= Swap lift of a contraction
{title2=$F(x,y)=(y,f(x)),\quad F^2(x,y)=(f(x),f(y))$}
On the <maximum product metric>, the displayed map is nonexpansive when $f$ is a <contraction mapping>. If the space has two distinct points, changing only the second input coordinate shows that one step preserves some nonzero distance exactly, so it is not itself a strict contraction. Two iterations apply $f$ to both coordinates and therefore contract with the same constant as $f$. This provides a direct example of a map whose iterate is contractive although the map is not.
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