Mappings of finite distortion: discreteness and openness for quasi-light mappings (Q2485739)

By a result of Yu. Reshetnyak a quasiregular mapping is either discrete and open or a constant. Several extensions of this result, weakening the quasiregular assumption, have been obtained [see \textit{T. Iwaniec} and \textit{G. Martin}, Geometric Function Theory and Nonlinear Analysis, Oxford Mathematical Monographs. Oxford: Oxford University Press (2001; Zbl 1045.30011)]. In this paper quasi-light (the inverse image of a point is compact) mappings of finite distortion (the dilatation coefficient \(K(x)\) finite a.e) are considered. In this class of mappings it is shown that a mapping is discrete and open provided that \(K(x)^{n-1}/(\phi (x)\log (e + K(x)))\) is in \(L^1\) for some function \(\phi\) with \(\int^{\infty}_1 1/\phi \,dt = \infty\). The last condition is sharp. For closely related results see Math. Ann. 324, No. 3, 451--464 (2002; Zbl 1017.30030).