Manuscript X-J-ST@1 · X-J-ST.tex · proposition · 8f0d1a36

Proposition: Transitive-orbit erasure bound

Selectivity, orbit bounds, and non-acquisition › The encoding-independent orbit statement · lines 1573–1591 · proof included or linked

X-J-ST@1#prop:orbit

Let $T : R \to D$ take values in a metric space $(D, d)$ . If

T (z + t) = T (z) (z, t \in R),

then $T$ is constant. If instead

z, t \in R sup d (T (z + t), T (z)) \leq η,

then the diameter of its image is at most $η$ .

Proof. For any $z, z^{'}$ , take $t = z^{'} - z$ .

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