x(0), y(0)) contained in $B_K^{n_x}\times B_K^{n_y}$ has length

$$\begin{aligned} \begin{aligned} \int _{T^-}^{T^+}\Vert (x(t),y(t))\Vert dt&\le K_0 \int _{T^-}^{T^+}\Vert (x(t),y(t))\Vert _a\, dt\\&\le K_0 \int _{0}^{+\infty }\Vert x(t+T^-)\Vert _{a_x}\, dt + K_0 \int _{-\infty }^{0}\Vert y(t+T^+)\Vert _{a_y}dt \\&\le 2KK_0\int _{0}^{+\infty }e^{-kt}dt = \frac{2K K_0}{k}=:C_z, \end{aligned} \end{aligned}$$

where $\Vert \,\cdot \,\Vert$ is induced by the Riemannian metric from the statement of the lemma and the constant $K_0$ satisfies $\frac{1}{K_0}\Vert \,\cdot \,\Vert _a \le \Vert \,\cdot \,\Vert \le K_0 \Vert \,\cdot \,\Vert _a$ on $B_K^{n_x}\times B_{K}^{n_y}$.¹⁸ Defining $\tilde{U}_z$ to be the interior of the set $\psi ^{-1}(B_{K}^{n_x}\times B_K^{n_y})$, it follows that every connected component of $\gamma (\mathbb {R})\cap \tilde{U}_z$ for every maximal $\textbf{v}$-integral curve $\gamma$ has length smaller than $C_z$.

We now construct an open neighborhood $U_z\subset \tilde{U}_z$ of z such that $\gamma (\mathbb {R})\cap U_z$ has at most one connected component for any maximal $\textbf{v}$-integral $\gamma$. Define $D^u{:=} \psi ^{-1}\left( \partial B_{K}^{n_x}\times B_K^{n_y}\right)$ and $D^s:=\psi ^{-1}\left( B_{K}^{n_x}\times \partial B_K^{n_y}\right)$ and let $\Phi :\mathbb {R}\times M \rightarrow M$ be the flow of $\textbf{v}$. Let $h\in C^\infty (M)$ be a smooth [FP19] complete Lyapunov function [Con78] for $\textbf{v}$.¹⁹ Let $N\subset \tilde{U}_z$ be a small neighborhood of z and define $U_z$ to be the connected component containing z of $\tilde{U}_z\cap \Phi ^{\mathbb {R}}(N)$. By the first and fourth inequalities in (153), there exists $\delta > 0$ and N sufficiently small such that $h|_{\partial U_z\cap D^u}$ is bounded below by $h(z)+\delta$ and $h|_{\partial U_z\cap D^s}$ is bounded above by $h(z)-\delta$. Fix such an N. From the same inequalities in (153) and the definition of $U_z$ it follows that every integral curve of $\textbf{v}$ entering $U_z$ does so at some point p satisfying $h(p) > h(z)+\delta$ and leaves at some point q satisfying $h(q) < h(z)-\delta$. Since h is nonincreasing along trajectories it follows that $\gamma (\mathbb {R})\cap U_z$ has at most one connected component for any maximal integral curve $\gamma$ of $\textbf{v}$, as desired.

Note that $\textbf{v}^{-1}(0) = R(\textbf{v})$ is a finite set by assumption. We construct open sets $U_z$ and associated constants $C_z$ as above for each $z\in \textbf{v}^{-1}(0)$, and we define $U:=\bigcup _{z\in \textbf{v}^{-1}(0)}U_z$ and $\bar{C}_1:=\sum _{z\in \textbf{v}^{-1}(0)}C_z$. Since the intersection of any maximal integral curve with each $U_z$ has at most one connected component it follows that, for any maximal integral curve $\gamma$ of $\textbf{v}$,

$$\begin{aligned} \text {length}(\gamma (\mathbb {R})\cap U) \le \bar{C}_1 < \infty . \end{aligned}$$

(154)

It remains to bound $\text {length}(\gamma (\mathbb {R}){\setminus } U)$. Since every $x\in M$ converges to $\textbf{v}^{-1}(0) = R(\textbf{v})$ in both forward and backward time, for every $x\in M$ there exists $t_x$ such that both $\Phi ^{-t_x}(x), \Phi ^{t_x}(x)\in U$. By continuity, x has a neighborhood $W_x$ such that both $\Phi ^{-t_x}(W_x), \Phi ^{t_x}(W_x)\subset U$. Since $M{\setminus } U$ is compact, we extract a finite subcover $W_{x_1},\ldots , W_{x_k}$ of $M{\setminus } U$ and define $T:=\max \{t_{x_1},\ldots t_{x_k}\}$. It follows that, for every $x\in M{\setminus } U$, both $\Phi ^{[0,T]}(x)\cap U\ne \varnothing$ and $\Phi ^{[-T,0]}(x)\cap U\ne \varnothing$. Since any maximal integral curve $\gamma$ intersects each of the $\#(\textbf{v}^{-1}(0))$ connected components of U at most once, $\gamma (\mathbb {R}){\setminus } U$ contains at most $\#(\textbf{v}^{-1}(0))$ segments, and these segments are defined on intervals of length at most T. Since

$$\begin{aligned} \int _{T_1}^{T_2} \Vert \dot{\gamma }(t)\Vert \,dt = \int _{T_1}^{T_2} \Vert \textbf{v}(\gamma (t))\Vert \, dt \le (T_2-T_1)\max _{x\in M}\Vert \textbf{v}(x)\Vert <\infty \end{aligned}$$

for a $\textbf{v}$-integral curve $\gamma$, it follows that

$$\begin{aligned} \text {length}(\gamma (\mathbb {R}){\setminus } U) \le T \#(\textbf{v}^{-1}(0)) \max _{x\in M}\Vert \textbf{v}(x)\Vert =:\bar{C}_2 < \infty \end{aligned}$$

(155)

for any maximal integral curve $\gamma$.

