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Siberian Mathematical Journal

, Volume 59, Issue 6, pp 1090–1093 | Cite as

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

  • V. V. KozlovEmail author
Article
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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.

Keywords

Monge–Kantorovich problem smooth endomorphisms small denominators 

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References

  1. 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).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bogachev V. I., Fundamentals of Measure Theory. Vol. 2 [Russian], Nauchno-Issled. Tsentr “Regulyarnaya i Khaoticheskaya Dinamika,” Moscow and Izhevsk (2003).Google Scholar
  3. 3.
    Moser J., “On the volume elements on a manifold,” Trans. Amer. Math. Soc., vol. 120, No. 2, 286–294 (1965).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Banyaga A., “Formes-volume sur les variétés `a bord,” Enseign. Math., vol. 20, No. 2, 127–131 (1974).zbMATHGoogle Scholar
  5. 5.
    Greene R. E. and Shiohama K., “Diffeomorphisms and volume-preserving embeddings of noncompact manifolds,” Trans. Amer. Math. Soc., vol. 255, 403–414 (1979).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 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).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 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).MathSciNetCrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Steklov Institute of MathematicsMoscowRussia

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