* $\mathfrak{c}$ contains exactly one forward corner at each pure saddle singularity, and no corner at quadratic-like singularities,

* If $(c,c')$ is a forward double-corner, then $\{c,c'\} \cap \mathfrak{c} = \varnothing$.

The relation $\prec^\mathrm{cor}$ depends on the choice of an admissible collection of forward corners $\mathfrak{c} \subset \mathfrak{C}^+$.

This choice will be crucial later when we extend the weak prefoliation structure to completed envelopes, see Section <ref> below.

We first assume that $f_\star$ and $g_\star$ are not comparable by $\prec^\mathrm{geo} \cup \prec^\mathrm{inc}$ and have an initial corner of overlap $c$. After possibly renaming them, we further assume that they meet along $c$ in the way described in Section <ref>. Then, we set $f_\star \prec^\mathrm{cor}_p g_\star$ if one of the following holds:

* $f_\star$ and $g_\star$ are geometrically noncoherent at $c$, and $\widehat{\Sigma}(c) \subsetneq \widehat{\Sigma}'(c)$,

* $f_\star$ and $g_\star$ are geometrically coherent at $c$, and $c \in \mathfrak{c}$.

Otherwise, we set $g_\star \prec^\mathrm{cor}_p f_\star$.

We now assume that $f_\star$ and $g_\star$ are not comparable by $\prec^\mathrm{geo} \cup \prec^\mathrm{inc}$ and have an initial double corner of overlap $(c,c')$ where they meet in the way described in Section <ref>. Then, we set $f_\star \prec^\mathrm{cor}_p g_\star$ if $f_\star$ and $g_\star$ are geometrically noncoherent at $c$ and $\widehat{\Sigma}_0(c) \subsetneq \widehat{\Sigma}'_0(c)$. Otherwise, we set $g_\star \prec^\mathrm{cor}_p f_\star$.

The relation along unstable components $\prec^{\mathrm{un}}$.

This relation has no equivalent in <cit.> and is a feature of the presence of source and quadratic singularities. We need to introduce some terminology and make some auxiliary choices before defining it.

Let $L \hookrightarrow M$ be a leaf of $\eta$ in $M^\mathrm{un}$. A forward boundary component of $L$ is a connected component of the complement of the backward edges of $L$ in $\partial L$. We denote by $\mathcal{B}_L^+$ the set of forward boundary components of $L$. A forward boundary component might be closed or might contain one or two corners. Moreover, a boundary component might contain two consecutive flow lines of $X$ with opposite orientations, corresponding to pure unstable branches $\gamma^\pm_u$ at a quadratic-like singularity $q$. We say that the pure stable branch $\gamma_s$ at $q$ is a spike for $L$ if it is connected to a (pure) saddle singularity in negative times. In that case, $q$ is a saddle singularity as well. Since $X$ has no broken triple saddle connections (see Definition <ref>), $L$ has at most one spike. The forward boundary component $\gamma^-_u$ of $L$ will be called a distinguished boundary component and will be denoted by $b^\dagger_L \in \mathcal{B}_L^+$. A total order on $\mathcal{B}_L^+$ for which $b^\dagger_L$ (if it exists) is maximal will be called admissible. We define $\mathfrak{O}^+_L$ as the collection of all admissible total orders on $\mathcal{B}_L^+$, and we set

$$\mathfrak{O}^+ \coloneqq \prod_{L} \mathfrak{O}^+_L,$$

where the product runs over all the leaves of $\eta$ in $M^\mathrm{un}$. The admissibility condition will become relevant in the proof of Lemma <ref>, to ensure transitivity of the full preorder.

The relation $\prec^\mathrm{un}$ depends on the choice of a collection of admissible orders $\mathfrak{o} \in \mathfrak{O}^+$.

Let $U_c \subset \mathcal{D}_{f_\star}$ be an unstable component of $f$ which is not initial if $f$ is of unstable type, and let $h \coloneqq f_{\vert U_c}$ be the corresponding unstable tile, which covers a leaf $L \hookrightarrow M$ of $\eta$ in $M^\mathrm{un}$. An entrance edge of $f_\star$ to $U_c$ is a forward edge of $U_c$ which belongs to $\mathcal{D}_{f_\star}$. It exists since $U_c$ is not initial, but it is not necessarily unique. However, there are at most three of those. If there are two, then they correspond to consecutive stable and unstable branches of a singularity of $X$, and if there are three, then they correspond to a succession of a stable branch of a saddle singularity $q$, a connection between $q$ and another saddle singularity $q'$, and an unstable branch of $q'$. Such an entrance edge projects to $\overline{L}$ via the universal covering map, and  to a forward boundary component $b_{f_\star} \in \mathcal{B}_L^+$. Obviously, the latter is independent on the choice of entrance edge.

We set $f_\star \prec_p^{\mathrm{un}} g_\star$ if $f_\star$ and $g_\star$ are not comparable by $\prec^\mathrm{geo}_p \cup \prec^\mathrm{inc}_p \cup \prec^\mathrm{cor}_p$, and the following holds. There exists an initial unstable component of overlap $U_c \subset \mathcal{D}_{f_\star \cap g_\star}$ between $f_\star$ and $g_\star$ corresponding to a leaf $L \hookrightarrow M$ as above, together with forward boundary components $b_{f_\star}, b_{g_\star} \in \mathcal{B}^+_L$ induced by an entrance edge of $f_\star$ to $U_c$ and of $g_\star$ to $\varphi_{\mathcal{D}_{f_\star \cap g_\star}}(U_c)$, respectively, which satisfy $b_{f_\star} <^\mathfrak{o} b_{g_\star}$. Here, $<^\mathfrak{o}$ denotes the total order on $\mathcal{B}^+_L$ coming from $\mathfrak{o} \in \mathfrak{O}^+$. Notice that $b_{f_\star}$ and $b_{g_\star}$ are necessarily distinct, since $f_\star$ and $g_\star$ don't have an initial (double) corner of overlap.

The $X$-lines orientation relation $\prec^{\mathrm{or}}$.

This relation corresponds to the relation $>_3$ from <cit.>. We define $f_\star \prec^\mathrm{or}_p g_\star$ if $f_\star$ and $g_\star$ are not comparable by $\prec^\mathrm{geo}_p \cup \prec^\mathrm{inc}_p \cup \prec^\mathrm{cor}_p \cup \prec^\mathrm{un}_p$, and the following holds. There exist maximal lower bounds $\ell_{f_\star} \in \mathcal{I}_{f_\star}$ and $\ell_{g_\star} \in \mathcal{I}_{g_\star}$ of $\mathcal{I}_{f_\star \cap g_\star} \subset \mathcal{I}_{f_\star}$ and $\mathcal{I}_{g_\star \cap f_\star} \subset \mathcal{I}_{g_\star}$, respectively, and two forward curves $\gamma_{f_\star} : [0,1] \rightarrow U$ and $\gamma_{g_\star} : [0,1] \rightarrow V$ such that

* $\gamma_{f_\star}(0) \in \ell_{f_\star}$ and $\gamma_{g_\star}(0) \in \ell_{g_\star}$,

* $\gamma_{f_\star}$ and $\varphi_{g_\star \cap f_\star}\big({\gamma_{g_\star}}_{\vert (0,1]} \big)$ are disjoint,

* There exist a pure $X$-line $\ell \in \mathcal{I}_{f_\star \cap g_\star}$ and real numbers $s < t$ such that $\ell(s) \in \gamma_{f_\star}$ and $\varphi_{f_\star \cap g_\star} \big( \ell(t)\big) \in \gamma_{g_\star}$.

For short, we use the orientation on the pure $X$-lines near $\ell_{f_\star}$ and $\ell_{g_\star}$ to define the relation in this situation, see Figure <ref>.

Strictly speaking, this definition depends on an overall choice of orientation on $\R$: we could have replaced the condition $s < t$ with $t<s$ in the above third bullet.

Overlap of $f_\star$ and $g_\star$ for $\prec^\mathrm{or}$. The blue region corresponds to $\mathcal{D}_{f_\star}$, the green region corresponds to $\mathcal{D}_{g_\star}$, and their region of overlap is in teal.

Combining the five relations. We are now in the position to define a relation $\precsim_p$ on $\mathscr{U}_p$ as follows. First, we set

$f_\star \prec_p g_\star$ if and only if $f_\star \prec^{\heartsuit}_p g_\star$ for some $\heartsuit \in \{\mathrm{geo}, \mathrm{inc}, \mathrm{cor}, \mathrm{un}, \mathrm{or}\}$.

We say that $f_\star$ and $g_\star$ are forward-equivalent if there exists an equivalence $\varphi : f_\star \overset{\sim}{\rightarrow} g_\star$ satisfying $\Sigma' \circ \varphi = \Sigma$. In particular, $f_\star \approx_p g_\star$. Then, we define:

$f_\star \precsim_p g_\star$ if and only if $f_\star \prec_p g_\star$, or $f_\star$ and $g_\star$ are forward-equivalent.

So far, the relation $\precsim_p$ is only defined for $p \in M^\mathrm{good}$. We extend it to every $p \in M$ as follows: $f_\star \precsim_p g_\star$ if and only if there exists $q \in M^\mathrm{good}$ and a continuous path $\gamma : [0,1] \rightarrow M$ with $\gamma(0) = p$, $\gamma(1) = q$, which lifts to both $f_\star$ and $g_\star$, and $f_{\gamma \star} \precsim_q g_{\gamma \star}$. Here, $f_{\gamma \star}$ and $g_{\gamma \star}$ are obtained from $f_\star$ and $g_\star$ by sliding the marked points along $\gamma$ in the obvious way.

The collection of homogeneous relations $\{ \precsim_p\}_{p \in M}$ naturally induces a homogeneous relation $\precsim$ on $\mathscr{U}_\star$.

§.§.§ Proof of the weak prefoliation axioms

This section is dedicated to the proof of

The set $\mathscr{U}$ of forward envelopes equipped with the relation $\precsim$ is a weak prefoliation.

We split the proof into the following four lemmas.

The relation $\precsim$ is coherent along intersections.

The relation $\precsim$ is trivially coherent along unstable components and along pure $X$-lines by definition. It is also coherent along forward curves for the same reasons as in the proof of <cit.>. We are left to show that it is coherent at singularities in $M^\mathrm{bad}$. If $p \notin M^\mathrm{good}$ and $f_\star \in \mathscr{U}_p$, then the marked point of $f_\star$ lies in the interior of the domain of $f$. Therefore, coherence of $\precsim$ along intersections contained in $M^\mathrm{good}$ immediately implies coherence at $p$.

For every $p \in M$, $\precsim_p$ it total. Moreover, for all $f_\star, g_\star \in \mathscr{U}_\star$,  $f_\star \simeq g_\star$ if and only if $f_\star$ and $g_\star$ are forward-equivalent.