Finally, for any $\textbf{v}$-integral curve $\gamma$ we obtain from (154) and (155)

$$\begin{aligned} \text {length}(\gamma (\mathbb {R})) = \text {length}(\gamma (\mathbb {R}) \cap U) + \text {length}(\gamma (\mathbb {R}){\setminus } U) \le \bar{C}_1+ \ \bar{C}_2=:C < \infty , \end{aligned}$$

as desired.

We now prove a preliminary lemma in preparation for the proofs of Lem. A.4 and Prop. 7.7. The following notation will be used for the statements and proofs of the remaining results in this appendix. Given a continuous vector field $\textbf{w}$ on M, we denote by $\mathcal {S}^\textbf{w}$ the action functional defined according to (41) but with the vector field $\textbf{w}$ replacing $\textbf{v}$, and we denote by $Q_{\textbf{w}}$ the associated quasipotential. If $\textbf{w}$ is a vector field on the universal cover $\tilde{M}$ of M, we still use the notations $\mathcal {S}^{\textbf{w}}$ and $Q_{\textbf{w}}$ for the actional functional and quasipotential associated to a vector field $\textbf{w}$ on $\tilde{M}$, with context dictating whether the notation refers to M or $\tilde{M}$.

Lemma A.3

Denote by $\mathfrak {X}^0(M)$ the space of $C^0$ vector fields on the Riemannian manifold M equipped with the $C^0$ topology. Let $\Pi (M)$ have the topology induced by its bijection with the smooth manifold $(\tilde{M}\times \tilde{M})/\text {Aut}(\pi )$, where $\pi :\tilde{M}\rightarrow M$ is the universal cover and the deck transformation group $\text {Aut}(\pi )$ acts diagonally on $\tilde{M}\times \tilde{M}$ (Rem. 7.6). Then the map

$$\begin{aligned} (\textbf{v},e)\in \mathfrak {X}^0(M)\times \Pi (M) \mapsto Q_{\textbf{v}}(e)\in [0,+\infty ) \quad \text {is upper semicontinuous}. \end{aligned}$$

Proof

Our task is equivalent to proving that the map

$$\begin{aligned} (\textbf{w},x,y)\in \mathfrak {X}^0(\tilde{M})\times \tilde{M}\times \tilde{M}\mapsto Q_{\textbf{w}}(x,y)\in [0,+\infty ) \end{aligned}$$

is upper semicontinuous, where $Q_{\textbf{w}}$ is defined with respect to the pullback metric on $\tilde{M}$. Fix $\textbf{w}\in \mathfrak {X}^0(\tilde{M})$, let $B\subset \tilde{M}$ be a precompact open set, and let $C_0$ be an upper bound for $\Vert \textbf{w}\Vert$ on B. Then if $\textbf{u}\in \mathfrak {X}^0(\tilde{M})$ satisfies $\Vert \textbf{u}-\textbf{w}\Vert \le 1$ on B and $\varphi _a$ is a unit speed length minimizing geodesic from $a\in B$ to $x\in B$, a computation using the definition (41) of $\mathcal {S}^{\textbf{u}}$ yields $\mathcal {S}^{\textbf{u}}(\varphi _a)\le L\text {dist}( a, x )$, where $L = (1/4)(2+C_0)^2$. Similarly, $\mathcal {S}^{\textbf{u}}(\varphi _b)\le L\text {dist}( y, b )$ if $\varphi _b$ is a unit speed length minimizing geodesic from $y\in B$ to $b\in B$. Thus, $Q_{\textbf{u}}(a,b)\le Q_{\textbf{u}}(x,y) + L\text {dist}( a, x ) + L\text {dist}( y, b )$ if $a,x\in B$ and $y,b\in B$ are sufficiently close. Hence it suffices to prove that the map

$$\begin{aligned} \textbf{w}\in \mathfrak {X}^0(\tilde{M})\mapsto Q_{\textbf{w}}(x,y)\in [0,+\infty ) \end{aligned}$$

(156)

is upper semicontinuous for each fixed $x,y\in \tilde{M}$.

Fix $\textbf{w}\in \tilde{M}$ and $\varepsilon > 0$. Let $\varphi :[0,T]\rightarrow \tilde{M}$ be a continuous path from x to y satisfying $\mathcal {S}^{\textbf{w}}_T(\varphi ) < Q_{\textbf{w}}(x,y) + \varepsilon .$ It is immediate from the definition (41) that also $\mathcal {S}^{\textbf{u}}_T(\varphi ) < Q_{\textbf{w}}(x,y) + \varepsilon$ if $\textbf{u}$ is sufficiently close to $\textbf{w}$ on the compact set $\varphi ([0,T])$. Since $Q_{\textbf{u}}(x,y) \le \mathcal {S}^{\textbf{u}}_T(\varphi )$, it follows that the map in (156) is upper semicontinuous.

