A finite saddle omega-graph is strongly connected and is a finite union of polycycles

At a saddle $q\in \mathrm{\Gamma }$ use the $C^1$ coordinates of [F2] in which the local transverse function of the disk map is $u=xy$; writing the pulled-back area form as $\mu =mdx\wedge dy$ with $m$ continuous and positive and the characteristic covector as $\beta =adu$ with $a$ continuous and nonzero, the field is $X=(a\mathrm{/}m)(x{\mathrm{\partial }}_x-y{\mathrm{\partial }}_y)$; replacing $(x,y)$ by $(y,x)$ if necessary (which changes $x{\mathrm{\partial }}_x-y{\mathrm{\partial }}_y$ to its negative) makes the coefficient $c(x,y)>0$, continuous on a smaller compact chart, and then $\dot{x}=cx$, $\dot{y}=-cy$ give two incoming half-branches (both $y$-axis rays) and two outgoing half-branches (both $x$-axis rays) with exponential rate bounds $\mathrm{\mid }\mathrm{c}\mathrm{o}\mathrm{o}\mathrm{r}\mathrm{d}(t)\mathrm{\mid }\in [r_0{e}^{-c_{+}t},r_0{e}^{-c_{-}t}]$ for the contracting branch and the reciprocal bounds for the expanding branch on the compact chart; since $Y=X$ on $\mathrm{\Gamma }$, every edge of $\mathrm{\Gamma }$ through $q$ follows one of these four half-branches, and by uniqueness of [F3] each half-branch is contained in exactly one maximal trajectory, so at most two edges leave each vertex and there are at most twice as many edges as vertices.

Every regular point of $\mathrm{\Gamma }$ lies on a maximal trajectory whose alpha- and omega-limits are saddle points of $\mathrm{\Gamma }$: since a singleton does not separate two points of the plane, alternative (i) of [F1] fails; alternative (ii) fails because $\mathrm{\Gamma }$ contains the given equilibrium; hence alternative (iii) holds; each such maximal trajectory is contained in $\mathrm{\Gamma }$ by invariance and closedness of $\mathrm{\Gamma }$, has no interior equilibrium by uniqueness [F3], and its closure is one of the edge closures of step 1.1, so the edge closures together with the saddle vertices exhaust $\mathrm{\Gamma }$, distinct edges meet only at common saddle endpoints, and the four half-branches at each saddle give the local embedded-graph structure.

With exact endpoints, $\mathrm{\Gamma }$ is internally chain-transitive: fix $p,q\in \mathrm{\Gamma }$, $ϵ>0$ and $T>0$; by joint continuity of the flow [F3] on the compact set $K_0$ and the time interval $[T,2T]$, choose $\delta \in (0,ϵ\mathrm{/}3)$ so that $\delta$-close points have $ϵ\mathrm{/}3$-close images throughout $[T,2T]$; since $\mathrm{\Gamma }={\omega }^{+}(y)$ and the orbit tail approaches $\mathrm{\Gamma }$ uniformly, choose $s$ so late that ${\mathrm{\Phi }}_u(y)$ is within $\delta$ of $\mathrm{\Gamma }$ for all $u\ge s$, then choose $s$ with ${\mathrm{\Phi }}_s(y)$ within $\delta$ of $p$ and $t>s+2T$ with ${\mathrm{\Phi }}_t(y)$ within $\delta$ of $q$; write $t-s-T=N\tau$ with an integer $N\ge 1$ and $\tau \in [T,2T]$; the finitely many intermediate orbit points ${\mathrm{\Phi }}_{s+T+k\tau }(y)$, $0\le k<N$, lie within $\delta$ of $\mathrm{\Gamma }$, so finitely many nearby points $z_k\in \mathrm{\Gamma }$ exist, and $x_0=p$, $x_1=z_0$, ..., $x_N=z_{N-1}$, $x_{N+1}=q$ together with the times $T,\tau ,\dots ,\tau$ form an $(ϵ,T)$-chain because each flowed image is within $ϵ\mathrm{/}3$ of the next orbit point and each jump is at most $ϵ\mathrm{/}3+\delta <ϵ$.

The directed graph is strongly connected: $\mathrm{\Gamma }$ is connected, being the intersection of the decreasing family of the connected closures of the orbit tails, since a separation of $\mathrm{\Gamma }$ into disjoint nonempty compact pieces has positive distance and would force a sufficiently late tail closure, which is connected, into a neighbourhood of one piece and away from the other [F6]; if the graph had more than one strongly connected component, its finite condensation would have a proper terminal component $S$, and if no edge entered $S$ from outside then no edge would leave it either, so the compact carriers of $S$ and of its complement would express the connected $\mathrm{\Gamma }$ as two disjoint nonempty closed sets, which is impossible; hence some edge enters $S$, and the compact set $N$ consisting of the vertices of $S$, all edges internal to $S$ and a short terminal segment of every entering edge with a trimming point in its regular part satisfies $S\subseteq {\mathrm{int}}_{\mathrm{\Gamma }}N$ with no edge leaving $S$; points of $N$ flow strictly toward $S$ and never reach a saddle in finite time by uniqueness [F3], so ${\mathrm{\Phi }}_t(N)\subseteq N$ for $t\ge 0$ and ${\mathrm{\Phi }}_T(N)\subseteq {\mathrm{int}}_{\mathrm{\Gamma }}N$, whence ${\delta }_0:=\mathrm{dist}({\mathrm{\Phi }}_T(N),\mathrm{\Gamma }\setminus {\mathrm{int}}_{\mathrm{\Gamma }}N)>0$ by [F6]; a chain with $ϵ<{\delta }_0$ starting at a point of $S$ stays in $N$ by induction, because ${\mathrm{\Phi }}_t(N)\subseteq {\mathrm{\Phi }}_T(N)$ for $t\ge T$ and a jump of size less than ${\delta }_0$ cannot leave the ${\delta }_0$-neighbourhood of ${\mathrm{\Phi }}_T(N)$; choosing the terminal point $q$ in the omitted middle part of an entering edge gives $q\mathrm{\notin }N$, contradicting the exact-endpoint chain transitivity of step 3.1, so all vertices lie in one strongly connected component.

Consequently, for each directed edge $e:v\to w$ strong connectivity supplies a directed path from $w$ back to $v$, and adjoining $e$ gives a closed directed edge walk containing $e$ and repeating vertices only as allowed; there are finitely many edges by step 1.1, so finitely many such walks cover $\mathrm{\Gamma }$; the construction used only finitely many points and paths of a finite graph plus the finite flow-box and compactness arguments, hence no choice principle, so the statement holds and its standing ${\mathrm{A}\mathrm{C}}_{\omega }$ hypothesis is not invoked.