The Monge Problem of “Piles and Holes” on the Torus and the Problem of Small Denominators
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We discuss the problem of existence of a smooth endomorphism of a closed n-dimensional manifold carrying a differential n-form into a prescribed volume form. Of course, we assume that the integrals of these forms over the whole manifold are equal. The solution of this problem for the n-dimensional torus reduces to the problem of small denominators well known in analysis.
KeywordsMonge–Kantorovich problem smooth endomorphisms small denominators
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- 2.Bogachev V. I., Fundamentals of Measure Theory. Vol. 2 [Russian], Nauchno-Issled. Tsentr “Regulyarnaya i Khaoticheskaya Dinamika,” Moscow and Izhevsk (2003).Google Scholar