First assume $p \in M^\mathrm{good}$, and consider $f_\star, g_\star \in \mathscr{U}_p$ with our usual notations for their domains, shadows, and marked points.

If $f_\star$ and $g_\star$ are comparable by $\prec^\mathrm{geo}_p$, say $f_\star \prec^\mathrm{geo}_p g_\star$, then $f_\star$ and $g_\star$ are not equivalent and $g_\star \not\prec^\mathrm{geo}_p f_\star$. Indeed, arguing as in <cit.> by considering forward curves from/to $a_0$, it is easy to see that for every $x \in \mathcal{D}_{f_\star \cap g_\star}$, $\Sigma(x) \subset \Sigma'\big( \varphi_{f_\star \cap g_\star}(x)\big)$ and $(f,x)$ is not locally strictly above $\big(g,\varphi_{f_\star \cap g_\star}(x)\big)$.

If $f_\star$ and $g_\star$ are not comparable by $\prec^\mathrm{geo}_p$, then for every $x \in \mathcal{D}_{f_\star \cap g_\star}$, $\Sigma(x) = \Sigma'\big(\varphi_{f_\star \cap g_\star}(x)\big)$ and $(f,x)$ and $\big(g,\varphi_{f_\star \cap g_\star}(x)\big)$ locally coincide.[Notice that they automatically locally coincide in the forward direction as soon as their shadows coincide, because of basic properties of forward envelopes.] We now analyze how $f_\star$ and $g_\star$ `first meet' along $\mathcal{D}_{f_\star \cap g_\star}$ depending on which case from Lemma <ref> holds:

(I0) $\mathcal{I}_{f_\star \cap g_\star} = \varnothing$. In that case, $f_\star$ and $g_\star$ have an initial unstable component of overlap $U_c \subset \mathcal{D}_{f_\star \cap g_\star}$.

If both $U_c$ and $\varphi\big(U_c\big)$ correspond to the initial unstable components of $f_\star$ and $g_\star$, respectively, then $f_\star$ and $g_\star$ are forward-equivalent by Proposition <ref>.

If $U_c$ corresponds the initial unstable component of $f_\star$, but $\varphi\big(U_c\big)$ does not correspond to the initial unstable component of $g_\star$, then $f_\star$ and $g_\star$ are comparable by $\prec^\mathrm{inc}$ and $f_\star \prec^\mathrm{inc} g_\star$. The case where $U_c$ and $\varphi\big(U_c\big)$ are swapped is similar.

Otherwise, $f_\star$ and $g_\star$ are comparable by $\prec^\mathrm{un}$, since $U_c$ does not correspond to their initial unstable components and they don't have an initial (double) corner of overlap. Obviously, either $f_\star \prec^\mathrm{un}_p g_\star$ of $g_\star \prec^\mathrm{un}_p f_\star$ holds, but not both.

(I1) $\mathcal{I}_{f_\star \cap g_\star}$ has minimal elements. If $f_\star$ and $g_\star$ have an initial (double) corner of overlap, then they are comparable by $\prec^\mathrm{cor}_p$, and only one of $f_\star \prec^\mathrm{cor}_p g_\star$ or $g_\star \prec^\mathrm{cor}_p f_\star$ holds. We now assume that they do not have an initial (double) corner of overlap.

If they have an initial unstable component of overlap, we proceed as in the last step of the previous item. Otherwise, $f_\star$ and $g_\star$ are comparable by $\prec^\mathrm{inc}_p$. Indeed, if $\mathcal{I}_{f_\star \cap g_\star}$ has a unique minimal element, which is a minimum, then this minimum corresponds to the initial edge of either $f_\star$ or $g_\star$, depending on whether it is the minimum of $\mathcal{I}_{f_\star}$ or not. If $\mathcal{I}_{f_\star \cap g_\star}$ has strictly more than one minimal element, then it is easy to see that $g_\star \prec^\mathrm{inc}_p f_\star$.

(I2) $\mathcal{I}_{f_\star \cap g_\star}$ has no minimal elements. Observe that if $f_\star$ and $g_\star$ are comparable by $\prec^\mathrm{cor} \cup \prec^\mathrm{inc}_p \cup \prec^\mathrm{un}_p$, then $\mathcal{I}_{f_\star \cap g_\star}$ is either empty or has a minimal element. Therefore, we need to show that $f_\star$ and $g_\star$ are comparable by $\prec^\mathrm{or}_p$. For that, we consider a forward curve $\gamma_0 : [0,1] \rightarrow U$ from the initial component of $f$ to $a$, and a forward curve $\gamma_1 : [0,1] \rightarrow V$ from the initial component of $g$ to $b$. There exist $t_0, t_1 \in [0,1]$ such that $\gamma_0(t_0)$ and $\gamma_1(t_1)$ belong $X$-lines $\ell_0 \subset U$ and $\ell_1 \subset V$ which are maximal lower bounds for $\mathcal{I}_{f_\star \cap g_\star} \subset \mathcal{I}_{f_\star}$ and $\mathcal{I}_{g_\star \cap f_\star} \subset \mathcal{I}_{g_\star}$, respectively. For $t > t_0$, $\gamma_0(t) \in \mathcal{D}_{f_\star \cap g_\star}$ and for $t > t_1$, $\gamma_1(t) \in \mathcal{D}_{g_\star \cap f_\star}$. Moreover, there exists $\epsilon > 0$ such that $\gamma_{f_\star} \coloneqq {\gamma_0}_{[t_0, t_0+ \epsilon)}$ and  $\gamma_{g_\star} \coloneqq {\gamma_1}_{[t_1, t_1+ \epsilon)}$ are strictly forward, and $\gamma_{f_\star}$ and $\varphi_{g_\star \cap f_\star}\big({\gamma_{g_\star}}_{\vert(t_1, t_1+ \epsilon)}\big)$ are disjoint. Since $f_\star$ and $g_\star$ do not have an initial unstable domain of overlap, there exists either

* A pure $X$-line $\ell$ intersecting both $\gamma_{f_\star}$ and $\gamma_{g_\star}$,

* A special triple $(\ell_s, \ell^-_u, \ell^+_u)$ such that $\ell_s$ intersects one of $\gamma_{f_\star}$ and $\gamma_{g_\star}$ and $\ell^-_u$ or $\ell^+_u$ intersects the other,

* Two special triples $(\ell_s, \ell^-_u, \ell^+_u)$ and $(\ell'_s, \ell'^-_u, \ell'^+_u)$ with $\ell'_s \in \{\ell^-_u, \ell^+_u\}$ such that $\ell_s$ intersects one of $\gamma_{f_\star}$ and $\gamma_{g_\star}$ and $\ell'^-_u$ or $\ell'^+_u$ intersects the other.

In cases 1 and 2, we can modify $\gamma_{f_\star}$ or $\gamma_{g_\star}$ by performing suitable $T$-moves to reduce to case 0 while keeping the two curves disjoint. This implies that $f_\star$ and $g_\star$ are comparable by $\prec^\mathrm{or}_p$. It remains to show that $f_\star \prec^\mathrm{or}_p g_\star$ and $g_\star \prec^\mathrm{or}_p f_\star$ cannot hold simultaneously. If both inequalities hold, then after performing suitable $T$-moves and forward isotopies, we would obtain the following situation: there exists two forward curves $\gamma_{f_\star}$ and $\gamma_{g_\star}$ as above, together with two pure $X$-lines or pure `broken $X$-lines' (as in cases 1 and 2 above), one from $\gamma_{f_\star}$ to $\gamma_{g_\star}$ and the other one from $\gamma_{g_\star}$ to $\gamma_{f_\star}$. This would imply the existence of a pure $X$-line which is `trapped' (in positive or negative times) in a compact region of $U$, which is impossible. The details are left to the reader.

Finally, we assume that $p \notin M^\mathrm{good}$. In particular, $p \in \Delta_\mathrm{sa} \cup Q$, and the marked points of $f_\star$ and $g_\star$ belong the the interior of their domains of definition. We can slide these marked points along a stable or unstable branch at $p$ to obtain marked tiles $(f, a'), (g, b') \in \mathscr{U}_{p'}$ where $p' \in M^\mathrm{good}$. Therefore, $(f, a')$ and $(g, b')$ are comparable by $\precsim_{p'}$ by the above, so $f_\star$ and $g_\star$ are comparable by $\precsim_p$ by definition. Finally, if $f_\star \precsim_p g_\star$ and $g_\star \precsim_p f_\star$, then the same holds at some nearby $p' \in M^\mathrm{good}$ by sliding the marked point $a$ and $b$ to nearby $a'$ and $b'$, hence $(f, a') \approx_{p'} (g,b')$. Sliding the marked points back yields $f_\star \approx_p g_\star$.

For all $f_\star, g_\star \in \mathscr{U}_\star$, if $f_\star \precsim g_\star$, then $g_\star$ is locally above $f_\star$.

The proof is essentially the same as the one of <cit.>. Let $p \in M$ and $f_\star, g_\star \in \mathscr{U}_p$. Assume first that $p \in M^\mathrm{good}$. If $f_\star$ and $g_\star$ are comparable by $\prec^\mathrm{geo}_p$, say $f_\star \prec^\mathrm{geo}_p g_\star$, then we connect the point $a_0$ given by the definition to the marked point $a$ of $f$ by a forward curve. Properties of upper enveloping tiles (see Definition <ref>) and one-dimensional envelopes (see <cit.>) immediately imply that $g_\star$ is locally above $f_\star$. If $f_\star$ and $g_\star$ are not comparable by $\prec^\mathrm{geo}_p$, then $f_\star$ is not locally strictly above $g_\star$, and since one of these two forward tiles is locally above the other (for the same reasons as in <cit.>), then $g_\star$ is locally above $f_\star$. If $p \notin M^\mathrm{good}$ and $f_\star$ is locally strictly above $g_\star$, then there exists $p' \in M^\mathrm{good}$ on one of the stable or unstable branches at $p$ such that if $(f, a'), (g, b') \in \mathscr{U}_{p'}$ denote the marked tiles obtained by sliding the marked points of $f_\star$ and $g_\star$, respectively, then $(f,a')$ is locally strictly above $(g,b')$. Therefore, $(g, b') \prec^\mathrm{geo}_{p'} (f, a')$, hence $g_\star \prec f_\star$.

We finish the proof of Proposition <ref> with the trickiest property:

The relation $\precsim$ is transitive.

Let $p \in M$. Notice that if $f_\star, g_\star, h_\star \in \mathscr{U}_p$ and $f_\star$ and $g_\star$ are forward-equivalent, then $f_\star \precsim_p h_\star$ if and only if $g_\star \precsim_p h_\star$, by definition. Therefore, in view of Lemma <ref>, transitivity of $\precsim_p$ is equivalent to the statement: every triple of elements in $\mathscr{U}_p$ admits an extremum (maximum or minimum). Moreover, by Lemma <ref>, it is enough to consider the case $p \in M^\mathrm{good}$. By Lemma <ref>, we can assume that $\mathcal{I}_{f_\star \cap g_\star } \subset \mathcal{I}_{f_\star \cap h_\star}$ and $\mathcal{D}_{f_\star \cap g_\star } \subset \mathcal{D}_{f_\star \cap h_\star}$. We enumerate all the possible cases for the order between $f_\star$ and $g_\star$.

