Hyperbolic return-map stability is orbital stability

The periodic solution over $[0,2T+1]$ is a compact trajectory segment by [F6]–[F7]. Cover it by finitely many neighborhoods on which the local flow in [F2] exists, subdivide $[0,2T+1]$ into finitely many pieces contained in those flow times, and compose the corresponding local flows. After shrinking the initial-state neighborhood, this gives a jointly $C^1$ map $(t,y)\mapsto {\varphi }_t(y)$ for $0\le t\le 2T+1$ and $y$ near $a$. Set $\mathrm{\Psi }(t,y)=\rho ({\varphi }_t(y))$. At $(T,a)$, $\mathrm{\Psi }(T,a)=0$ and ${\mathrm{\partial }}_t\mathrm{\Psi }(T,a)=D\rho (a)f(a)\mathrm{\ne }0$. Apply [F3] with $k=1$ to obtain a $C^1$ return time $\tau (y)$ near $T$ and hence a $C^1$ return map $P(y)={\varphi }_{\tau (y)}(y)$ on a section neighborhood.

For each $s\in [0,T]$, put $t_s=2T-s$. Then ${\varphi }_{t_s}(x_{\ast }(s))=a$ and $T\le t_s\le 2T$. Cover the compact segment from $x_{\ast }(s)$ to $a$ by finitely many flow neighborhoods from [F2] and compose them; this defines the flow near $(t_s,x_{\ast }(s))$. Apply [F3] to $\mathrm{\Psi }(t,x)=\rho ({\varphi }_t(x))$ there; its time derivative is $D\rho (a)f(a)\mathrm{\ne }0$. This gives a $C^1$ positive hitting time near each $x_{\ast }(s)$, bounded by $2T+1$, whose flow endpoint lies on $\mathrm{\Sigma }$ near $a$. Compactness of $\mathrm{\Gamma }$ from [F6]–[F7] gives finitely many such neighborhoods and a uniform neighborhood of $\mathrm{\Gamma }$ on which this forward-to-section property holds. Continuity of the finitely composed flow on compact time subintervals also makes each initial segment, up to its hitting time, uniformly close to the corresponding segment of $\mathrm{\Gamma }$ when the initial point is close to $\mathrm{\Gamma }$.

Suppose all eigenvalues of $DP(a)$ lie inside the unit circle. By [F4], after shrinking the section neighborhood, every return iterate of a sufficiently close section point stays in that neighborhood and converges to $a$. The return times are continuous and bounded below by $T\mathrm{/}2$ after another shrink. The flow is uniformly continuous on the compact time-section product by [F6] and [F8], so the orbit segment from each return point approaches one traversal of $\mathrm{\Gamma }$ uniformly as the return point tends to $a$. Since the return points converge to $a$, distance to $\mathrm{\Gamma }$ tends to zero along the full forward trajectory.

For Lyapunov stability, fix $\epsilon >0$. For a relatively compact section patch $\bar{V}$, the product $[0,T+1]\times \bar{V}$ is closed and bounded in Euclidean space and hence compact by [F6]. The flow on this set is continuous by [F2] and the finite-composition construction in step 1.1, so [F8] makes it uniformly continuous. Shrink $V$ so every orbit segment starting in $V$ stays within $\epsilon$ of $\mathrm{\Gamma }$ until its next return. Fixed-point stability in [F4] gives a smaller neighborhood whose return iterates all remain in $V$. Step 2.1 then supplies a neighborhood of $\mathrm{\Gamma }$ whose points first reach the section inside that smaller neighborhood while their initial segments stay within $\epsilon$ of $\mathrm{\Gamma }$. Thus all forward trajectories from this neighborhood remain within $\epsilon$ of $\mathrm{\Gamma }$. Together with step 3.1 this proves orbital asymptotic stability.

Suppose instead that $DP(a)$ has an eigenvalue outside the unit circle. By [F4], there is a fixed section neighborhood $V$ and section points arbitrarily close to $a$ whose return iterates leave $V$. Choose a relatively compact local section patch $W$ with $\bar{W}\cap \mathrm{\Gamma }=\{a\}$ and then choose $V$ with $\bar{V}\subset W$ and $P(V)\subset W$. The set $K=\bar{W}\setminus V$ is closed, bounded, and nonempty, hence compact by [F6]; it is disjoint from compact $\mathrm{\Gamma }$ by construction. For each $x\in K$, compactness of $\mathrm{\Gamma }$ gives a closest point, so $d_{\mathrm{\Gamma }}(x):=\mathrm{dist}(x,\mathrm{\Gamma })>0$. The triangle inequality gives $\mathrm{\mid }{d}_{\mathrm{\Gamma }}(x)-d_{\mathrm{\Gamma }}(y)\mathrm{\mid }\le \mathrm{\parallel }x-y\mathrm{\parallel }$, so $d_{\mathrm{\Gamma }}$ is continuous; [F6] gives it a positive minimum on $K$. Therefore the first return point outside $V$ lies a fixed positive distance from $\mathrm{\Gamma }$. Such initial points can be chosen arbitrarily close to $a\in \mathrm{\Gamma }$, so $\mathrm{\Gamma }$ is not Lyapunov stable as a set. It is orbitally unstable.