A flat transverse drift realizes the period-annulus frontier as an omega-limit set

Set ${\mathrm{\Omega }}_s=\mathrm{int}{D}_s$ and $\mathrm{\Omega }={\bigcup }_{0<s<1}{\mathrm{\Omega }}_s$, so that $\mathrm{\Gamma }=\mathrm{\partial }\mathrm{\Omega }$; strict nesting gives $\mathrm{cl}{\mathrm{\Omega }}_s\subseteq {\mathrm{\Omega }}_t$ and $C_s\subseteq {\mathrm{\Omega }}_t$ for $s<t$, and the sets $T_r=\mathrm{cl}{\bigcup }_{s\ge r}{C}_s$ are nonempty compact subsets of $D$ decreasing in $r$, so their tail intersection $K_{\mathrm{\infty }}$ lies in $\mathrm{cl}\mathrm{\Omega }$, meets no ${\mathrm{\Omega }}_t$ because a ball about a point of ${\mathrm{\Omega }}_t$ is avoided by all $C_s$ with $s>t$, and therefore lies in $\mathrm{\Gamma }$; if arbitrarily late $C_s$ had points at distance at least $ϵ$ from $\mathrm{\Gamma }$ the nested compact sets $T_r\cap \{\mathrm{dist}(\cdot ,\mathrm{\Gamma })\ge ϵ\}$ would have a common point, so ${\sup }_{x\in C_s}\mathrm{dist}(x,\mathrm{\Gamma })\to 0$ by [F2], while conversely for fixed $p\in \mathrm{\Gamma }$ and $ϵ>0$ a point $x\in \mathrm{\Omega }\cap B_{ϵ\mathrm{/}2}(p)$ and an index $t$ with $x\in {\mathrm{\Omega }}_t$ force every $C_s$ with $s>t$ to meet the segment from $x$ to $p$, and a finite cover of the compact $\mathrm{\Gamma }$ by such balls makes $\mathrm{\Gamma }$ everywhere within $ϵ$ of $C_s$; hence $C_s\to \mathrm{\Gamma }$ in Hausdorff distance and $\mathrm{\Gamma }\subseteq \mathrm{cl}A$.

Fix $s_1\in (0,1)$ and set $m(s)={\min }_{\theta }a(s,\theta )>0$, $L(s)={\max }_{\theta }\mathrm{\parallel }{\mathrm{\partial }}_s\mathrm{\Psi }(s,\theta )\mathrm{\parallel }$ and $q(s)=\min \{(1-s{)}^2,(1-s{)}^{2}m(s)\mathrm{/}(1+L(s))\}$, which are continuous and positive on compact subintervals; with $\tau (s)=\log ((1-s_1)\mathrm{/}(1-s))$ and the explicit bump $\eta (t)=e^{-1\mathrm{/}(1-t^2)}$ for $\mathrm{\mid }t\mathrm{\mid }<1$ and $\eta =0$ otherwise, the functions ${\rho }_n(s)=\eta (\tau (s)-n-\frac{1}{2})\mathrm{/}{\sum }_{j\ge 0}\eta (\tau (s)-j-\frac{1}{2})$ form a smooth locally finite partition of $[s_1,1)$ with positive sum, uniformly finite overlap, compact supports in $(0,1)$ and active indices tending to infinity as $s\to 1$; for each support the compact set $B_n=\mathrm{\Psi }(S^1\times \mathrm{supp}{\rho }_n)$ is disjoint from $\mathrm{\Gamma }$ with positive distance $d_n$, while $q_n={\min }_{\mathrm{supp}{\rho }_n}q>0$ and $M_n=1+{\sup }_{B_n}(\mathrm{\mid }{\rho }_{n}W\mathrm{\mid }+\mathrm{\parallel }D({\rho }_{n}W)\mathrm{\parallel })<\mathrm{\infty }$ are attained finite extrema of continuous functions on nonempty compacta by [F2], so these are uniquely specified real numbers and no sequence of witnesses is selected.