* $f_\star$ and $g_\star$ are comparable by $\prec^\mathrm{geo}$. Assume that there exist $a_0 \in \mathcal{D}_{f_\star \cap g_\star}$ and $b_0 = \varphi_{f_\star \cap g_\star}(a_0)$ such that $\Sigma_f(a_0) \neq \Sigma_g(b_0)$, and consider $c_0 \coloneqq \varphi_{f_\star \cap h_\star}(a_0) \in \mathcal{D}_{h_\star}$. This is well-defined since $a_0 \in \mathcal{D}_{f_\star \cap g_\star} \subset \mathcal{D}_{f_\star \cap h_\star}$. Then, among the three sections $\Sigma_f(a_0)$, $\Sigma_g(b_0)$ and $\Sigma_h(c_0)$ of $\mathscr{U}_p$, there must exist a strict maximum or minimum for the inclusion, implying that the corresponding marked tiles is an extremum in $\{f_\star, g_\star, h_\star\}$. Otherwise, there exists $a_0$ and $b_0$ as above such that $(g, b_0)$ is locally strictly above $(f, a_0)$. In that case, $p \in M^\mathrm{pure}$. Intersecting $f_\star$, $g_\star$ and $h_\star$ with a small embedded disk in $M$ passing through $p$ and transverse to $X$, we obtain three $1$-dimensional envelopes for the restriction of $\mathscr{A}$ to that disk, two of which are strictly comparable by the geometric order. We further consider two cases.

* If $c_0 = \varphi_{f_\star \cap h_\star}(a_0)$ does not belong to the initial edge of $h$ (if it exists), then one of the three $1$-dimensional envelopes is strictly below or above the two others, and the corresponding marked forward envelope is an extremum in $\{f_\star, g_\star, h_\star\}$.

* Otherwise, $f_\star$ and $g_\star$ `begin to overlap' along the initial edge of $h_\star$. If $\Sigma_h(c_0) \neq \Sigma_f(a_0)$, then since $\Sigma_f(a_0)=\Sigma_g(b_0)$ by assumption, $h_\star$ is comparable by $\prec^\mathrm{geo}$ with both $f_\star$ and $g_\star$. Moreover, it is either below or above both of them for $\prec^\mathrm{geo}$, so $h_\star$ is an extremum of $\{f_\star, g_\star, h_\star\}$. If $\Sigma_h(c_0) = \Sigma_f(a_0) = \Sigma_g(b_0)$, then $h_\star$ is comparable by $\prec^\mathrm{inc}$ with both $f_\star$ and $g_\star$, and $h_\star$ is similarly an extremum of $\{f_\star, g_\star, h_\star\}$.

* $f_\star$ and $g_\star$ are comparable by $\prec^\mathrm{inc}$. We first assume that $g_\star$ is a strict subenvelope of $f_\star$; in particular, $\mathcal{D}_{g_\star} = \mathcal{D}_{g_\star \cap f_\star}$. Notice that $\mathcal{D}_{h_\star \cap g_\star} \subset \mathcal{D}_{h_\star \cap f_\star}$ by Lemma <ref>. By the previous case, we can assume that $g_\star$ and $h_\star$ are not comparable by $\prec^\mathrm{geo}$. We also have $\mathcal{D}_{g_\star} = \mathcal{D}_{g_\star \cap h_\star}$, so either $g_\star$ and $h_\star$ are order-equivalent, or $g_\star$ is a strict subenvelope of $h_\star$. Hence, either $f_\star$ or $g_\star$ is an extremum in $\{f_\star, g_\star, h_\star\}$. We now assume that $f_\star$ is a strict subenvelope of $g_\star$. Hence, $\mathcal{D}_{f_\star} = \mathcal{D}_{f_\star \cap g_\star} \subset \mathcal{D}_{f_\star \cap h_\star} = \mathcal{D}_{f_\star}$. In particular, $\mathcal{D}_{g_\star \cap f_\star} = \mathcal{D}_{g_\star \cap h_\star}$ and $\mathcal{D}_{h_\star \cap f_\star} = \mathcal{D}_{h_\star \cap g_\star}$, and by case 1 above, we can assume that $f_\star$ and $g_\star$ are not comparable by $\prec^\mathrm{geo}$, and $f_\star$ and $h_\star$ are not comparable by $\prec^\mathrm{geo}$. In this situation, $f_\star$ is a (not necessarily strict) subenvelope of $g_\star$ and $h_\star$, so $\{f_\star, g_\star, h_\star\}$ has an extremum by the previous argument.

* $f_\star$ and $g_\star$ are comparable by $\prec^\mathrm{cor}$. We consider three cases.

* $f_\star$ and $g_\star$ have a simple initial corner of overlap $c$, which is not connected to another forward corner. Since we do not consider forward envelopes starting at the pure stable branch of a quadratic-like singularity, $c$ is a pure corner. We consider two subcases.

* $\mathcal{D}_{f_\star \cap g_\star} \subsetneq \mathcal{D}_{f_\star \cap h_\star}$. Then $c$ is also the initial corner of overlap of $g_\star$ and $h_\star$. In that case, $\mathcal{D}_{g_\star \cap h_\star} = \mathcal{D}_{g_\star \cap f_\star}$ so by case 1 above, we can assume that $g_\star$ and $h_\star$ are not comparable by $\prec^\mathrm{geo}$, hence they are comparable by $\prec^\mathrm{cor}$. Let $a_0 \in U$ be a point on the $X$-line corresponding to the branch at $c$ bounding $\mathcal{D}_{f_\star \cap g_\star}$ in $\mathcal{D}_{f_\star}$, and $c_0 \coloneqq \varphi_{f_\star \cap h_\star}(a_0)$. Note that $f_\star$ and $h_\star$ play a symmetric role here. If $\Sigma_f(a_0) \neq \Sigma_h(c_0)$, then $\widehat{\Sigma}_f(c) \neq \widehat{\Sigma}_h(c)$ (see Section <ref> for the notation), and either $f_\star$ or $h_\star$ is an extremum of $\{f_\star, g_\star, h_\star\}$. Otherwise, $\widehat{\Sigma}_f(c) = \widehat{\Sigma}_h(c)$, and the order of $f_\star$ with $g_\star$ for $\prec^\mathrm{cor}$ is the same as the one of $h_\star$ with $g_\star$, implying that $g_\star$ is an extremum of $\{f_\star, g_\star, h_\star\}$.

* $\mathcal{D}_{f_\star \cap g_\star} = \mathcal{D}_{f_\star \cap h_\star}$. By case 1 above, we can assume that $f_\star$ and $h_\star$ are not comparable by $\prec^\mathrm{geo}$, and moreover they are not comparable by $\prec^\mathrm{inc}$. We consider three further subcases.

* $f_\star$ and $h_\star$ are comparable by $\prec^\mathrm{cor}$. Then $c$ is also the initial corner of overlap between $f_\star$ and $g_\star$, and $\mathcal{D}_{g_\star \cap f_\star} \subsetneq \mathcal{D}_{g_\star \cap h_\star}$ so we can appeal to case 3(a)i above.

* $f_\star$ and $h_\star$ are comparable by $\prec^\mathrm{un}$. Then one easily sees that $h_\star$ and $g_\star$ are also comparable by $\prec^\mathrm{un}$ and share the same initial unstable domain of overlap as $h_\star$ and $f_\star$. Since the entrance edges of $f_\star$ and $g_\star$ lie on the same forward boundary component of the leaf corresponding to that unstable component, but the entrance edge(s) of $h_\star$ necessarily lies on a different forward boundary component, the order of $f_\star$ with $h_\star$ for $\prec^\mathrm{un}$ is the same as the one of $g_\star$ with $h_\star$, implying that $h_\star$ is an extremum of $\{f_\star, g_\star, h_\star\}$.

* $f_\star$ and $h_\star$ are comparable by $\prec^\mathrm{or}$. Then $g_\star$ and $h_\star$ are comparable by $\prec^\mathrm{or}$ as well, and it follows from the definition of $\prec^\mathrm{or}$ and the behavior of $X$-lines near a forward corner that the order of $f_\star$ with $h_\star$ for $\prec^\mathrm{or}$ is the same as the one of $g_\star$ with $h_\star$, implying that $h_\star$ is an extremum of $\{f_\star, g_\star, h_\star\}$.

* $f_\star$ and $g_\star$ have a simple initial corner of overlap $c$, which is connected to another forward corner $c'$. We again consider two subcases.

* $c'$ is a corner at a pure saddle singularity. We further assume that $f_\star$ and $h_\star$ are comparable by $\prec^\mathrm{cor}$ and $c'$ is their initial corner of overlap, as all the other cases can be treated as in case 3(a).

Up to permuting $f_\star$, $g_\star$ and $h_\star$, we can assume that $(c,c')$ is a double corner and $f_\star$ and $h_\star$ meet along $(c,c')$. We will use the notations of Section <ref>. We consider the sections $\widehat{\Sigma}_f(c), \widehat{\Sigma}_g(c) \subset \mathscr{A}_{q}$, $\widehat{\Sigma}_g(c'), \widehat{\Sigma}_h(c') \subset \mathscr{A}_{q'}$, and $\Sigma_g(b_0), \widehat{\Sigma}_{f,0}(c), \widehat{\Sigma}_{h,0}(c') \subset \mathscr{A}_{q_0}$. Here, $b_0$ lies on the boundary of $\mathcal{D}_{g_\star \cap f_\star} \subset \mathcal{D}_{g_\star}$ and $g(b_0) = q_0$. Tracing through the definitions of these sets, one easily shows:

\begin{align}
\widehat{\Sigma}_f(c) \subset \widehat{\Sigma}_g(c) &\implies \widehat{\Sigma}_{f,0}(c) \subset \Sigma_g(b_0), \label{eq:sigma1} \\
\widehat{\Sigma}_g(c) \subset \widehat{\Sigma}_f(c) &\iff \Sigma_g(b_0) \subset \widehat{\Sigma}_{f,0}(c), \label{eq:sigma2}
\end{align}

and similarly,

\begin{align*}
\widehat{\Sigma}_h(c') \subset \widehat{\Sigma}_g(c') &\implies \widehat{\Sigma}_{h,0}(c') \subset \Sigma_g(b_0), \\
\widehat{\Sigma}_g(c') \subset \widehat{\Sigma}_h(c') &\iff \Sigma_g(b_0) \subset \widehat{\Sigma}_{h,0}(c).
\end{align*}

This immediately implies the following:

* If $\widehat{\Sigma}_f(c) \subset \widehat{\Sigma}_g(c)$ and  $\widehat{\Sigma}_g(c') \subsetneq \widehat{\Sigma}_h(c')$, then $\widehat{\Sigma}_{f,0}(c) \subsetneq \widehat{\Sigma}_{h,0}(c')$.