Lemma A.4

Let $(\textbf{u}_n)_{n\in \mathbb {N}}$ be a sequence of $C^0$ vector fields on the closed Riemannian manifold M converging uniformly to a continuous vector field $\textbf{v}$. Let $k > 0$ and $(\varphi ^{(n)}\in C([T_1,T_2],M))_{n\in \mathbb {N}}$ be a sequence of paths with fixed domain $[T_1,T_2]$ satisfying $\mathcal {S}^{\textbf{u}_n}_{T_1,T_2}(\varphi ^{(n)})\le k$ for all n. Then the family $(\varphi ^{(n)})$ is uniformly equicontinuous, and there is a subsequence $(\varphi ^{(n_k)})_{k\in \mathbb {N}}$ which converges uniformly to a path $\varphi \in C([T_1,T_2],M)$ satisfying $\mathcal {S}^{\textbf{v}}_{T_1,T_2}(\varphi )\le k$.

Proof

By a translation of $\mathbb {R}$ we may assume that $T_1 = 0$ and $T_2 = T$. By the Nash embedding theorem we may assume that M is isometrically embedded in some $\mathbb {R}^N\supset M$, so the Riemannian metric on M is the restriction of the Euclidean inner product [Nas56, Thm 2], and we may view $\varphi , \varphi ^{(n)}, \dot{\varphi }^{(n)}\in C([0,T],\mathbb {R}^N)$ as $\mathbb {R}^N$-valued. We may also view $\textbf{v}$ and $\textbf{u}_n$ as $\mathbb {R}^N$-valued, and we arbitrarily extend the $\textbf{u}_n$ and $\textbf{v}$ to $C^0$ maps $\mathbb {R}^N\rightarrow \mathbb {R}^N$.

We first show uniform equicontinuity of the family $(\varphi ^{(n)})$ and convergence of a subsequence $\varphi ^{(n_k)}$ to some $\varphi \in C([0,T],M)$. Given $t,h\ge 0$ satisfying $t+h \le T$, the triangle and Cauchy-Schwarz inequalities imply that

$$\begin{aligned} \begin{aligned} \Vert \varphi ^{(n)}(t+h)-\varphi ^{(n)}(t)\Vert&\le \int _t^{t+h}\Vert \dot{\varphi }^{(n)}(s)\Vert ds\le \int _t^{t+h}\Vert \dot{\varphi }^{(n)}-\textbf{u}_n(\varphi ^{(n)})\Vert ds \\&\quad + \int _t^{t+h}\Vert \textbf{u}_n(\varphi ^{(n)})\Vert ds \\&\le \sqrt{h\int _t^{t+h}\Vert \dot{\varphi }^{(n)}-\textbf{u}_n(\varphi ^{(n)})\Vert ^2ds} + h\Vert \textbf{u}_n\Vert _{\infty }\\&\le \sqrt{4h\mathcal {S}^{\textbf{u}_n}_T(\varphi ^{(n)})} + h\Vert \textbf{u}_n\Vert _{\infty } \le \sqrt{4hk} + 2hK, \end{aligned} \end{aligned}$$

(157)

where $\Vert \,\cdot \,\Vert _\infty$ is the supremum norm on the restrictions of functions to $M\subset \mathbb {R}^N$ and $K>0$ is an upper bound on the convergent sequence $(\Vert \textbf{u}_n\Vert _\infty )$. Thus, the resulting family of paths $(\varphi ^{(n)})$ is uniformly equicontinuous. Since $M\subset \mathbb {R}^N$ is compact and hence bounded, the Arzelà-Ascoli theorem and closedness of $C([0,T],M)\subset C([0,T],\mathbb {R}^N)$ imply the existence of a subsequence of $(\varphi ^{(n)})$ converging uniformly to some $\varphi \in C([0,T],M)$ as claimed. For simplicity we relabel the subsequence so as to use the same notation $(\varphi ^{(n)})$ below, so that $\varphi ^{(n)}\rightarrow \varphi$, and we continue to relabel in this same way after passing to further subsequences.

It remains to show that $\mathcal {S}_T^\textbf{v}(\varphi )\le k$. Define the $L^2$ inner product $\langle f, g \rangle _2:=\int _0^T \langle f, g \rangle dt$ and norm $\Vert g\Vert _2:=\sqrt{\langle g, g \rangle _2}$ of measurable functions $[0,T]\rightarrow \mathbb {R}^m$ for some m. Using the Cauchy-Schwarz inequality we note (cf. Rem. 5.2) that

$$\begin{aligned} \begin{aligned} 4\mathcal {S}_{T}^{\textbf{u}_n}(\varphi ^{(n)})&= \Vert \dot{\varphi }^{(n)}\Vert _2^2 - 2\langle \dot{\varphi }^{(n)}, \textbf{u}_n(\varphi ^{(n)}) \rangle _2 + \Vert \textbf{u}_n(\varphi ^{(n)})\Vert _2^2\\&\ge \Vert \dot{\varphi }^{(n)}\Vert _2^2 - 2\Vert \dot{\varphi }^{(n)}\Vert _2\Vert \textbf{u}_n(\varphi ^{(n)})\Vert _2 + \Vert \textbf{u}_n(\varphi ^{(n)})\Vert _2^2 \\&= (\Vert \dot{\varphi }^{(n)}\Vert _2-\Vert \textbf{u}_n(\varphi ^{(n)})\Vert _2)^2. \end{aligned} \end{aligned}$$

