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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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##### References:
 [1] Bogachev, V. I.; Kolesnikov, A. V., The Monge-Kantorovich problem: achievements, connections, and perspectives, Russian Math. Surveys, 67, 785-890, (2012) · Zbl 1276.28029 [2] Bogachev V. I., Fundamentals of Measure Theory. Vol. 2 [Russian], Nauchno-Issled. Tsentr “Regulyarnaya i Khaoticheskaya Dinamika,” Moscow and Izhevsk (2003). [3] Moser, J., On the volume elements on a manifold, Trans. Amer. Math. Soc., 120, 286-294, (1965) · Zbl 0141.19407 [4] Banyaga, A., Formes-volume sur les variétés ‘a bord, Enseign. Math., 20, 127-131, (1974) · Zbl 0281.58001 [5] Greene, R. E.; Shiohama, K., Diffeomorphisms and volume-preserving embeddings of noncompact manifolds, Trans. Amer. Math. Soc., 255, 403-414, (1979) · Zbl 0418.58002 [6] Yagasaki, T., Groups of volume-preserving diffeomorphisms of noncompact manifolds and mass flow toward ends, Trans. Amer. Math. Soc., 362, 5745-5770, (2010) · Zbl 1206.57041 [7] Arnold, V. I., Small denominators and problems of stability of motion in classical and celestial mechanics, Russian Math. Surveys, 18, 85-191, (1963) · Zbl 0135.42701
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