* If $\widehat{\Sigma}_g(c) \subset \widehat{\Sigma}_f(c)$ and $\widehat{\Sigma}_h(c') \subset \widehat{\Sigma}_g(c')$, then $\widehat{\Sigma}_{f,0}(c) \subset \widehat{\Sigma}_{h,0}(c')$.

By the definition of $\prec^\mathrm{cor}$, we have:

\begin{align}
\widehat{\Sigma}_h(c) \subset \widehat{\Sigma}_g(c) \implies h_\star \prec^\mathrm{cor} g_\star, \notag \\
\widehat{\Sigma}_g(c) \subset \widehat{\Sigma}_f(c) \implies g_\star \prec^\mathrm{cor} f_\star, \label{eq:impl1}\\
\widehat{\Sigma}_{h,0}(c) \subset \widehat{\Sigma}_{f,0}(c) \implies h_\star \prec^\mathrm{cor} f_\star. \label{eq:impl2}
\end{align}

It is easy to conclude that in all the possible cases for the inclusions between $\widehat{\Sigma}_f(c)$ and $\widehat{\Sigma}_g(c)$, and between $\widehat{\Sigma}_g(c)$ and $\widehat{\Sigma}_h(c)$, $\{f_\star, g_\star, h_\star\}$ has an extremum.

* $c'$ is a corner at a quadratic-like singularity. If $(c',c)$ is a double forward corner, all the possible cases can be treated as 3(a), so we assume that $(c,c')$ is a double forward corner. For the same reasons, we assume that $f_\star$ and $h_\star$ are comparable by $\prec^\mathrm{cor}$ and $(c,c')$ is their initial corner of overlap, so that $\mathcal{D}_{g_\star \cap f_\star} \subseteq \mathcal{D}_{g_\star \cap h_\star}$. Here, $f_\star$ and $g_\star$ intersect as in case (C2)(i) of Section <ref>, see Figure <ref>, while $f_\star$ and $h_\star$ intersect as in case (C1) (ii), see Figure <ref> (in particular the bottom one).

We use the same notations as before. We also consider points $y_0 \in \mathcal{D}_{g_\star}$ and $z_0 \in \mathcal{D}_{h_\star}$ such that $\varphi_{h_\star \cap g_\star}(z_0) = y_0$, and $z_0$ lies on the $X$-line corresponding to the minimum of $\mathcal{I}_{h_\star \cap g_\star}$ in $\mathcal{I}_{h_\star}$, so that $h(z_0)$ belongs to the right unstable branch of the singularity at $c'$.

The implications (<ref>) and (<ref>) still hold, as well as

\begin{align*}
\Sigma_h(z_0) \subset \Sigma_g(y_0) \implies \widehat{\Sigma}_{h,0}(c') \subset \Sigma_g(b_0).
\end{align*}

The implications (<ref>) and (<ref>) hold for similar reasons, as well as

\begin{align*}
\Sigma_h(z_0) \subset \Sigma_g(y_0) \implies h_\star \prec g_\star.
\end{align*}

Indeed, if $\Sigma_h(z_0) \subsetneq \Sigma_g(y_0)$, then $g_\star$ and $h_\star$ are comparable by $\prec^\mathrm{geo}$ and $h_\star \prec^\mathrm{geo} g_\star$. If $\Sigma_h(z_0) = \Sigma_g(y_0)$, then $g_\star$ and $h_\star$ are not comparable by $\prec^\mathrm{geo}$, and  two possibilities arise. Either $z_0$ belong to the initial edge of $h_\star$, and $h_\star \prec^\mathrm{inc} g_\star$, or $g_\star$ and $h_\star$ are comparable by $\prec^\mathrm{un}$ and their initial unstable domain of overlap corresponds to the unstable tile $L$ at $q'$ (recall that $f_\star$ and  $h_\star$ do not meet along $c$). Moreover, the connection in $c \cap c'$ is a spike for $L$, and the entrance edge for $g_\star$ is the distinguished boundary component of $L$. By our conventions for admissible orders on unstable tiles, we have $h_\star \prec^\mathrm{un} g_\star$.

Examining all the possible inclusions between $\widehat{\Sigma}_f(c)$ and $\widehat{\Sigma}_g(c)$, and between $\Sigma_g(y_0)$ and $\Sigma_h(z_0)$, it is easy to show that $\{f_\star, g_\star, h_\star\}$ always has an extremum.

* $f_\star$ and $g_\star$ have an initial double corner of overlap $(c,c')$. The cases where $h_\star$ is comparable by $\prec^\mathrm{geo}$, $\prec^\mathrm{inc}$, or $\prec^\mathrm{cor}$ with either $f_\star$ or $g_\star$ can be treated as in 3(a) or 3(b), up to permutation. In the remaining cases, $h_\star$ is comparable with both $f_\star$ and $g_\star$ by one of the two relations $\prec^\mathrm{un}$ or $\prec^\mathrm{or}$, and the strategy of case 3(a)ii applies.

* $f_\star$ and $g_\star$ are comparable by $\prec^\mathrm{un}$. The only case that has not been treat yet, up to permutation, is when $f_\star$, $g_\star$, and $h_\star$ are all comparable by $\prec^\mathrm{un}$ and share the same initial unstable domain of overlap. In this situation, transitivity immediately follows from the definition of $\prec^\mathrm{un}$.

* $f_\star$ and $g_\star$ are comparable by $\prec^\mathrm{or}$. The only case that has not been treated yet, up to permutation, is when $f_\star$, $g_\star$, and $h_\star$ are all comparable by $\prec^\mathrm{or}$, and $\mathcal{D}_{f_\star \cap g_\star} = \mathcal{D}_{f_\star \cap h_\star}$ and $\mathcal{D}_{g_\star \cap f_\star} = \mathcal{D}_{g_\star \cap h_\star}$. Without loss of generality, let us assume that $f_\star \prec^\mathrm{or} g_\star$ and $g_\star \prec^\mathrm{or} h_\star$. We consider forward curves $\gamma_{f_\star}$, $\gamma_{g_\star}$, and $\gamma_{h_\star}$ as in the definition of $\prec^\mathrm{or}$; by item (I2) in the proof of Lemma <ref>, we can assume that there exist a pure $X$-line $\ell_0 \in \mathcal{I}_{f_\star \cap g_\star}$ from $\gamma_{f_\star}$ to $\gamma_{g_\star}$ and a pure $X$-line $\ell_1 \in \mathcal{I}_{g_\star \cap h_\star}$ from $\gamma_{g_\star}$ to $\gamma_{h_\star}$. Let $\ell'_1 \in \mathcal{I}_{f_\star \cap g_\star}$ denote $\alpha_{g_\star \cap f_\star}(\ell_1)$. If $\ell'_1 = \ell_0$, we readily obtain a pure $X$-line from $\gamma_{f_\star}$ to $\gamma_{h_\star}$ and $f_\star \prec^\mathrm{or} h_\star$. Otherwise, $\varphi_{g_\star \cap f_\star}({\gamma_{g_\star}}_{\vert (0,1]})$ induces a strictly forward curve intersecting $\ell_0$ and $\ell'_1$, so either $\ell_0 \lhd \ell'_1$ or $\ell'_1 \lhd \ell_0$. Let us consider the case $\ell_0 \lhd \ell'_1$ first. Using Lemma <ref>, it is easy to show that either $\ell'_0$ intersects $\gamma_{f_\star}$, or is part of a broken $X$-line intersecting $\gamma_{f_\star}$ as in cases 1 and 2 in of item (I2) in the proof of Lemma <ref>. It follows that $f_\star \prec^\mathrm{or} h_\star$. The case $\ell_0 \lhd \ell'_1$ can be treated similarly by considering $\ell'_0 \coloneqq \alpha_{f_\star \cap g_\star}(\ell_0)$ and $\ell_1$ instead, and showing that $\ell'_0$ is part of a (broken) pure $X$-line intersecting $\gamma_{h_\star}$.

§.§.§ Completed forward envelopes

In this section, we extend the previously constructed weak prefoliation structure on $\mathscr{U}$ to the collection of completed forward envelopes $\overline{\mathscr{U}}$. Notice that since $\mathscr{U}$ covers $M^\mathrm{good}$ (minus the stable branches at quadratic-like singularities), $\overline{\mathscr{U}}$ covers $M$.

Let $f, g \in \mathscr{U}$ be forward envelopes (with our usual notations), with completions $\bar{f}, \bar{g} \in \overline{\mathscr{U}}$, and $a \in \overline{U}$ and $b \in \overline{V}$ such that $\bar{f}(a) = \bar{g} (b) \eqqcolon p$. We write $\bar{f}_\star = (\bar{f}, a)$ and $\bar{g}_\star = (\bar{g}, b)$.

If $p \in M^\mathrm{good}$, then the extension of $\precsim_p$ to $\overline{\mathscr{U}}_p$ is entirely determined by axioms 1 and 2 of Definition <ref>. Indeed, $a$ either belongs to $U$ or to a forward edge of $\overline{U}$, and similarly for $b$. Near $p$, $\bar{f}_\star$ and $\bar{g}_\star$ locally look like a disk or a half-disk bounded by the flow line of $X$ passing through $p$. Therefore, we can distinguish two cases:

* $\bar{f}_\star$ and $\bar{g}_\star$ locally coincide. Then we can slide the marked points along a (possibly trivial) small curve $\gamma$ from $p$ to $p' \in M$ such that the new marked points satisfy $a' \in U$ and $b' \in V$. Then, we set $\bar{f}_\star \dot{\precsim}_p \bar{g}_\star$ if $(f, a') \precsim_{p'} (g,b')$, and $\bar{g}_\star \dot{\precsim}_p \bar{f}_\star$ otherwise. Notice that this does not depend on the choice of $\gamma$, since $f$ and $g$ intersect along a path-connected set near $p$.

* One of $\bar{f}_\star$ and $\bar{g}_\star$ is locally strictly above the other, since $f$ and $g$ cannot topologically cross. We set $\bar{f}_\star \dot{\precsim}_p \bar{g}_\star$ if $\bar{g}_\star$ is locally strictly above $\bar{f}_\star$, and $\bar{g}_\star \dot{\precsim}_p \bar{f}_\star$ otherwise.

The previous arguments can easily be adapted to extend $\precsim_p$ to $\overline{\mathscr{U}}_p$ for $p \in Q^*$, because of the shape of forward tiles near a quadratic-like singularities, see Figure <ref>. Crucially, the interiors of two such tiles intersect along a path-connected set near $p$ when they locally coincide. It is then straightforward to check that $\dot{\precsim}_p$ is an extension of $\precsim_p$ to $\overline{\mathscr{U}}_p$ and that the axioms of Definition <ref> are satisfied on $M\setminus \Delta^\mathrm{pure}_\mathrm{sa}$.

