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We construct an a.e. approximately differentiable homeomorphism of a unit$n$-dimensional cube onto itself which is orientation preserving, has the Lusinproperty (N) and has the Jacobian determinant negative a.e. Moreover, thehomeomorphism together with its inverse satisfy a rather general sub-Lipschitzcondition, in particular it can be bi-H\"older continuous with an arbitraryexponent less than $1$.

Modulus of continuity of orientation preserving approximately differentiable homeomorphisms with a.e. negative Jacobian