A center period annulus has an orbit or polycycle frontier

Writing $\omega$ for the defining form, the characteristic covector $\beta =h^{\ast }\omega$ is $C^1$ with kernel exactly the characteristic line field and with the characteristic singularities as zeros, and equivalently it is obtained by summing finitely many pulled-back local transverse covectors, which differ on overlaps by positive nowhere-vanishing factors; choosing an oriented area form $\mu$ and defining $X$ by ${\iota }_X\mu =\beta$ makes $X$ a $C^1$ planar field whose regular line foliation has a $C^2$ first-integral atlas with local first integrals $u=z\circ h$, and by [F1] the singularities of $X$ in the disk are finitely many nondegenerate interior centers and saddles with nonvanishing $\beta$ on a boundary collar; in the transversal boundary case $\beta$ does not vanish on the boundary tangent and in the leafwise case $\beta$ does, so then $X$ is tangent to and nonvanishing along $\mathrm{\partial }{D}^2$ and the boundary circle is one regular closed $X$-orbit; finally [F8] extends $X$ to a $C^1$ field on an open neighborhood $U$ of the disk.

Fix a center $c$ of $X$ and let $P_c$ be the set of points of regular periodic $X$-orbits contained in $\mathrm{int}{D}^2$ whose bounded Jordan interior contains $c$; a nonconstant periodic orbit is an embedded circle whose period set is a closed additive subgroup of $\mathbb{R}$ and therefore has a least positive period, so the orbit is a $C^2$ embedded circle because an individual trajectory of a $C^1$ field is $C^2$ in time ($x\mathrm{\prime }\mathrm{\prime }=DX(x)X(x)$), the center picture of [F1] makes $\mathrm{int}{P}_c$ nonempty, and the local flow carries $P_c$ and $\mathrm{int}{P}_c$ to themselves; the component $A$ of $\mathrm{int}{P}_c$ containing the small center collar is open and saturated, because components are unions of connected subsets and each orbit is connected.

The circles of $A$ are strictly nested and consistently oriented: two distinct orbits are disjoint by uniqueness of trajectories and their bounded interiors both contain $c$; a connected circle disjoint from a Jordan curve lies in one complementary component by [F7], so if $C_2$ lay in the exterior of $C_1$ then the bounded domain of $C_1$, being connected, disjoint from $C_2$ and containing $c\in \mathrm{int}{D}_{C_2}$, would lie in the bounded domain of $C_2$, giving strict nesting; and the sign in which the $X$-orientation of an orbit agrees with the boundary orientation of its bounded domain is locally constant on the connected $A$, hence one global sign.

By [F2] applied to the $C^2$ first-integral atlas of step 1.1 on the connected set $A$ with its strictly nested consistently oriented compact circle leaves, $A$ is an open annulus with a $C^2$ product $\mathrm{\Psi }:S^1\times (0,1)\to A$ taking circles onto leaves and increasing in the nested order; $A$ is maximal among connected open regular circle families continuing the chosen center collar, because any such family lies in $P_c$, hence in $\mathrm{int}{P}_c$, hence in this component.

Write $C_s=\mathrm{\Psi }(S^1\times \{s\})$, let ${\mathrm{\Omega }}_s$ be the bounded Jordan domain of $C_s$ and $\mathrm{\Omega }={\bigcup }_{0<s<1}{\mathrm{\Omega }}_s$; the closed bounded domains lie in $D^2$ because their exteriors contain the connected complement of the disk, for $s<t$ one has $\mathrm{cl}{\mathrm{\Omega }}_s\subseteq {\mathrm{\Omega }}_t$ and $C_s\subseteq {\mathrm{\Omega }}_t$ by strict nesting, $\mathrm{\Gamma }=\mathrm{\partial }\mathrm{\Omega }$ by [F10] is a nonempty compact subset of $D^2$ by [F6], and $\mathrm{\Gamma }$ separates $c$ from any fixed point $q$ outside the closed disk, since $c$ lies in some ${\mathrm{\Omega }}_s$, $q$ lies outside $\mathrm{cl}\mathrm{\Omega }$, and every path between them has a first exit from $\mathrm{\Omega }$, on $\mathrm{\Gamma }$.