(158)

Since $\mathcal {S}_{T}^{\textbf{u}_n}(\varphi ^{(n)})$ and $\Vert \textbf{u}_n(\varphi ^{(n)})\Vert _2\le T \Vert \textbf{u}_n\Vert _\infty \rightarrow T\Vert \textbf{v}\Vert _\infty$ are uniformly bounded, it follows from (158) that so are the $\Vert \dot{\varphi }^{(n)}\Vert _2$. Writing $\varphi ^{(n)} = (\varphi ^{(n)}_1, \ldots , \varphi ^{(n)}_N)$, it follows that the $L^2$ norms of the derivatives of each of the components $\dot{\varphi }^{(n)}_i\in L^2([0,T],\mathbb {R})$ are uniformly bounded. Hence the Banach-Alaoglu theorem [Sho94, p. 23, Thm 6.2] implies that, after passing to a subsequence, each component sequence $\dot{\varphi }^{(n)}_i$ converge weakly in the Hilbert space $L^2([0,T],\mathbb {R})$ to some $g_i\in L^2([0,T],\mathbb {R})\subset L^1([0,T],\mathbb {R})$. This means that

$$\begin{aligned} \forall i\in \{1,\ldots , N\}:\forall f\in L^2([0,T],\mathbb {R}):\lim _{n\rightarrow \infty } \langle \dot{\varphi }^{(n)}_i, f \rangle _2 = \langle g_i, f \rangle _2. \end{aligned}$$

(159)

Taking f to be the indicator functions $\textbf{1}_{[0,t]}$ in (159) yields, for all $1\le i\le N$:

$$\begin{aligned} \begin{aligned} \forall t\in [0,T]:\varphi _i(t)&= \lim _{n\rightarrow \infty }\varphi _i^{(n)}(t)= \lim _{n\rightarrow \infty }\varphi ^{(n)}_i(0) + \int _0^t \dot{\varphi }_i^{(n)}(s)ds = \varphi _i(0) \\&\quad + \lim _{n\rightarrow \infty } \langle \dot{\varphi }_i^{(n)}, \textbf{1}_{[0,t]} \rangle _2 \\&= \varphi _i(0) + \langle g_i, \textbf{1}_{[0,t]} \rangle _2 = \varphi _i(0) + \int _0^t g_i(s)ds. \end{aligned} \end{aligned}$$

Since each $g_i\in L^1([0,T],\mathbb {R})$, it follows that $\varphi$ is absolutely continuous with derivative $\dot{\varphi }= (g_1,\ldots , g_n)$ almost everywhere [Fol99, Thm 3.35].

Since Hilbert space norms are weakly lower semicontinuous [RS80, p. 355] and since the $\dot{\varphi }^{(n)}$ converge weakly to $g = \dot{\varphi }$, we also have $\Vert \dot{\varphi }\Vert _2\le \liminf _{n\rightarrow \infty }\Vert \dot{\varphi }_n\Vert _2$. Thus,

$$\begin{aligned} \begin{aligned} \mathcal {S}_T^\textbf{v}(\varphi )&= \Vert \dot{\varphi }\Vert _2^2 - 2\langle \dot{\varphi }, \textbf{v}(\varphi ) \rangle _2 + \Vert \textbf{v}(\varphi )\Vert _2 = \Vert \dot{\varphi }\Vert _2^2 - 2\lim _{n\rightarrow \infty }\langle \dot{\varphi }^{(n)}, \textbf{v}(\varphi ) \rangle _2 + \lim _{n\rightarrow \infty } \Vert \textbf{u}_n(\varphi ^{(n)})\Vert _2\\&= \Vert \dot{\varphi }\Vert _2^2 + \lim _{n\rightarrow \infty }\left[ - 2\langle \dot{\varphi }^{(n)}, \textbf{u}_n(\varphi ^{(n)}) \rangle _2 + \Vert \textbf{u}_n(\varphi ^{(n)})\Vert _2 + 2\langle \dot{\varphi }^{(n)}, \textbf{u}_n(\varphi ^{(n)})-\textbf{v}(\varphi ) \rangle _2\right] \\&\le \liminf _{n\rightarrow \infty }\Vert \dot{\varphi }^{(n)}\Vert _2^2+ \lim _{n\rightarrow \infty }\left[ - 2\langle \dot{\varphi }^{(n)}, \textbf{u}_n(\varphi ^{(n)}) \rangle _2 + \Vert \textbf{u}_n(\varphi ^{(n)})\Vert _2 \right. \\&\quad \left. + 2\langle \dot{\varphi }^{(n)}, \textbf{v}(\varphi )-\textbf{u}_n(\varphi ^{(n)}) \rangle _2\right] \\&\le \liminf _{n\rightarrow \infty } \mathcal {S}_T^{\textbf{u}_n}(\varphi ^{(n)}) + 2\limsup _{n\rightarrow \infty } \langle \dot{\varphi }^{(n)}, \textbf{v}(\varphi )-\textbf{u}_n(\varphi ^{(n)}) \rangle _2, \end{aligned} \end{aligned}$$

where the second equality follows from (159) and the uniform convergence of $\textbf{u}_n(\varphi ^{(n)})$ to $\textbf{v}(\varphi )$, and the second inequality follows from the first equality in (158) and the general fact that

$$\begin{aligned} \liminf _{n\rightarrow \infty }a_n+\liminf _{n\rightarrow \infty }(b_n+c_n)\le \liminf _{n\rightarrow \infty } (a_n+b_n+c_n) \le \liminf _{n\rightarrow \infty }(a_n+b_n)+\limsup _{n\rightarrow \infty }c_n. \end{aligned}$$

