# Generalization of one Poletskii lemma to classes of space mappings

## Authors

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- Received:

DOI: 10.1007/s11253-009-0267-0

- Cite this article as:
- Sevost’yanov, E.A. Ukr Math J (2009) 61: 1151. doi:10.1007/s11253-009-0267-0

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* → ℝ^{n}, *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.