Non-trapping and geometric control from infinity

Put $Z=\bar{\mathrm{\Omega }}$ with the Euclidean metric and let $\mathcal{F}$ be the family of forward spatial restrictions of all complete unit-speed generalized rays whose initial bases lie in ${\bar{B}}_R$. For each member of any countable sequence in $\mathcal{F}$, Countable Choice in F1 selects one normalized complete compressed lift. Fix an integer $a\ge 1$. Apply the finite-interval compactness in F2 to these lifts on $[0,a]$. Their values lie in a compact normalized phase set over the closed base bound $\mathrm{\mid }x\mathrm{\mid }\le R+a$. The continuous base projection is uniformly continuous by F8, so the resulting compressed-phase subsequence convergence implies uniform spatial convergence on $[0,a]$. This works for every sequence and every subsequence in $\mathcal{F}$, verifying the first condition of the abstract hitting lemma with this spatial metric.

Now suppose $(x_j)$ in $\mathcal{F}$ has a locally uniform forward spatial limit $x$; choose complete normalized lifts $q_j$ as in step 1.1. For every integer $a\ge 1$, F8 applied on the interval from $-a$ to $0$ gives $\mathrm{\mid }{x}_j(-a)\mathrm{\mid }\le R+a$. Translate the restrictions $q_j{\mathrm{\mid }}_{[-a,a]}$ to $[0,2a]$ by stationarity. The compactness supplier in F2 applies with initial radius $R+a$ and length $2a$; its compact target may be taken over $\mathrm{\mid }x\mathrm{\mid }\le R+3a$. Dependent Choice successively selects nested infinite subsequences, converging in compressed phase on $[-1,1]$, then on $[-2,2]$, and so on, with admissible finite-interval limits $p_a$. On $[-a,a]$ the sequence selected at stage $a+1$ is a subsequence of the one selected at stage $a$. Both phase limits can be compared in the larger compact phase set, since the topology is the intrinsic compressed topology. Uniqueness of metric limits in F8 therefore gives $p_{a+1}{\mathrm{\mid }}_{[-a,a]}=p_a$. Equivalently, choosing successive original indices from these nested subsequences, each larger than the previous index and the stage number, gives one diagonal sequence converging on every centered finite interval. Define $q(t)=p_a(t)$ for any $a>\mathrm{\mid }t\mathrm{\mid }$; consistency makes this independent of $a$. Each point has a neighborhood on which $q$ is an admissible finite ray segment, so $q$ is continuous and satisfies the local compressed ray rules of F1 for all $t\in \mathbb{R}$. It is a normalized complete admissible ray, without selecting a unique continuation through any initial covector. Its spatial projection on $[0,a]$ is $x$: that projection is the uniform limit of the same subsequence of $x_j$, and $x_j\to x$ there. Thus every locally uniform forward limit considered by the hitting lemma is the forward restriction of a complete ray and belongs to $\mathcal{F}$. This proves the second condition without assuming an extension theorem for an arbitrary forward curve.

Apply the hitting lemma in F2 with $Z=\bar{\mathrm{\Omega }}$, $B=Z$, $\pi$ the identity, $\mathcal{A}=\mathcal{F}$ and the relatively open set $U=\{x\in Z:\mathrm{\mid }x\mathrm{\mid }>R\}$. Steps 1.1 and 2.1 verify its two compactness and admissibility conditions. Non-trapping in F1 applies to the complete ray constructed in step 2.1, so its forward restriction must hit $U$ at a finite time. The lemma gives a common finite bound $L$, which may be enlarged to be nonnegative, for all members of $\mathcal{F}$. Set $L_R=L+1$. An initial point in ${\bar{B}}_R$ is not in $U$, so its hit occurs at a time in $(0,L_R)$. If a complete ray starts outside ${\bar{B}}_R$, spatial continuity gives a hit of $U$ at a sufficiently small positive time below $L_R$. Hence every complete ray has the required hit and GCC holds.

By F3, after an outward exit from a ball containing the obstacle a complete ray follows a Euclidean straight line tending to infinity and cannot return. Conversely, suppose that the uniform GCC property holds for every sufficiently large containing ball, without assuming non-trapping for this implication. Choose any such ball. Its hitting bound applies in particular to every complete ray initially in its closure, so the no-return conclusion in F3 makes all those rays tend to spatial infinity. The same supplier treats complete rays initially outside that ball and makes them eventually tend to infinity as well. Every compact spatial set lies inside a finite-radius ball; eventual divergence therefore gives the required pointwise non-trapping predicate for every complete ray. Together with step 3.1 this establishes the stated geometric equivalence.

Apply F5 to the controlled systems and endpoint classes specified in F4, and F6 to each fixed cubic trajectory satisfying both (W) and (UCPw). Each result quantifies over every $T>0$ within its own hypotheses: arbitrary linear endpoints, sufficiently small nonlinear endpoints near zero, or sufficiently close trajectory endpoints, respectively. Their conclusions impose no positive lower bound $L_R$ on $T$. This application uses those control suppliers; the ray-hitting argument alone supplies no observability inequality.

The geometry, Dirichlet realization, cutoff and orders in F7 meet its independent estimate suppliers' hypotheses, so the stated cutoff resolvent and local smoothing estimates are available. Their role as the article's analytic geometric input is attributed in the Source notes. They are separate analytic premises: steps 1.1–4.1 establish the geometric statement, and step 4.2 records the control assertions with their own supplied arguments. This completes all assertions with their respective hypotheses retained.