With $c_n=2^{-n-1}\min (q_n,d_n\mathrm{/}{M}_n)$ and $b_0={\sum }_{n\ge 0}{c}_n{\rho }_n$ one has $b_0>0$ and $b_0\le q$, while the fields $V_n=c_n{\rho }_{n}W$ satisfy $\mathrm{\mid }{V}_n(z)\mathrm{\mid }\le 2^{-n-1}\mathrm{dist}(z,\mathrm{\Gamma })$ and $\mathrm{\parallel }D{V}_n(z)\mathrm{\parallel }\le 2^{-n-1}{d}_n$ on $B_n$ and vanish elsewhere, because $c_n{M}_n\le 2^{-n-1}{d}_n$ and points of $B_n$ have distance to $\mathrm{\Gamma }$ at least $d_n$; near $\mathrm{\Gamma }$ only indices $n\ge N$ contribute for $N$ arbitrarily large, so $V=b_{0}W$ satisfies $\mathrm{\mid }V(z)\mathrm{\mid }\le 2^{-N}\mathrm{dist}(z,\mathrm{\Gamma })$ and $\mathrm{\parallel }DV(z)\mathrm{\parallel }\le 2^{-N}{\sup }_{n\ge N}{d}_n$, giving $V=o(\mathrm{dist}(z,\mathrm{\Gamma }))$ and $DV\to 0$ at $\mathrm{\Gamma }$; therefore the extension of $V$ by zero across $\mathrm{\Gamma }$ is $C^1$ with zero derivative there, and multiplying $b_0$ by one fixed smooth cutoff flat at $s_1$, positive for $s>s_1$ and equal to one near $1$, produces a $C^1$ function $b$ with $0\le b\le q$ that extends the drift by zero across the inner edge.

Define $Y=X+b(s)W$ on the outer subannulus $\{s>s_1\}$ and $Y=X$ elsewhere on a neighborhood of $D$; since $b$ is a $C^1$ function of the $C^2$ leaf coordinate $s$ and $W$ is $C^1$, the field $Y$ is $C^1$, agrees with $X$ off the outer subannulus and on $\mathrm{\Gamma }$, and in product coordinates reads $Y=a(s,\theta ){\mathrm{\partial }}_{\theta }+b(s){\mathrm{\partial }}_s$ with $a>0$ and $b\ge 0$, so no new zero is created in the drift region.

Let $y$ be the maximal $Y$-trajectory starting at $\mathrm{\Psi }(0,s_0)$ with $s_1<s_0<1$: along it $\dot{s}=b(s)$ and $\dot{\theta }=a(s,\theta )$, so $s$ increases strictly, $ds\mathrm{/}d\theta \le (1-s{)}^2\mathrm{/}(1+L(s))$, and ${\int }_{s_2}^{s}du\mathrm{/}b(u)\ge {\int }_{s_2}^{s}du\mathrm{/}(1-u{)}^2\to \mathrm{\infty }$ as $s\to 1$, so $s$ tends to $1$ only at infinite time with $\theta (s)-\theta (s_2)\ge {\int }_{s_2}^s(1+L(u))\mathrm{/}(1-u{)}^{2}du\to \mathrm{\infty }$; during each full phase turn starting at parameter $s_k$ the parameter increases by at most one fixed normalization constant times $(1-s_k{)}^2$, and ${\int }_{s_k}^{s}L(u)du=\int L(s(\theta ))(ds\mathrm{/}d\theta )d\theta \le (1-s_k{)}^2$ up to the same constant, so the corresponding fixed-phase ambient displacement from the leaf $C_{s_k}$ is bounded by that integral and every complete turn stays uniformly within that distance of the whole reference circle, while every phase is visited during the turn; as $k\to \mathrm{\infty }$ the leaves $C_{s_k}$ converge to $\mathrm{\Gamma }$ in Hausdorff distance by step 1.1, so every point of $\mathrm{\Gamma }$ is a limit of the orbit and the orbit tail approaches $\mathrm{\Gamma }$, giving ${\omega }_Y^{+}(y)=\mathrm{\Gamma }$; the orbit remains in the compact set $\mathrm{cl}\mathrm{\Omega }$, never meets $\mathrm{\Gamma }$ because $C_s\cap \mathrm{\Gamma }=\mathrm{\varnothing }$ for all $s<1$, and is defined for all positive times by [F3].

Consequently $Y$ is a $C^1$ field on a neighborhood of $D$ that equals $X$ on $\mathrm{\Gamma }$ and off the outer subannulus and has the positive orbit $y$ with ${\omega }_Y^{+}(y)=\mathrm{\Gamma }$; every selection in the construction was an explicit band function or a uniquely determined extremum of a continuous function on a compact set, so no choice principle is used.