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The Monge problem of “piles and holes” on the torus and the problem of small denominators. (English. Russian original) Zbl 1412.57024
Sib. Math. J. 59, No. 6, 1090-1093 (2018); translation from Sib. Mat. Zh. 59, No. 6, 1370-1374 (2018).
The article discusses the problem of existence of a smooth (analytic) map \(\varphi : M \rightarrow M\) of a closed \(n\)-dimensional manifold \(M\) carrying a differential \(n\)-form into a prescribed volume form (assuming that the integrals of these forms on the whole manifold coincide). For the \(n\)-dimensional torus it is then shown how the problem can be reduced to the problem of small denominators [V. I. Arnol’d, Russ. Math. Surv. 18, No. 6, 85–191 (1963; Zbl 0135.42701); translation from Usp. Mat. Nauk 18, No. 6(114), 91–192 (1963)].
MSC:
57R50 Differential topological aspects of diffeomorphisms
57R35 Differentiable mappings in differential topology
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