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The Monge Problem of “Piles and Holes” on the Torus and the Problem of Small Denominators

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Abstract

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.

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Authors and Affiliations

  1. Steklov Institute of Mathematics, Moscow, Russia

    V. V. Kozlov

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  1. V. V. Kozlov
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Correspondence to V. V. Kozlov.

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Original Russian Text © 2018 Kozlov V.V.

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Moscow. Translated from Sibirskii Matematicheskii Zhurnal, vol. 59, no. 6, pp. 1370–1374, November–December, 2018; DOI: 10.17377/smzh.2018.59.611.

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Kozlov, V.V. The Monge Problem of “Piles and Holes” on the Torus and the Problem of Small Denominators. Sib Math J 59, 1090–1093 (2018). https://doi.org/10.1134/S0037446618060113

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  • DOI: https://doi.org/10.1134/S0037446618060113

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