Using $\Vert \textbf{v}(\varphi )-\textbf{u}_n(\varphi ^{(n)})\Vert \le \Vert \textbf{v}(\varphi )-\textbf{v}(\varphi ^{(n)})\Vert + \Vert \textbf{v}(\varphi ^{(n)}) -\textbf{u}_n(\varphi ^{(n)})\Vert$, it follows that

$$\begin{aligned} \begin{aligned} \mathcal {S}_T^\textbf{v}(\varphi )&\le \liminf _{n\rightarrow \infty } \mathcal {S}_T^{\textbf{u}_n}(\varphi ^{(n)}) + 2\sqrt{T} \limsup _{n\rightarrow \infty } \Vert \dot{\varphi }^{(n)}\Vert _2\sup _{t\in [0,T]}\Vert \textbf{v}(\varphi (t))-\textbf{u}_n(\varphi ^{(n)}(t))\Vert \\&\le \liminf _{n\rightarrow \infty } \mathcal {S}_T^{\textbf{u}_n}(\varphi ^{(n)}) + 2\sqrt{T} \limsup _{n\rightarrow \infty } \Vert \dot{\varphi }^{(n)}\Vert _2 \\&\times \left( \sup _{t\in [0,T]}\Vert \textbf{v}(\varphi (t))-\textbf{v}(\varphi ^{(n)}(t))\Vert + \Vert \textbf{v}-\textbf{u}_n\Vert _\infty \right) . \end{aligned} \end{aligned}$$

The $L^2$ norms $\Vert \dot{\varphi }^{(n)}\Vert _2$ are bounded (as noted following (158)) and $\textbf{v}$ is uniformly continuous on the compact M, so the uniform convergence of $\varphi ^{(n)}$ to $\varphi$ and of $\textbf{u}_n$ to $\textbf{v}$ implies that the $\limsup$ is zero. Since $\mathcal {S}_T^{\textbf{u}_n}(\varphi ^{(n)}) \le k$ for all n by assumption, this proves the desired remaining claim

$$\begin{aligned} \mathcal {S}_T^\textbf{v}(\varphi )\le \liminf _{n\rightarrow \infty } \mathcal {S}_T^{\textbf{u}_n}(\varphi ^{(n)}) \le k. \end{aligned}$$

We now prove Prop. 7.7. For convenience we restate the proposition.

Proposition 7.7

Denote by $\mathfrak {X}^1(M)$ the space of $C^1$ vector fields on the closed Riemannian manifold M equipped with the $C^1$ topology. Let $\Pi (M)$ have the topology induced by its bijection with the smooth manifold $(\tilde{M}\times \tilde{M})/\text {Aut}(\pi )$, where $\pi :\tilde{M}\rightarrow M$ is the universal cover and the deck transformation group $\text {Aut}(\pi )$ acts diagonally on $\tilde{M}\times \tilde{M}$. Let $\textbf{v}_0\in \mathfrak {X}^1(M)$ be a Morse–Smale vector field without nonstationary periodic orbits. Then for any $e_0\in \Pi (M)$, the map

$$\begin{aligned} (\textbf{v},e)\in \mathfrak {X}^1(M)\times \Pi (M) \mapsto Q_{\textbf{v}}(e)\in [0,+\infty ) \quad \text {is continuous at }(\textbf{v}_0,e_0). \end{aligned}$$

Proof

Our task is equivalent to proving that the map

$$\begin{aligned} (\textbf{w},x,y)\in \mathfrak {X}^1(\tilde{M})\times \tilde{M}\times \tilde{M}\mapsto Q_{\textbf{w}}(x,y)\in [0,+\infty ) \end{aligned}$$

is continuous at $(\tilde{\textbf{v}}_0,x,y)$ for each $x,y\in \tilde{M}$, where $\tilde{\textbf{v}}_0\in \mathfrak {X}^1(\tilde{M})$ is the unique lift of $\textbf{v}_0$ to $\tilde{M}$ ($\pi _*\tilde{\textbf{v}}_0 = \textbf{v}_0$) and $Q_{\textbf{v}}$ is defined with respect to the pullback metric on $\tilde{M}$.

Fix $x,y\in \tilde{M}$. As shown in the proof of Lem. A.3, if $\Vert \textbf{u}-\tilde{\textbf{v}}_0\Vert \le 1$ on some compact neighborhood B of (x, y), then there exists $L > 0$ such that $Q_{\textbf{u}}(a,b)\le Q_{\textbf{u}}(x,y) + L\text {dist}( a, x ) + L\text {dist}( y, b )$ for all $a,b\in B$ with (a, b) sufficiently close to (x, y). Reversing the roles of (a, b) and (x, y) yields $Q_{\textbf{u}}(x,y)\le Q_{\textbf{u}}(a,b) + L\text {dist}( a, x ) + L\text {dist}( y, b )$, so

$$\begin{aligned} |Q_{\textbf{u}}(x,y)-Q_{\textbf{u}}(a,b)| \le L\text {dist}( a, x ) + L\text {dist}( y, b ) \rightarrow 0 \quad \text {as} \quad \text {dist}( a, x ) + \text {dist}( y, b )\rightarrow 0. \end{aligned}$$

