Swap lift of a contraction ($F(x,y)=(y,f(x)),F^2(x,y)=(f(x),f(y))$)

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.