Squeezing the Sierpiński sponge (Q2773415)

A mapping \(f\) from a domain \(\Omega\) of \(\mathbb{R}^n\) into \(\mathbb{R}^n\) is of finite distortion if \(f\in W^{1,1}_{\text{loc}}(\Omega)\), the Jacobian determinant \(J(x,f)\) of \(f\) is non-negative and locally integrable and \(|f'(x)|^n\leq K(x)J(x,f)\) for some function \(K(x)\geq 1\) finite a.e. If \(K\in L^\infty\), then \(f\) is quasiregular and \(f\) satisfies the Lusin condition \((N)\). If \(f\) is a homeomorphism and of the class \(W^{1,n}_{\text{loc}}(\Omega)\), then \(f\) again satisfies the condition \((N)\),see [\textit{J. Malij} and \textit{O. Martio}, Lusin's condition \((N)\) and mappings of the class \(W^{1,n}\),J. Reine Angew. Math. 485, 19-36 (1995; Zbl 0812.30007)]. In general, finite distortion does not present cavitation and the authors exhibit an example of a homeomorphism \(f\) of a finite distortion such that \(f\in W^{1,p}\) for \(p<n\), \(K(n) \in L^p\) for \(p<n\) and \(f\) does not satisfy the condition \((N)\). Moreover, the inverse mapping \(f^{-1}\) of \(f\) is a Lipschitz map, and obtained by sequeezing a special Sierpinski sponge. The result is quite optimal.