The exhaustion $T_r=\mathrm{cl}{\bigcup }_{s\ge r}{C}_s$ shows $C_s\to \mathrm{\Gamma }$ in Hausdorff distance and $\mathrm{\Gamma }\subseteq \mathrm{cl}A$: the tail intersection lies in $\mathrm{cl}\mathrm{\Omega }$ and misses $\mathrm{\Omega }$ because a ball about a point of ${\mathrm{\Omega }}_t$ is avoided by all $C_s$ with $s>t$, hence lies in $\mathrm{\partial }\mathrm{\Omega }=\mathrm{\Gamma }$; if arbitrarily late $C_s$ had points at distance at least $ϵ$ from $\mathrm{\Gamma }$ the nested compacta $T_r\cap \{\mathrm{dist}(\cdot ,\mathrm{\Gamma })\ge ϵ\}$ would meet, contradiction; and conversely each $p\in \mathrm{\Gamma }$ and $ϵ>0$ admit $x\in \mathrm{\Omega }\cap B_{ϵ\mathrm{/}2}(p)$ and $t$ with $x\in {\mathrm{\Omega }}_t$, so every $C_s$ with $s>t$ meets the segment from $x$ to $p$ within $ϵ$ of $p$, and a finite cover of the compact $\mathrm{\Gamma }$ gives $\mathrm{\Gamma }$ everywhere within $ϵ$ of $C_s$.

No center lies on $\mathrm{\Gamma }$: the fixed center $c$ lies in the open ${\mathrm{\Omega }}_t$, and for any other center $d$ the center picture of [F1] supplies a small saturated disk $V$ disjoint from $c$; every regular orbit meeting $V$ is a complete small level circle inside $V$ by uniqueness, so it does not enclose $c$ and is not in $A$, whence $V$ is disjoint from $\mathrm{cl}A$ and from $\mathrm{\Gamma }$; consequently every equilibrium on $\mathrm{\Gamma }$ is among the finitely many nondegenerate saddles of the disk.

In the transversal boundary case the period annulus does not meet the boundary: in an inward collar coordinate $r\ge 0$ the radial component of $X$ is nonzero on $\mathrm{\partial }{D}^2$ because $\beta$ is nonzero on the boundary tangent, its sign is constant along the connected boundary circle, and continuity gives $\delta >0$ and $k>0$ with $\mathrm{\mid }Xr\mathrm{\mid }\ge k$ and one fixed sign throughout $0\le r\le \delta$; if a periodic orbit met $\{r<\delta \}$ then its radial coordinate has a minimum below $\delta$, attained on the compact periodic curve, where its derivative along $X$ must be zero, contradicting $\mathrm{\mid }Xr\mathrm{\mid }\ge k$, so no orbit of $A$ meets that collar and, in particular, the period annulus cannot meet the boundary.