The extension of $\precsim_p$ to $\overline{\mathscr{U}}_p$ for $p \in \Delta^\mathrm{pure}_\mathrm{sa}$ is more subtle. We consider two cases:

* $\bar{f}_\star$ and $\bar{g}_\star$ intersect along a stable or unstable branch at $p$. We then slide the marked points along this branch to obtain $(\bar{f}, a'), (\bar{g}, b') \in \overline{\mathscr{U}}_{p'}$ and we set $\bar{f}_\star \dot{\precsim}_p \bar{g}_\star$ if $(\bar{f}, a') \dot{\precsim}_{p'} (\bar{g}, b')$, and $\bar{g}_\star \dot{\precsim}_p \bar{f}_\star$ otherwise. Notice that $\dot{\precsim}_{p'}$ is already defined.

* $\bar{f}_\star$ and $\bar{g}_\star$ do not intersect along the stable and unstable branches at $p$. In that case, they form two opposite (forward) quadrants at $p$, and they only intersect at $p$ near $p$. Let $\Sigma^\llcorner_f \subset \mathscr{A}_p$ denotes the set of marked tiles $h_\star \in \mathscr{A}_p$ such that either $h_\star$ is locally geometrically below $f$, or they locally coincide (relative to the domain of $f$) and after sliding the marked points inside the interiors of the domain of $f$ and $h$, $(h, z') \in \Sigma_f(x')$. Here, $x'$ and $z'$ denote the slid marked points of $f$ and $h$, respectively, satisfying $f(x') = h(z')$. We further denote by $\widehat{\Sigma}^\llcorner_f \subset \mathscr{A}_p$ the section generated by $\Sigma^\llcorner_f$. We similarly define $\widehat{\Sigma}^\llcorner_g \subset \mathscr{A}_p$. If $\widehat{\Sigma}^\llcorner_f \subsetneq \widehat{\Sigma}^\llcorner_g$ (resp. $\widehat{\Sigma}^\llcorner_g \subsetneq \widehat{\Sigma}^\llcorner_f$), we set $\bar{f}_\star \dot{\precsim}_p \bar{g}_\star$ (resp. $\bar{g}_\star \dot{\precsim}_p \bar{f}_\star$). This corresponds to the `geometrically coherent' case in the definition of $\prec^\mathrm{cor}$.

Furthermore, if there exists $\bar{h}_\star \in \overline{\mathscr{U}}_p$ such that

* $\bar{f}_\star$ and $\bar{h}_\star$ intersect as in case 1 and $\bar{f}_\star \dot{\precsim}_p \bar{h}_\star$ (resp. $\bar{h}_\star \dot{\precsim}_p \bar{f}_\star$),

* $\bar{g}_\star$ and $\bar{h}_\star$ intersect as in case 1 and $\bar{h}_\star \dot{\precsim}_p \bar{g}_\star$ (resp. $\bar{g}_\star \dot{\precsim}_p \bar{h}_\star$),

then necessarily $\widehat{\Sigma}^\llcorner_f \subset \widehat{\Sigma}^\llcorner_g$ (resp. $\widehat{\Sigma}^\llcorner_g \subset \widehat{\Sigma}^\llcorner_f$). If moreover $\widehat{\Sigma}^\llcorner_f = \widehat{\Sigma}^\llcorner_g$, then we set $\bar{f}_\star \dot{\precsim}_p \bar{g}_\star$ (resp. $\bar{g}_\star \dot{\precsim}_p \bar{f}_\star$).

If $\widehat{\Sigma}^\llcorner_f = \widehat{\Sigma}^\llcorner_g$ and no such $\bar{h}_\star \in \overline{\mathscr{U}}_p$ exists, we set $\bar{f}_\star \dot{\precsim}_p \bar{g}_\star$ if $\bar{f}_\star$ is bounded by the forward corner in $\mathfrak{c}$, and $\bar{g}_\star \dot{\precsim}_p \bar{f}_\star$ otherwise.[Alternatively, we could use the forward corner at $p$ not in $\mathfrak{c}$. This choice has to be the same for all completed forward envelopes at $p$, but it does not need to be the same for all pure saddle singularities.]

It is not a priori clear that $\dot{\precsim}$ is coherent at $p \in \Delta^\mathrm{pure}_\mathrm{sa}$ in case 1, since $f$ and $g$ can locally intersect along a disconnected set. This corresponds to the situation that we now describe. Let us pick local coordinates $(x,y,z)$ about $p$ adapted to $\eta$ such that the stable branches at $p$ are contained in $\{x=0\}$ and the unstable branches at $p$ are contained in $\{y=0\}$. In these coordinates, we identify $f$ and $g$ with the graphs of two functions, still denoted by $f$ and $g$, over domains $D_f, D_g \subset \R^2_{x,y}$. After possibly shrinking the coordinate neighborhood of $p$, we further assume

\begin{align*}
D_f &= \{- \epsilon < x < 0\} \cup \{- \epsilon < y < 0\}, \\
D_g &= \{0 < x < \epsilon\} \cup \{0 < y < \epsilon\},
\end{align*}

for some $\epsilon > 0$. Moreover, $f$ and $g$ continuously extend to $0$ along $\{x = 0\} \cup \{y=0\}$. Writing

\begin{align*}
Q_- &\coloneqq \{ 0 < x < \epsilon, - \epsilon < y< 0\}, \\
Q_+ &\coloneqq \{ - \epsilon < x < 0, 0 < y < \epsilon\},
\end{align*}

we have

$$D_f \cap D_g = Q_-  \sqcup Q_+,$$

and this set is disconnected. We choose arbitrary points $x_- \in Q_-$ and $x_+ \in Q_+$, and we set $f \prec_\pm g$ if one of the following holds:

(a) $f_{\vert Q_\pm} \leq  g_{\vert Q_\pm}$ and $f_{\vert Q_\pm} \neq g_{\vert Q_\pm}$,

(b) $f_{\vert Q_\pm} = g_{\vert Q_\pm}$, and $(f,x_\pm) \prec (g, x_\pm)$.

Otherwise, we set $g \prec_\pm f$. In the latter case, $g_{\vert Q_\pm} \leq  f_{\vert Q_\pm}$ and $(g,x_\pm) \prec (f, x_\pm)$ if $f_{\vert Q_\pm} = g_{\vert Q_\pm}$, since $f$ and $g$ are forward envelopes and $(f, x_\pm) \not\approx (g, x_\pm)$. Moreover, when $f_{\vert Q_\pm} = g_{\vert Q_\pm}$, $(f, x_\pm)$ and $(g, x_\pm)$ are comparable by $\prec^\mathrm{geo}$ or $\prec^\mathrm{cor}$, and in he latter case their initial corner over overlap $c_\pm$ bounds $Q_\pm$. See Figure <ref>. The coherence at $p$ of the relation $\dot{\precsim}$ between $\bar{f}$ and $\bar{g}$ can be stated as follows:

Overlap of $f$ and $g$ near $p$. The blue region corresponds to $D_f$, the green region corresponds to $D_g$, and their overlap is in teal. The quadrant with gray dots represents $Q_-$ while the quadrant with white dots represents $Q_+$.

$f \prec_- g$ if and only if $f \prec_+ g$.

Without loss of generality, we assume $c_+ \in \mathfrak{c}$. By sliding marked points to the corners and completing, we obtain sections $\widehat{\Sigma}_f(c_\pm), \widehat{\Sigma}_g(c_\pm) \subset \mathscr{A}_p$ as in Section <ref>. Because of the shape of backward tiles passing though $p$ and of the flow of $X$ near $p$, one easily shows

\begin{align} \label{eq:sigmafg}
\widehat{\Sigma}_f(c_-) = \widehat{\Sigma}_f(c_+), \qquad
\widehat{\Sigma}_g(c_-) = \widehat{\Sigma}_g(c_+).
\end{align}

Moreover, if $\widehat{\Sigma}_f(c_-) \subset \widehat{\Sigma}_g(c_-)$ then $f_{\vert Q_-} \leq g_{\vert Q_+}$ (and similarly for $f$ and $g$ swapped and for $c_+$). Indeed, we consider two cases.

* There exists $y_- \in Q_-$ such that $\Sigma_f(y_-)\neq \varnothing$. Then every backward tile in $\Sigma_f(y_-)$ intersects $\{x=y=0\}$ and is geometrically below $f$. If it passes through $0$, then it induces an element in $\widehat{\Sigma}_f(c) \subset \widehat{\Sigma}_g(c)$, and this tile is also geometrically below $g$ over $Q_-$, implying $f(y'_-) \leq g(y'_-)$ for every $y'_- \in Q_-$ in the forward region of $Q_-$ containing $y_-$ by the definition of one-dimensional envelopes (see <cit.>). Moreover, for every $y'_- \in Q_-$ in the backward region of $Q_-$ containing $y_-$, $\Sigma_f(y'_-) \neq \varnothing$ and the same argument shows that $f(y_-) \leq g(y_-)$. Therefore, $f_{\vert Q_-} \leq g_{\vert Q_-}$.

* Otherwise, $\Sigma_f(y_-)= \varnothing$ for every $y_- \in Q_-$. Then no tile in $\widehat{\Sigma}_f(c_-)$ intersects $Q_- \times (-\epsilon, \epsilon)$, and $f$ is necessarily geometrically below $g$ over $Q_-$. Indeed, $f_{\vert Q_-}$ coincides with the restriction to $Q_-$ of (the underlying tile of) the forward envelope starting at $\big((\epsilon/2, 0), \varnothing\big)$ which is geometrically the lowest such tile.

We now show:

\begin{align}
\widehat{\Sigma}_g(c_\pm) \subset \widehat{\Sigma}_f(c_\pm) &\implies g\prec_\pm f, \label{eq:precpm1} \\
\widehat{\Sigma}_f(c_\pm) \subsetneq \widehat{\Sigma}_g(c_\pm) &\implies f\prec_\pm g \label{eq:precpm2},
\end{align}

which will immediately imply the desired result. To show (<ref>), we assume $\widehat{\Sigma}_g(c_\pm) \subset \widehat{\Sigma}_f(c_\pm)$ and we consider the order for $f$ and $g$ with respect to $\prec_-$.

* $f_{\vert Q_-} \neq g_{\vert Q_-}$. Then since $g_{\vert Q_-} \leq f_{\vert Q_-}$ by the above argument, $g \prec_- f$.