Since the triangle inequality yields

$$\begin{aligned} |Q_{\textbf{w}}(x,y)-Q_{\textbf{u}}(a,b)|\le |Q_{\textbf{w}}(x,y)-Q_{\textbf{u}}(x,y)| + |Q_{\textbf{u}}(x,y) - Q_{\textbf{u}}(a,b)|, \end{aligned}$$

(160)

we see it suffices to prove that the map

$$\begin{aligned} \textbf{w}\in \mathfrak {X}^1(\tilde{M})\mapsto Q_{\textbf{w}}(x,y)\in [0,+\infty ) \end{aligned}$$

is continuous at $\tilde{\textbf{v}}_0$ for each fixed $x,y\in \tilde{M}$, and this is in turn equivalent to proving that the map

$$\begin{aligned} \textbf{v}\in \mathfrak {X}^1(M)\mapsto Q_{\textbf{v}}(e)\in [0,+\infty ) \end{aligned}$$

(161)

is continuous at $\textbf{v}_0$ for each fixed $e\in \Pi (M)$.

Since the $C^1$ topology is finer than the $C^0$ topology, upper semicontinuity at $\textbf{v}_0$ follows from Lem. A.3. It remains to establish lower semicontinuity.

Since Morse–Smale vector fields are open in the $C^1$ topology [Pal68, Thm 3.5] and structurally stable [PS70, Thm 5.2], there exists a neighborhood $\mathcal {N}\subset \mathfrak {X}^1(M)$ of $\textbf{v}_0$ such that every $\textbf{v}\in \mathcal {N}$ is Morse–Smale without nonstationary periodic orbits. Suppose (to obtain a contradiction) that the map in (161) is not lower semicontinuous at $\textbf{v}_0$ for arbitrary $e\in \Pi (M)$. Then there exists $e\in \Pi (M)$, $k > 0$, and a sequence $(\textbf{u}_n)_{n\in \mathbb {N}}\subset \mathcal {N}$ with $\textbf{u}_n\rightarrow \textbf{v}_0$ in $\mathfrak {X}^1(M)$ such that $Q_{\textbf{u}_n}(e) < Q_{\textbf{v}_0}(e)-2k$ for all n. Hence for each n there exists a path $\varphi ^{(n)}\in C_e([0,T_n],M)$ with

$$\begin{aligned} \mathcal {S}^{\textbf{u}_n}(\varphi ^{(n)})< Q_{\textbf{v}_0}(e)-2k \end{aligned}$$

(162)

for all n.

If $(T_n)$ is bounded, then by passing to a subsequence we may assume that $T_n\rightarrow T\ge 0$. In this case we define $\psi ^{(n)}:=\varphi ^{(n)}|_{[0,T]}$ if $T_n\ge T$ and otherwise we define $\psi ^{(n)}$ to be the extension of $\varphi ^{(n)}|_{[0,T_n]}$ by the constant path $[T_n, T]\rightarrow \{\mathfrak {t}(e)\}$. Since $T_n\rightarrow T$ and $\Vert \textbf{u}_n-\textbf{v}_0\Vert \rightarrow 0$, $\mathcal {S}_T^{\textbf{u}_n}(\psi ^{(n)})< Q_{\textbf{v}_0}(e)-2k$ for all n large enough. After passing to a subsequence if necessary, it follows from Lem. A.4 that the $\psi ^{(n)}$ converge to a path $\varphi \in C_e([0,T],M)$ satisfying $\mathcal {S}_T^{\textbf{v}_0}(\varphi )\le Q_{\textbf{v}_0}(e)-2k$, a contradiction.

It remains to consider the case that $(T_n)$ is unbounded. We first make some preliminary observations. Define $E\subset \Pi (M)$ via $E:=\mathfrak {s}^{-1}(\textbf{v}^{-1}(0)) \cap \mathfrak {t}^{-1}(\textbf{v}^{-1}(0))$, and define $E_0\subset E$ via

$$\begin{aligned} E_0:=\{e\in E:\mathfrak {s}(e)=\mathfrak {t}(e) \text { and } e\text { is not a constant path homotopy class}\}. \end{aligned}$$

The chain recurrent set $R(\textbf{v})$ consists of a finite number of hyperbolic zeros since $\textbf{v}$ is Morse–Smale without nonstationary periodic orbits, so Prop. 6.5 and Lem. 6.8 imply the existence of $C_0 > 0$ such that, for any finite sequence $e_1,\ldots , e_n \in E$ such that $e_1\cdots e_n\in E_0$,

$$\begin{aligned} \min _{i\in \{1,\ldots , n\}}Q_{\textbf{v}}(e_i) > C_0. \end{aligned}$$

(163)

