Bi-Sobolev homeomorphism with zero Jacobian almost everywhere

Let $$N\ge 3$$ . We construct a homeomorphism $$f$$ in the Sobolev space $$W^{1,1}((0,1)^N,(0,1)^N)$$ such that $$f^{-1}\in W^{1,1}((0,1)^N,(0,1)^N)$$ , $$J_f=0$$ a.e. and $$J_{f^{-1}}=0$$ a.e. It follows that $$f$$ maps a set of full measure to a null set and a remaining null set to a set of full measure.We also show that such a pathological homeomorphism cannot exist in dimension $$N=2$$ or with higher regularity $$f\in W^{1,N-1}$$ .