Kozlov, V. V. 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)]. Reviewer: Alexander Schmeding (Berlin) MSC: 57R50 Differential topological aspects of diffeomorphisms 57R35 Differentiable mappings in differential topology Keywords:Monge-Kantorovich problem; smooth endomorphisms; small denominators Citations:Zbl 0135.42701 PDFBibTeX XMLCite \textit{V. V. Kozlov}, Sib. Math. J. 59, No. 6, 1090--1093 (2018; Zbl 1412.57024); translation from Sib. Mat. Zh. 59, No. 6, 1370--1374 (2018) Full Text: DOI References: [1] Bogachev V. I. and Kolesnikov A. V., “The Monge-Kantorovich problem: achievements, connections, and perspectives,” Russian Math. Surveys, vol. 67, No. 5, 785-890 (2012). · Zbl 1276.28029 · doi:10.1070/RM2012v067n05ABEH004808 [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., vol. 120, No. 2, 286-294 (1965). · Zbl 0141.19407 · doi:10.1090/S0002-9947-1965-0182927-5 [4] Banyaga A., “Formes-volume sur les variétés ‘a bord,” Enseign. Math., vol. 20, No. 2, 127-131 (1974). · Zbl 0281.58001 [5] Greene R. E. and Shiohama K., “Diffeomorphisms and volume-preserving embeddings of noncompact manifolds,” Trans. Amer. Math. Soc., vol. 255, 403-414 (1979). · Zbl 0418.58002 · doi:10.1090/S0002-9947-1979-0542888-3 [6] Yagasaki T., “Groups of volume-preserving diffeomorphisms of noncompact manifolds and mass flow toward ends,” Trans. Amer. Math. Soc., vol. 362, No. 11, 5745-5770 (2010). · Zbl 1206.57041 · doi:10.1090/S0002-9947-2010-05101-3 [7] Arnold V. I., “Small denominators and problems of stability of motion in classical and celestial mechanics,” Russian Math. Surveys, vol. 18, No. 6, 85-191 (1963). · Zbl 0135.42701 · doi:10.1070/RM1963v018n06ABEH001143 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.