* $f_{\vert Q_-} = g_{\vert Q_-}$. Then $f_\star = (f, x_-)$ and $g_\star = (g, x_-)$ are comparable by either $\prec^\mathrm{geo}$ or $\prec^\mathrm{cor}$. First, for every $y_- \in Q_-$, $\Sigma_g(y_-) \subset \Sigma_f(y_-)$. Otherwise, there would exist a marked backward tile $a_\star \in \Sigma_g(y_-) \setminus \Sigma_f(y_-)$ which would geometrically coincide with $f$ and $g$ over $Q_-$, since $f \leq a \leq g$ over $Q_-$ by elementary properties of one-dimensional envelopes and their shadows (see <cit.>). However, sliding the marked point of $a_\star$ to the singularity at $c_-$ would produce a marked tile in $\widehat{\Sigma}_g(c_-) \setminus \widehat{\Sigma}_f(c_-)$, a contradiction. This implies that if $f_\star$ and $g_\star$ are comparable by $\prec^\mathrm{geo}$, then $g_\star \prec^\mathrm{geo} f_\star$. Otherwise, $f_\star$ and $g_\star$ are comparable by $\prec^\mathrm{cor}$ and since $c_- \notin \mathfrak{c}$, we have $g_\star \prec^\mathrm{cor} f_\star$ independently of the geometric (in)coherence of $f_\star$ and $g_\star$ at $c_-$.

Notice that the same argument applies to $\prec_+$, since $c_+ \in \mathfrak{c}$ so the order $\prec^\mathrm{cor}$ for geometrically coherent forward envelopes meeting at $c_+$ compared to $c_-$ is reversed. Here, the choice of admissible corners `twisting' $\prec^\mathrm{cor}$ in the geometrically coherent case is essential.

The proof of (<ref>) is the same up to swapping $f$ and $g$, and the geometrically coherent case for $\prec^\mathrm{cor}$ does not arise.

It is now rather straightforward to check that the collection of relations $\{ \dot{\precsim}_p\}_{p \in M}$ induces a relation $\dot{\precsim}$ on $\overline{\mathscr{U}}_\star$ which is coherent along intersections. This implies that $\dot{\precsim}$ satisfies the axioms of Definition <ref> since $\precsim$ does; the details are left to the reader. To summarize, we obtain:

The set $\overline{\mathscr{U}}$ of completed forward envelopes equipped with $\dot{\precsim}$ is a weak prefoliation. It extends $(\mathscr{U}, \precsim)$ and covers $M$.

§ PROOF OF PROPOSITION <REF>

Let $\big(\mathscr{A}, {\precsim^\mathscr{A}}\big)$ be a (possibly empty) complete and backward weak prefoliation. In the previous sections, we constructed a complete and forward weak prefoliation $\big(\overline{\mathscr{U}}, \dot{\precsim}\big)$ which covers $M$. Our construction depends on auxiliary choices that are omitted from the notations. To construct an extension $\mathscr{A} \hookrightarrow \mathscr{B}$, we proceed in two steps.

* First, we construct a weak prefoliation structure on $\mathscr{A} \cup \overline{\mathscr{U}}$ such that the inclusion $\mathscr{A} \hookrightarrow \mathscr{A} \cup \overline{\mathscr{U}}$ is an extension.

* Then, we construct an extension $\mathscr{A} \cup \overline{\mathscr{U}} \hookrightarrow \mathscr{B}$ by gluing the edges and corners of tiles in $\mathscr{A}$ to suitable completed forward envelopes (the upper neighbors), while keeping copies of completed forward envelopes.

§.§ First extension

We define a relation $\precsim^\cup$ on $\mathscr{A} \cup \overline{\mathscr{U}}$ by adapting <cit.> and the proof of <cit.>. It restricts to $\precsim^\mathscr{A}$ on $\mathscr{A}_\star$ and to
$\dot{\precsim}$ on $\overline{\mathscr{U}}_\star$. We define  $\precsim^\cup_p$ in two steps. The first one is of a geometric nature, while the second one is purely formal.

§.§.§ Step 1: geometric extension

Let $p \in M$, $f_\star \in \mathscr{A}_p$, and $\bar{u}_\star \in \overline{\mathscr{U}}_p$. Here, $\bar{u}$ is the completion of a forward envelope $u \in \mathscr{U}$, $u : U \rightarrow M$ with shadow $\Sigma$.

We first use the shadow $\Sigma$ and the geometric order to define a relation $\precsim^{\cup_\circ}_p$:

* If the marked point $a \in \overline{U}$ of $\bar{u}_\star$ belongs to $U$, we set $f_\star \precsim^{\cup_\circ}_p \bar{u}_\star$ if $f_\star \in \Sigma(a)$, and $\bar{u}_\star \precsim^{\cup_\circ}_p f_\star$ otherwise.

* If $a \in \overline{U} \setminus U$ and one of $f_\star$ or $\bar{u}_\star$ is locally strictly above the other, their relation with respect to $\precsim^{\cup_\circ}_p$ is defined as to match the geometric order.

We then extend $\precsim^{\cup_\circ}$ along intersections at certain boundary components of $\overline{U}$ in a coherent way. More precisely, suppose that there exists a continuous path $\gamma : [0,1] \rightarrow \overline{U}$ starting at $a \in \overline{U} \setminus U$ which lifts to $f_\star$ so that $f$ and $\bar{u}$ are comparable by $\precsim^{\cup_\circ}$ at $p' = f\circ \gamma(1)$ by Step 1, after sliding the marked points along $\gamma$ and its lift. We define the relation between $f_\star$ and $\bar{u}_\star$ with respect to $\precsim^{\cup_\circ}_p$ as to match the one with respect to $\precsim^{\cup_\circ}_{p'}$ after sliding the marked points. Notice that this step applies if the marked point of $\bar{u}_\star$ belongs to an edge connected to the initial edge of $u$ via a (double) corner, and this (double) corner lifts to $f$.

This defines a relation $\precsim^{\cup_\circ}$ on $\mathscr{A}_\star \cup \overline{\mathscr{U}}_\star$ that restricts to $\precsim^\mathscr{A}$ on $\mathscr{A}_\star$ and to
$\dot{\precsim}$ on $\overline{\mathscr{U}}_\star$. By definition, it is coherent along intersections and is compatible with the geometric order. Moreover, it satisfies the following `pseudo-transitivity' property:

Let $a_\star, b_\star, c_\star \in \mathscr{A}_\star \cup \overline{\mathscr{U}}_\star$. If $a_\star \precsim^{\cup_\circ} b_\star$, $b_\star \precsim^{\cup_\circ} c_\star$, and $a_\star$ and $c_\star$ are comparable by $\precsim^{\cup_\circ}$, then $a_\star \precsim^{\cup_\circ} c_\star$.

Let $p \in M$ and consider $a_\star, b_\star, c_\star \in \mathscr{A}_p \cup \overline{\mathscr{U}}_p$ such that $a_\star \precsim^{\cup_\circ}_p b_\star$, $b_\star \precsim^{\cup_\circ}_p c_\star$, and $a_\star$ and $c_\star$ are comparable by $\precsim^{\cup_\circ}_p$. The case $p \in M^\mathrm{un}$ is immediate. We then assume that $p \in M^\mathrm{pure}$ and we consider three cases, depending on the nature of the flow line $\gamma$ of $X$ passing through $p$.

* $\gamma$ is not connected to a singularity of $X$. Notice that if $f_\star \in \mathscr{A}_p$ and $\bar{u}_\star \in \overline{\mathscr{U}}_p$ are comparable by $\precsim^{\cup_\circ}_p$, and if they only overlap along $\gamma$, then the marked point of $\bar{u}_\star$ necessarily belongs to the initial edge of $u$. Then, by considering all the possible cases depending on whether the three marked tiles belong to $\mathscr{A}_p$ or $\overline{\mathscr{U}}_p$, it is easy to check that $a_\star \precsim^{\cup_\circ}_p c_\star$.

* $\gamma$ is connected to a single singularity of $X$, which is not itself connected to a saddle singularity. This case is slightly more complicated because of the following: if $f_\star \in \mathscr{A}_p$ and $\bar{u}_\star \in \overline{\mathscr{U}}_p$ are comparable by $\precsim^{\cup_\circ}_p$, and if they only overlap along a (concave or convex) corner $c$ containing $\gamma$, then the marked point of $\bar{u}_\star$ belongs to one of the edges at $c$, but not necessarily to the initial edge of $u$.

Let us consider the following particular case: $a_\star = \bar{u}_\star \in \overline{\mathscr{U}}_p$, $b_\star = f_\star \in \mathscr{A}_p$, and $c_\star = \bar{v}_\star \in \overline{\mathscr{U}}_p$. Moreover, the marked point of $f_\star$ belongs to a concave corner on in the boundary of $f$, the marked point of $\bar{u}_\star$ belongs to the domain of $u$, but the marked point of $\bar{v}_\star$ does not belong to the domain of $v$. Arguing as in the proof of Lemma <ref>, we have that $\bar{u}_\star$ is locally geometrically below $\bar{v}_\star$. If they locally coincide, we can slide the marked points inside the forward domain of overlap of $u$ and $v$, and compare them there. They would be comparable by either $\prec^\mathrm{geo}$, or by $\prec^\mathrm{cor}$ and would be geometrically noncoherent at their initial corner of overlap. Both situations yield $\bar{u}_\star \dot{\precsim} \bar{v}_\star$ as desired.

The other cases can be treated in a similar way.

* $\gamma$ is connected to a saddle singularity of $X$ which is itself connected to another saddle singularity. This case is more tedious than the previous one, but can be treated in a similar way. The details are left to the reader.

We are now left with the case where $p$ is a pure saddle or a quadratic-like singularity. This includes the case of a saddle connected to another saddle singularity.

Let us consider the following particular case: $p \in \Delta_\mathrm{sa}^\mathrm{pure}$ is not connected to another saddle singularity, $a_\star = \bar{u}_\star \in \overline{\mathscr{U}}_p$, $b_\star = f_\star \in \mathscr{A}_p$, and $c_\star = \bar{v}_\star \in \overline{\mathscr{U}}_p$. Moreover, the marked point of $f_\star$ belongs to a concave corner on in the boundary of $f$, and $\bar{u}_\star$ and $\bar{v}_\star$ overlap with $f_\star$ near $p$ along two different edges. Then using the notations of Section <ref>, $f_\star \in \widehat{\Sigma}^\llcorner_v \setminus \widehat{\Sigma}^\llcorner_u$, hence $\widehat{\Sigma}^\llcorner_u \subsetneq \widehat{\Sigma}^\llcorner_v$ and $\bar{u}_\star \precsim^\cup_p \bar{v}_\star$.

The remaining cases can be treated in a similar way, and the details are left to the reader.

