= Asymptotic-pair obstruction to an invariant metric
If distinct points $x,y$ of a <dynamical system> on a compact <metric space> satisfy $d(T^nx,T^ny)\to0$ in one <compatible metric>, no <compatible metric> can make $T$ an <isometry>. On a compact space, all <compatible metrics> give the same asymptotic-pair property by <uniform continuity>, whereas an <isometry> preserves the strictly positive distance between distinct points. In a <full shift>, a constant <sequence> and a <sequence> differing at just one coordinate converge to each other under forward shifts, proving this obstruction directly in the <product topology>.
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