Let $\kappa _0 > 0$ be sufficiently small that the closed metric balls $B_\kappa (z)$ of radius $\kappa$ centered at each $z\in \textbf{v}^{-1}(0)$ are geodesically convex [Lee18, Thm 6.17] and pairwise disjoint, and define $B_\kappa :=\bigcup _{z\in \textbf{v}^{-1}(0)}B_\kappa (z)$ for $\kappa \in (0,\kappa _0)$. Given any path $\varphi \in C([T_1,T_2],M)$ with initial and terminal points in $B_\kappa$, we denote by $e(\varphi )\in E$ the path homotopy class of the path defined by first following the unique minimizing geodesic in $B_\kappa$ from $\textbf{v}^{-1}(0)$ to $\varphi (0)$, then following $\varphi$, then following the unique minimizing geodesic in $B_\kappa$ from $\varphi (T)$ to $\textbf{v}^{-1}(0)$. By [FW12, p. 143, Lem. 1.1] we may choose $\kappa$ small enough that any pair of points in the same component of $B_\kappa$ may be joined by a path $\varphi$ satisfying $\mathcal {S}(\varphi )< \varepsilon /2$. It follows that, for any $\varepsilon > 0$ such that $C_1:=C_0 - \varepsilon > 0$, all sufficiently small $\kappa > 0$, and any finite sequence of paths $\gamma _1,\ldots , \gamma _n$ with initial and terminal points in $B_{\kappa }$ satisfying $e(\gamma _1)\cdots e(\gamma _n)\in E_0$,

$$\begin{aligned} \begin{aligned} \mathcal {S}(\gamma _1)+\cdots + \mathcal {S}(\gamma _n) \ge \min _{i\in \{1,\ldots ,n\}}\mathcal {S}(\gamma _i) \ge \min _{i\in \{1,\ldots , n\}}Q_{\textbf{v}}(e(\gamma _i))-\varepsilon> C_1 > 0. \end{aligned} \end{aligned}$$

(164)

Fix $\varepsilon \in (0,k)$ (cf. (162)) and $\kappa _1\in (0,\kappa _0)$ small enough that (164) holds for some $C_1 > 0$ for all $\kappa \in (0,\kappa _1)$.

Next, the implicit function theorem implies that $(\textbf{u}_n)^{-1}(0) \subset \text {int}(B_{\kappa })$ for all sufficiently large n since $\textbf{u}_n\rightarrow \textbf{v}_0$ in $\mathfrak {X}^1(M)$ and $\textbf{v}^{-1}(0)$ consists of finitely many hyperbolic zeros. Let $\Phi _{\textbf{v}}:\mathbb {R}\times M\rightarrow M$ denote the flow of $\textbf{v}\in \mathfrak {X}^1(M)$. For any $\kappa \in (0,\kappa _1)$, since $R(\textbf{u}_n) = \textbf{u}_n^{-1}(0)\subset \text {int}(B_\kappa )$ for each n and since $\textbf{u}_n\rightarrow \textbf{v}_0$, joint continuity of the map

$$\begin{aligned} (t,x,\textbf{v})\in \mathbb {R}\times M\times \mathfrak {X}^1(M)\mapsto \Phi _{\textbf{v}}^t(x) \in M \end{aligned}$$

[DK00, Thm B.3] implies the existence of $T(\kappa ) > 0$ such that, for any $x\not \in \text {int}(B_\kappa )$ and sufficiently large $n\in \mathbb {N}$, there exists $t_x, t_{x,n}\in [0,T(\kappa )/2]$ such that $\Phi _{\textbf{v}_0}^{t_x}(x),\Phi _{\textbf{u}_n}^{t_x^{(n)}}(x)\in \text {int}(B_\kappa )$. Thus, $\mathcal {S}_{T(\kappa )}^{\textbf{v}_0}(\varphi ), \mathcal {S}_{T(\kappa )}^{\textbf{u}_n}(\varphi ) > 0$ for any $\varphi \in ([0,T(\kappa )],M{\setminus } \text {int}(B_\kappa ))$ for large enough n. Since $C([0,T(\kappa )],M{\setminus } \text {int}(B_\kappa ))\subset C([0,T(\kappa )],M)$ is closed, Lem. A.4 implies the existence of $C_2(\kappa )>0$ such that

$$\begin{aligned} \mathcal {S}_{T(\kappa )}^{\textbf{v}_0}(\varphi ), \mathcal {S}_{T(\kappa )}^{\textbf{u}_n}(\varphi ^{(n)})> C_2(\kappa ) > 0 \end{aligned}$$

(165)

for all $\varphi \in C([0,T(\kappa )],M{\setminus } \text {int}(B_\kappa ))$, all $\kappa \in (0,\kappa _1)$, and all n large enough.

Next, for each n we define the interval $[a^{(n)}_1, b^{(n)}_1]$ by the properties $\varphi ^{(n)}(a_1^{(n)}), \varphi ^{(n)}(b_1^{(n)}) \in \partial B_\kappa$, $e(\varphi ^{(n)}|_{[a_1^{(n)},b_1^{(n)}]})$ is not a constant path homotopy class, and $a^{(n)}_1, b^{(n)}_1$ are the smallest numbers in $[0,T_n]$ with these properties. For each n we then recursively define the intervals $[a^{(n)}_{i+1}, b^{(n)}_{i+1}]$ so that $b^{(n)}_{i+1}> a^{(n)}_{i+1} > b^{(n)}_i$ are the smallest numbers larger than $b^{(n)}_i$ with the same properties; it follows in particular that each $e(\varphi ^{(n)}|_{[b_{i}^{(n)},a_{i+1}^{(n)}]})$ is a constant path homotopy class. Suppose (to obtain a contradiction) that the numbers $N_n$ of such intervals $[a^{(n)}_{i}, b^{(n)}_{i}]$ are unbounded; after passing to a subsequence we may assume that $N_n\rightarrow +\infty$. Equation (165) implies that the numbers $(b_i^{(n)}-a_i^{(n)})$ are bounded, so after passing to a diagonal subsequence we may further assume that, for each i, $(b_i^{(n)}-a_i^{(n)})$ converges to some $c_i \ge 0$. For each i and large enough n we define $\psi _i^{(n)}:[0,c_i]\rightarrow M$ via $\psi _i^{(n)}(t):=\varphi ^{(n)}(a_i^{(n)}+t)$ for all $t\in [0,c_i]$ if $c_i \le (b_i^{(n)}-a_i^{(n)})$, and otherwise we define $\psi _i^{(n)}$ to be given by this formula for $t\in [0, b_i^{(n)}-a_i^{(n)}]$ and constant on $[b_i^{(n)}-a_i^{(n)},c_i]$. For each i we have that $\mathcal {S}^{\textbf{u}_n}(\psi _i^{(n)})$ is bounded, so Lem. A.4 and a diagonal argument imply that, after passing to a subsequence of $(\varphi ^{(n)})$, the $\psi _i^{(n)}$ converge to paths $\gamma _i$ such that $\mathfrak {t}(e(\gamma _i)) = \mathfrak {s}(e(\gamma _{i+1}))$ and $\mathcal {S}^{\textbf{u}_n}(\psi _i^{(n)})\rightarrow \mathcal {S}^{\textbf{v}_0}(\gamma _i)$. Hence