By [F3] applied to the product $\mathrm{\Psi }$ and the compact frontier $\mathrm{\Gamma }$ take the field $Y=X+V$ constructed by the flat-drift proof, and its positive orbit $y$ with compact closure in $D^2$ and ${\omega }_Y^{+}(y)=\mathrm{\Gamma }$. The derivative equality needed below follows from that construction, not merely from $Y=X$ on $\mathrm{\Gamma }$: its locally finite band terms $V_n$ are supported on compact sets $B_n\subset A$ of distance $d_n>0$ from $\mathrm{\Gamma }$, with $\mathrm{\mid }{V}_n(z)\mathrm{\mid }\le 2^{-n-1}\mathrm{dist}(z,\mathrm{\Gamma })$ and $\parallel D{V}_n(z)\parallel \le 2^{-n-1}{d}_n$. The fixed inner cutoff is identically one near $\mathrm{\Gamma }$. Any finite collection of band supports stays away from $\mathrm{\Gamma }$, so only $n\ge N$ contribute sufficiently near it; the $d_n$ are bounded by the diameter of the disk. Thus $V(z)=o(\mathrm{dist}(z,\mathrm{\Gamma }))$ and $DV(z)\to 0$. Extend $V=0$ on $\mathrm{\Gamma }$: for $p\in \mathrm{\Gamma }$, $\mathrm{dist}(z,\mathrm{\Gamma })\le \mathrm{\mid }z-p\mathrm{\mid }$ proves $DV(p)=0$, giving $Y=X$ and $DY=DX$ on $\mathrm{\Gamma }$. Hence $\mathrm{\Gamma }$ is compact, invariant under the flow, and connected because it is the intersection of the decreasing family of connected closures of the orbit tails, while its regular $Y$-trajectories are $X$-trajectories by equality of the fields on the invariant set and uniqueness.

Apply [F5] to the positive orbit of $y$, whose $\omega$-limit set $\mathrm{\Gamma }$ contains only the finitely many equilibria of step 7.1: alternative (i) fails because a singleton does not separate $c$ from $q$, since the complement of one point of the plane is path connected by explicit polygonal detours, so either $\mathrm{\Gamma }$ is one regular periodic orbit, or $\mathrm{\Gamma }$ contains equilibria and every regular point of it lies on a nonconstant trajectory whose alpha- and omega-limits are among those saddles.

In the second alternative of step 10.1, [F4] applies with the field $Y$, its positive orbit and the separating compact $\mathrm{\Gamma }$ whose equilibria are nondegenerate saddles, so $\mathrm{\Gamma }$ is a finite embedded strongly connected directed saddle multigraph whose edges are the closures of the distinct nonconstant saddle-to-saddle trajectories, and finitely many closed directed edge walks cover it; in the first alternative $\mathrm{\Gamma }$ is a single regular closed orbit, and if in the leafwise boundary case $\mathrm{\Gamma }$ meets $\mathrm{\partial }{D}^2$, then the boundary circle is itself a regular closed orbit inside $\mathrm{\Gamma }$, invariance forces the whole boundary orbit into $\mathrm{\Gamma }$, and alternatives (i) and the equilibrium alternative cannot hold because the boundary carries no equilibrium, so $\mathrm{\Gamma }$ equals that boundary orbit; thus the outer frontier is a regular closed orbit, the boundary orbit in the leafwise case, or a finite strongly connected saddle graph covered by finitely many polycycles, with loops, repeated vertices, shared and parallel edges allowed.

Finally the annulus parameter gives the prescribed trace: for each fixed phase ${\theta }_0$ the map $s\mapsto h(\mathrm{\Psi }({\theta }_0,s))$ is $C^2$ because $h$ and $\mathrm{\Psi }$ are, and it is transverse to $F$ because the pulled-back characteristic covector applied to ${\mathrm{\partial }}_s\mathrm{\Psi }$ is nonzero, as ${\mathrm{\partial }}_{\theta }\mathrm{\Psi }$ spans the characteristic direction and $({\mathrm{\partial }}_{\theta }\mathrm{\Psi },{\mathrm{\partial }}_s\mathrm{\Psi })$ is a basis; these closed traces are exactly the prescribed loops $C_s$, and no nontriviality of their return holonomy is assumed or used.

Therefore every center of a disk map in relative generic position is surrounded by a maximal period annulus whose outer frontier is one of the listed alternatives, the transversal boundary case cannot be met by the annulus, and the annulus parameter supplies the $C^2$ transverse trace of the prescribed closed characteristic loops; the only countable selections in the proof are those in the product supplier of step 4.1, made under the standing ${\mathrm{A}\mathrm{C}}_{\omega }$ of [F12], while all other steps use finitely many explicit objects.