§.§.§ Step 2: abstract extension

Even after the previous step, the relation $\precsim^{\cup_\circ}_p$ between $f_\star$ and $\bar{u}_\star$ might still remain undefined if the marked points of $f_\star$ and $\bar{u}_\star$ both belong to certain edges, or if they are singularities at certain corners intersecting along edges only. We proceed in three steps:

* Let us first assume that $p$ belongs to a pure flow line of $X$ which is not connected to any singularity of $X$. We extend $\precsim^{\cup_\circ}$ at $p$ by the following abstract procedure:

* If there exists $a_\star \in \mathscr{A}_p \cup \overline{\mathscr{U}}_p$ such that $f_\star \precsim^{\cup_\circ}_p a_\star$ and  $a_\star \precsim^{\cup_\circ}_p \bar{u}_\star$, we set $f_\star \precsim^\cup_p \bar{u}_\star$.

* Otherwise, we set $\bar{u}_\star \precsim^\cup_p f_\star$.

By Lemma <ref>, this definition ensures that $\precsim^\cup_p$ is a total preorder on $\mathscr{A}_p \cup \overline{\mathscr{U}}_p$.

* We now assume that $p$ belongs to a connection between two saddle singularities $q_\pm \in \Delta_\mathrm{sa}$. By sliding the marked points towards $q_\pm$ along this connection, we obtain injections

\begin{align*}
\varsigma_\pm : \mathscr{A}_p \cup \overline{\mathscr{U}}_p \longrightarrow \mathscr{A}_{q_\pm} \cup \overline{\mathscr{U}}_{q_\pm}
\end{align*}

which preserve $\precsim^{\cup_\circ}$.
We modify the previous abstract procedure to extend $\precsim^{\cup_\circ}$ on $\mathscr{A}_p \cup \overline{\mathscr{U}}_p$. With the same notations as before, we set $f_\star \precsim^\cup_p \bar{u}_\star$ if one of the following holds:

* There exists $a_\star \in \mathscr{A}_p \cup \overline{\mathscr{U}}_p$ such that $f_\star \precsim^{\cup_\circ}_p a_\star$ and  $a_\star \precsim^{\cup_\circ}_p \bar{u}_\star$.

* There exists $a^\pm_\star \in \mathscr{A}_{q_\pm} \cup \overline{\mathscr{U}}_{q_\pm}$ such that $\varsigma_\pm\big(f_\star\big) \precsim^{\cup_\circ}_{q_\pm} a^\pm_\star$ and  $a^\pm_\star \precsim^{\cup_\circ}_{q_\pm} \varsigma_\pm\big(\bar{u}_\star\big)$.

Otherwise, we set $\bar{u}_\star \precsim^\cup_p f_\star$. We further coherently extend $\precsim^\cup$ along the flow line of $X$ passing through $p$ and at $q_\pm$ by sliding the marked points along this flow line. As in the previous item, Lemma <ref> ensures that $\precsim^\cup_p$ is a total preorder on $\mathscr{A}_p \cup \overline{\mathscr{U}}_p$.

* Finally, we assume that $p \in \Delta_\mathrm{sa} \cup Q$ is a saddle or quadratic singularity. We use the same abstract procedure as in the first item to extend $\precsim^\cup$ at $p$, taking into account the extension in step 2 if $p$ is connected to another saddle singularity. As before, this yields a total preorder $\precsim^\cup_p$ on $\mathscr{A}_p \cup \overline{\mathscr{U}}_p$. We then extend $\precsim^\cup$ by coherence along the pure stable and unstable branches at $p$ (no extension is needed along a connection to another saddle singularity).

It is now rather straightforward to prove:

The set $\mathscr{A} \cup \overline{\mathscr{U}}$ equipped with the relation $\precsim^\cup$ is a weak prefoliation which extends $\big( \mathscr{A}, {\precsim^\mathscr{A}}\big)$.

Left to the reader.

§.§ Second extension

We now define an extension $\mathscr{A} \cup \overline{\mathscr{U}} \hookrightarrow \mathscr{B}$ by adapting the strategy of the proof of <cit.>. To that extent, we define a similar notion of upper neighbors along the edges of tiles in $\mathscr{A}$ (see <cit.>), and we extend every such tile by attaching its upper neighbors to its boundary components. We also keep copies of tiles in $\overline{\mathscr{U}}$ to ensure that $\mathscr{B}$ covers $M$. We then define a weak prefoliation structure on $\mathscr{B}$ in a purely formal way. As before, we will need to treat the corners with particular care.

§.§.§ Typology of backward corners

Let $f : U \rightarrow M$ be a complete backward tile. We denote by $\mathcal{B}$ the set of boundary components of $U$, and by $\mathcal{E}$ the set of edges of $U$. A boundary component of $U$ contains between one and three edges. Two edges $e_0, e_1 \in \mathcal{E}$ are adjacent if $c = (f(e_0), f(e_1))$ is a (backward or forward) corner in $M$. Note the importance of the order. We further distinguish two cases:

* The singularity at $c$ is a pure saddle singularity. If $c$ is a backward (resp. forward) corner, we say that $(e_0,e_1)$ is a pure convex (resp. concave) corner of $U$ (or $f$).

* The singularity at $c$ is a quadratic-like singularity. If $c$ is a backward (resp. forward) corner, we say that $(e_0,e_1)$ is an unstable convex (resp. concave) corner of $U$ (or $f$).

These four types of backward corners are illustrated in Figure <ref>.

Pure convex corner.

Pure concave corner.

Unstable convex corner.

Unstable concave corner.

The four types of corners of backward tiles.

We say that $(e_0,e_1)$ is an adjoining pair of edges if it is either a pure concave corner or an unstable (convex or concave) corner of $f$. If $e_2 \in \mathcal{E}$ and both $(e_0, e_1)$ and $(e_1, e_2)$ are adjoining pairs of edges, we call $(e_0,e_1,e_2)$ an adjoining triple of edges. We define a boundary row of $U$ (or $f$) as either

* A single edge which is not part of an adjoining pair,

* An adjoining pair of edges which is not part of an adjoining triple, or

* An adjoining triple of edges.

We denote by $\mathcal{R}$ the set of boundary rows of $f$. Since every edge is included in a row, and every row is included in a boundary component, we have natural surjections $\mathcal{E} \twoheadrightarrow \mathcal{R} \twoheadrightarrow \mathcal{B}$.

We also define a collection of distinguished edges as follows. Each boundary row $r$ contains exactly one distinguished edge, and moreover,

* If $r$ is an adjoining pair of edges forming a forward corner $c$, then the last edge of $r$ is distinguished if $c \notin \mathfrak{c}$, and the first one is if $c \in \mathfrak{c}$. Recall that $\mathfrak{c}$ denotes the chosen admissible collection of forward corners in the definition of the weak prefoliation structure on $\mathscr{U}$.

* If $r$ is an adjoining triple of edges, the last edge of $r$ is distinguished.

We denote by $\mathcal{E}^\bullet \subset \mathcal{E}$ the set of distinguished edges of $f$. A row made of a single edge will be called a trivial row, in which case this edge is automatically distinguished.

We now explain the relevance of these definitions. Any complete forward tile intersecting a distinguished edge of $f$ will also intersect all the other edges of $f$ in the same boundary row. Moreover, if it starts at such an edge, it will not intersect edges of $f$ in different boundary rows. We leave it to the reader to draw the corresponding pictures. Therefore, we have to be careful to define upper neighbors at edges of $f$ that are `coherent' along corners at adjoining edges.

§.§.§ Upper neighbors

Let $p \in M^\mathrm{pure}$, $f_\star \in \mathscr{A}_p$, $\bar{u}_\star \in \mathscr{U}_p$, and assume that the marked points of $f_\star$ and $\bar{u}_\star$ belong to edges of the respective domains for $f$ and $\bar{u}$. Following <cit.>, we say that $\bar{u}_\star$ is an upper neighbor of $f_\star$ in $\big(\mathscr{A} \cup \overline{\mathscr{U}}, {\precsim^\cup}\big)$ if $f_\star \prec^\cup_p \bar{u}_\star$, and $\bar{u}_\star$ is the lowest such element in $\mathscr{A}_p \cup \overline{\mathscr{U}}_p$, i.e.,

\begin{align}
\forall g_\star \in \mathscr{A}_p \cup \overline{\mathscr{U}}_p, \quad f_\star \prec^\cup_p g_\star \implies \bar{u}_\star \precsim^\cup_p g_\star.
\end{align}

With the previous notations, $f_\star$ admits an upper neighbor which is unique up to order-equivalence.

The uniqueness immediately follows from the definition. To show existence, we consider the section $\Sigma_0 \subset \mathscr{A}_p$ defined by

\begin{align} \label{eq:sigma0}
\Sigma_0 \coloneqq \left\{ g_\star \in \mathscr{A}_p \ \vert \ g_\star \precsim^\mathscr{A}_p f_\star \right\}.
\end{align}

If $p$ does not belong to the pure stable branch of a quadratic-like singularity, we define $\bar{u}_\star$ as the forward envelope starting at $\big(p, \Sigma_0\big)$ and with marked point corresponding to the initial basepoint of $\bar{u}$. We emphasize that this forward envelope might not be the upper neighbor of $f_\star$. However, we claim that if the marked point of $f_\star$ belong to the distinguished edge of a boundary row $r$, then $\bar{u}_\star$ is the upper neighbor of $f_\star$. To prove this, we consider three cases.

* $r$ is a trivial row. By definition of $\precsim^\cup$ in Step 1 of the previous section, $f_\star \precsim^\cup \bar{u}_\star$. Then, let $g_\star \in \mathscr{A}_p \cup \overline{\mathscr{U}}_p$ such that $f_\star \prec^\cup g_\star$. We consider two cases:

* $g_\star \in \mathscr{A}_p$. Then $g_\star \notin \Sigma_0$, so $\bar{u}_\star \precsim^\cup_p g_\star$ by definition.

* $g_\star = \bar{v}_\star \in \overline{\mathscr{U}}_p$. By definition, there exists $\bar{w}_\star \in \overline{\mathscr{U}}_p$ such that $f_\star \precsim^\cup_p \bar{w}_\star \precsim^\cup_p \bar{v}_\star$, and the marked point $z$ of $\bar{w}$ belongs to the domain of $w$. We must have $f_\star \in \Sigma_w(z)$, otherwise $\bar{w}_\star$ would be locally geometrically below $f_\star$, so $\Sigma_0 \subset \Sigma_w(z)$. If this inclusion is strict, then $u_\star \prec^{\mathrm{geo}}_p w_\star$ and $\bar{u}_\star \dot{\precsim}_p \bar{w}_\star$, hence $\bar{u}_\star \precsim^\cup_p \bar{w}_\star \precsim^\cup_p \bar{v}_\star$. If $\Sigma_0 = \Sigma_w(z)$, then $u_\star \prec^\mathrm{inc}_p w_\star$, hence $\bar{u}_\star \precsim^\cup_p \bar{v}_\star$ as well.