$$\begin{aligned} \begin{aligned} \sum _{i=1}^\infty \mathcal {S}^{\textbf{v}_0}(\gamma _i)&= \sum _{i=1}^\infty \lim _{n\rightarrow \infty }\mathcal {S}^{\textbf{u}_n}(\psi _i^{(n)}) = \sum _{i=1}^\infty \lim _{n\rightarrow \infty }\mathcal {S}^{\textbf{u}_n}(\varphi |^{(n)}_{[a_i^{(n)},b_i^{(n)}]}) \le \liminf _{n\rightarrow \infty }\sum _{i=1}^\infty \mathcal {S}^{\textbf{u}_n}(\varphi |^{(n)}_{[a_i^{(n)},b_i^{(n)}]})\\ {}&\le \liminf _{n\rightarrow \infty }\mathcal {S}^{\textbf{u}_n}(\varphi ^{(n)})\le Q_{\textbf{v}}(e)-2k, \end{aligned} \end{aligned}$$

(166)

where the second equality follows since $|\mathcal {S}^{\textbf{u}_n}(\psi _i^{(n)})-\mathcal {S}^{\textbf{u}_n}(\varphi ^{(n)}|_{[a^{(n)}_i,b^{(n)}_i]})| \rightarrow 0$, the first inequality follows from Fatou’s lemma, the second inequality follows since for each n the intervals $(a_i^{(n)},b_i^{(n)})$ are disjoint, and the final inequality follows from (162). But since $\mathfrak {t}(e(\gamma _i)) = \mathfrak {s}(e(\gamma _{i+1}))$ for all i, (164) and the pigeonhole principle imply that the left side of (166) is larger than $C_1 + C_1 + \dots = +\infty$, so we have arrived at a contradiction. It follows that there is an integer $N\ge 1$ such that

$$\begin{aligned} \forall n\in \mathbb {N}:\text { there are at most }\,N\,\text { such intervals}\, [a_i^{(n)},b_i^{(n)}]. \end{aligned}$$

(167)

Finally, observe that (167) holds with the same constant N for all $\kappa \in (0,\kappa _1)$ even though the constants $T(\kappa )$, $C_2(\kappa )$ in (165) depend on the specific value of $\kappa$. Additionally, (164) holds with the same constant $C_1$ for all $\kappa \in (0,\kappa _1)$. Using [FW12, p. 143, Lem. 1.1] again, there is $\kappa _2\in (0,\kappa _1)$ such that, for any $\kappa \in (0,\kappa _2)$, any pair of points in the same component of $B_\kappa$ may be joined by a short path $\varphi :[0,\tau ]\rightarrow B_{\kappa }$ satisfying $\mathcal {S}(\varphi )< \varepsilon /N$ and $\tau < \varepsilon /N$. For each n we modify $\varphi ^{(n)}$ by deleting each of the $N_n \le N$ restrictions $\varphi ^{(n)}|_{[b_i^{(n)},a_{i+1}^{(n)}]}$ and replacing them with such short paths from $\varphi ^{(n)}(b_i^{(n)})$ to $\varphi ^{(n)}(a_{i+1}^{(n)})$. Thus, after reparametrizing appropriately, we obtain well-defined paths $\theta ^{(n)}\in C_e([0,\tau _n],M)$ satisfying

$$\begin{aligned} \mathcal {S}(\theta ^{(n)})\le Q_{\textbf{v}}(e)-2k+\varepsilon < Q_{\textbf{v}}(e)-k \quad \text {and} \quad \tau _n \le \varepsilon + \sum _{i=1}^{N_n}(b_i^{(n)}-a_i^{(n)}). \end{aligned}$$

(168)

Equation (165) (now with new constants $C_2(\kappa ), T(\kappa )$) again implies that $(b_i^{(n)}-a_i^{(n)})$ is bounded for each $i\le N_n \le N$, so the second inequality in (168) implies that $(\tau _n)$ is bounded. Hence with k replacing 2k, $\theta ^{(n)}$ replacing $\varphi ^{(n)}$, and $\tau _n$ replacing $T_n$, we have reduced to the case of bounded $(T_n)$ which, as explained previously (following (162)), leads to a contradiction. This completes the proof.