The Monge problem of ``piles and holes'' on the torus and the problem of small denominators
type: Item
instance of: scholarly article
title: The Monge problem of ``piles and holes'' on the torus and the problem of small denominators
zbMATH Open document ID: 1412.57024
DOI: 10.1134/S0037446618060113
author: Xianqiang Yang
published in: Siberian Mathematical Journal
publication date: 2019-03-13T00:00:00Z
review text: 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 [\textit{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)].
reviewed by: Alexander Schmeding
Mathematics Subject Classification ID: 57R50
Mathematics Subject Classification ID: 57R35
zbMATH DE Number: 7035752
zbMATH Keywords: Monge-Kantorovich problem
zbMATH Keywords: smooth endomorphisms
zbMATH Keywords: small denominators
MaRDI profile type: Publication
cites work: The Monge-Kantorovich problem: achievements, connections, and perspectives
cites work: On the Volume Elements on a Manifold
cites work: Formes-volume sur les variétés à bord
cites work: Diffeomorphisms and Volume-Preserving Embeddings of Noncompact Manifolds
cites work: Groups of volume-preserving diffeomorphisms of noncompact manifolds and mass flow toward ends
cites work: SMALL DENOMINATORS AND PROBLEMS OF STABILITY OF MOTION IN CLASSICAL AND CELESTIAL MECHANICS
full work available at URL: s0037446618060113
OpenAlex ID: W2906122341
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description: scientific article; zbMATH DE number 7035752