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Given $1\\leq m_1,m_2\\leq n-2$, we construct a homeomorphism $f :\\Omega\\to \\Omega$ that is H\\\"older continuous, $f$ is the identity on $\\partial \\Omega$, the derivative $D f$ has rank $m_1$ a.e.\\ in $\\Omega$, the derivative $D f^{-1}$ of the inverse has rank $m_2$ a.e.\\ in $\\Omega$, $Df\\in W^{1,p}$ and $Df^{-1}\\in W^{1,q}$ for $p<\\min\\{m_1+1,n-m_2\\}$, $q<\\min\\{m_2+1,n-m_1\\}$. The proof is based on convex integration and laminates. 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Given $1\\leq m_1,m_2\\leq n-2$, we construct a homeomorphism $f :\\Omega\\to \\Omega$ that is H\\\"older continuous, $f$ is the identity on $\\partial \\Omega$, the derivative $D f$ has rank $m_1$ a.e.\\ in $\\Omega$, the derivative $D f^{-1}$ of the inverse has rank $m_2$ a.e.\\ in $\\Omega$, $Df\\in W^{1,p}$ and $Df^{-1}\\in W^{1,q}$ for $p<\\min\\{m_1+1,n-m_2\\}$, $q<\\min\\{m_2+1,n-m_1\\}$. The proof is based on convex integration and laminates. 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