The paper is devoted to investigations in the field of space mappings. We prove that open discrete mappings f ∈ W 1,n loc such that their outer dilatation K O (x, f) belongs to L n−1 loc and the measure of the set B f of branching points of f is equal to zero have finite length distortion. In other words, the images of almost all curves γ in the domain D under the considered mappings f : D → ℝn, n ≥ 2, are locally rectifiable, f possesses the (N)-property with respect to length on γ, and, furthermore, the (N)-property also holds in the inverse direction for liftings of curves. The results obtained generalize the well-known Poletskii lemma proved for quasiregular mappings.
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 61, No. 7, pp. 969–975, July, 2009.
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Sevost’yanov, E.A. Generalization of one Poletskii lemma to classes of space mappings. Ukr Math J 61, 1151–1157 (2009). https://doi.org/10.1007/s11253-009-0267-0
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DOI: https://doi.org/10.1007/s11253-009-0267-0