* $r = (e_0, e_1)$ is an adjoining pair of edges. We first assume that the corresponding corner $c$ is a pure forward corner. Let $g_\star \in \mathscr{A}_p \cup \overline{\mathscr{U}}_p$ such that $f_\star \prec^\cup g_\star$. As in the previous case, $g_\star \in \mathscr{A}_p$ implies $\bar{u}_\star \prec^\cup_p g_\star$. The case $g_\star \in \overline{\mathscr{U}}_p$ is slightly more complicated. As before, there exists $\bar{w}_\star \in \overline{\mathscr{U}}_p$ such that $f_\star \precsim^\cup_p \bar{w}_\star \precsim^\cup_p \bar{v}_\star$. However, the marked point if $\bar{w}_\star$ might not belong to the domain of $w$. If it does, then the previous argument still implies $\bar{u}_\star \precsim^\cup_p \bar{w}_\star$. Otherwise, we slide the marked point of $\bar{w}_\star$ along $c$, and the new marked point $z'$ now belongs to the domain of $w$. We also slide the marked point of $f_\star$ along $c$ to get a new tile $f'_\star$ passing through $p' \coloneqq w(z')$. We also define the corresponding section

$$\Sigma'_0 \coloneqq \left\{ g'_\star \in \mathscr{A}_{p'} \ \vert \ g'_\star \precsim^\mathscr{A}_{p'} f'_\star \right\}$$

so that $\Sigma'_0 \subset \Sigma_w(z')$. If this inclusion is strict, then $\widehat{\Sigma}_u(c) \subsetneq \widehat{\Sigma}_w(c)$, implying $\bar{u}_\star \precsim^\cup_p \bar{w}_\star$. If $\Sigma'_0 = \Sigma_w(z')$, then the definitions of $\prec^\mathrm{cor}$ and distinguished edges imply $\bar{u}_\star \precsim^\cup_p \bar{w}_\star$ as well.

The cases where $c$ is a (forward or backward) quadratic-like corner is simpler since no forward envelope starts at the pure stable branch of the corresponding singularity. The details are left to the reader.

* $r = (e_0, e_1, e_2)$ is an adjoining triple of edges. There are multiple cases to consider depending on the nature of the corners corresponding to $(e_0, e_1)$ and to $(e_1, e_2)$, but all of these cases can be treated as in case 2. The details are left to the reader.

If the marked point of $f_\star$ does not belong to a distinguished edge, we consider the previously constructed upper neighbor at (a point on) the distinguished edge in $r$, and we slide the marked point along the corner(s) to obtain a marked tile at $p$. By the previous arguments, it is easy to check that the latter is the upper neighbor of $f_\star$.

We emphasize that by construction, the upper neighbors of $f \in \mathscr{A}$ satisfy the following coherence condition along boundary rows. Let $(e_0, e_1)$ be an adjoining pair of edges for $f$ forming a corner $c$ in $M$, and $x_0 \in e_0$, $x_1 \in e_1$ be two boundary marked points. We denote by $\bar{u}^0_\star$ and $\bar{u}^1_\star$ the upper neighbors of $(f,x_0)$ and $(f,x_1)$, respectively. Then:

Up to order-equivalence, $\bar{u}^0_\star$ and $\bar{u}^1_\star$ differ by sliding the marked point along $c$.

Therefore, each boundary row $r \in \mathcal{R}$ of $f$ determines a completed forward envelope $\bar{u}^r \in \overline{\mathscr{U}}$, unique up to forward-equivalence, such that the upper neighbor of $f$ for a marked point on an edge of $r$ is $\bar{u}^r$, with a suitable marked point on the boundary component of the initial edge of $u^r$. Importantly, gluing $\bar{u}^r$ to $f$ along $r$ yields a tile.

Assume that $p \in M^\mathrm{pure}$ belongs to a closed orbit $\gamma$ of $X$ with period $T > 0$, and $\bar{u}_\star \in \overline{\mathscr{U}}_p$ is the upper neighbor of $f_\star \in \mathscr{A}_p$. In that case, $p$ belongs to the initial edge of $u$. Assume that $\bar{u}^T_\star \simeq_p \bar{u}_\star$, where we use the notation from Section <ref> for the $T$-translation. Then $u^T_\star \simeq_p u_\star$ and $\Phi^T\big(\Sigma_0\big) = \Sigma_0$, where $\Sigma_0$ is defined by (<ref>). It immediately follows that $f^T_\star \simeq_p f_\star$. Reciprocally, if $f^T_\star \simeq_p f_\star$ then $\Phi^T\big(\Sigma_0\big) = \Sigma_0$ and $u^T_\star \simeq_p u_\star$ by the uniqueness of forward envelopes (see Proposition <ref>), hence $\bar{u}^T_\star \simeq_p \bar{u}_\star$.

To summarize, the upper neighbor of a tile $f_\star$ in $\mathscr{A}_p$ is order-equivalent to its $T$-translation if and only if $f_\star$ is. However, it is possible that $f^T_\star \approx_p f_\star$ without $f^T_\star \simeq_p f_\star$ ($f$ is geometrically covering itself, but not from the viewpoint of the preorder), in which case we might not even have $\bar{u}^T_\star \approx_p \bar{u}_\star$. This is because our preorders contain strictly more information than the geometric order.

§.§.§ Gluing upper neighbors

We now define a complete and backward weak prefoliation $\mathscr{B}$ extending $\mathscr{A} \cup \overline{\mathscr{U}}$ by gluing the upper neighbors to the backward edges of the tiles in $\mathscr{A}$, while keeping copies of the tiles in $\overline{\mathscr{U}}$. More precisely, to each tile $f \in \mathscr{A}$ we associate a new tile $\widetilde{f}$ defined as follows. If $f$ has no edge, we set $\widetilde{f}=f$. Otherwise, for each $e \in \mathcal{E}^\bullet$, we choose a point $x_e \in e$ and write $p_e \coloneqq f(x_e)$. Then, we denote by $\bar{u}_e \in \overline{\mathscr{U}}_{p_e}$ the upper neighbor of $(f, x_e)$. It can be glued to $f$ along the row of edges containing $e$. Recall that by Lemma <ref> above, the upper neighbor of $f$ at adjoining edges to $e$ coincides with $\bar{u}_e$ with a slid marked point. We then define $\widetilde{f}$ as the complete and forward tile obtained by gluing all such upper neighbors along the (rows of) edges of $f$, without repetitions. By construction, there is a natural embedding $\varphi_f$ of the domain of $f$ into the domain of $\widetilde{f}$ as in Definition <ref>.

We now define

$$\widetilde{\mathscr{A}} \coloneqq \big\{ \widetilde{f} \ \vert \ f \in \mathscr{A}\big\}, \qquad \mathscr{B} \coloneqq \widetilde{\mathscr{A}} \cup \overline{\mathscr{U}}.$$

It comes with a natural map $i : \mathscr{A} \cup \overline{\mathscr{U}} \rightarrow \mathscr{B}$ which restricts to the identity on $\overline{\mathscr{U}}$. By construction, the tiles in $\mathscr{B}$ are complete and forward, and they cover $M$. It remains to define a weak prefoliation structure $\precsim^\mathscr{B}$ on $\mathscr{B}$ making $i$ an extension. To this end, we proceed as in the proof of <cit.> in three steps.

* We first define the restriction of $\precsim^\mathscr{B}$ to $\widetilde{\mathscr{A}}$. Let $\widetilde{f}_\star \in \widetilde{\mathscr{A}}_\star$, where $f \in \mathscr{A}$. We consider two cases:

* The marked point belongs to the domain of $f$. This induces a marked tile $f_\star \in \mathscr{A}_\star$.

* The marked point does not belong to the domain of $f$, but to the domain of a completed forward envelope $\bar{u} \in \overline{\mathscr{U}}$. This induces a marked tile $\bar{u}_\star \in \overline{\mathscr{U}}_\star$.

This allows us to define a `restriction' map $\rho : \widetilde{\mathscr{A}}_\star \rightarrow \mathscr{A}_\star \cup \overline{\mathscr{U}}_\star$ as

* $\rho \big(\widetilde{f}_\star\big) = f_\star$ in case (a),

* $\rho \big(\widetilde{f}_\star\big) = \bar{u}_\star$ in case (b).

Then, we define $\precsim^\mathscr{B}$ on $\widetilde{\mathscr{A}}$ as the pullback of $\precsim^\cup$ along $\rho$:

$$\forall \widetilde{f}_\star, \widetilde{g}_\star \in \widetilde{\mathscr{A}}_\star, \quad \widetilde{f}_\star \precsim^\mathscr{B} \widetilde{g}_\star \iff \rho\big(\widetilde{f}_\star\big) \precsim^\cup \rho\big(\widetilde{g}_\star\big).$$

* We now define $\precsim^\mathscr{B}$ between elements $\widetilde{f}_\star \in \widetilde{\mathscr{A}}_\star$ and $\bar{v}_\star \in \overline{\mathscr{U}}_\star$. If $\rho\big(\widetilde{f}_\star\big) = f_\star \in \mathscr{A}_\star$, we define the order between $\widetilde{f}_\star$ and $\bar{v}_\star$ for $\precsim^\mathscr{B}$ to match the one between $f_\star$ and $\bar{v}_\star$ for $\precsim^\cup$. If $\rho\big(\widetilde{f}_\star\big) = \bar{u}_\star \in \overline{\mathscr{U}}_\star$, we set

\begin{align*}
\widetilde{f}_\star \precsim^\mathscr{B} \bar{v}_\star \iff \bar{u}_\star \ \dot{\precsim} \ \bar{v}_\star, \\
\bar{v}_\star \precsim^\mathscr{B} \widetilde{f}_\star \iff \bar{v}_\star \ \dot{\prec} \ \bar{u}_\star.
\end{align*}

That way, if $\bar{u}_\star \simeq \bar{v}_\star$, then $\widetilde{f}_\star \prec^\mathscr{B} \bar{v}_\star$.

* Finally, the restriction of $\precsim^\mathscr{B}$ on $\overline{\mathscr{U}}_\star$ is defined to agree with $\dot{\precsim}$.

By virtue of the properties of upper neighbors, it is rather mechanical exercise to check:

The set $\mathscr{B}$ equipped with the relation $\precsim^\mathscr{B}$ is a weak prefoliation, and the map $i : \mathscr{A} \cup \overline{\mathscr{U}} \rightarrow \mathscr{B}$ is an extension.

Left to the reader.

§.§ Finishing the proof

We conclude the proof of Proposition <ref>. Starting from a complete and backward weak prefoliation $\mathscr{A}$, the desired forward extension $\mathscr{A} \hookrightarrow \mathscr{B}$ is obtained as the composition of the extensions $\mathscr{A} \hookrightarrow \mathscr{A} \cup \overline{\mathscr{U}}$ from Lemma <ref> and $\mathscr{A} \cup \overline{\mathscr{U}} \hookrightarrow \mathscr{B}$ from Lemma <ref>. Finally, the case where $\mathscr{A}$ is forward instead of backward is obtained by reversing the (co)orientation of $\eta$ and applying the previous results.