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Examples of $2$-unrectifiable normal currents

We construct new examples of normal (metric) currents using inverse systems of cube complexes. For any $N\ge 2$ we provide examples of $N$-dimensional normal currents whose associated vector fields are simple, and whose supports are purely $2$-unrectifiable and have Nagata dimension $N$. We show that in $l^\infty$ normal currents can be realized as limits in the flat distance of currents associated to cube complexes.