On the Existence and Stability of an Infinite-Dimensional Invariant Torus

We consider an annular set of the form , where is a closed ball of the Banach space , is the infinite-dimensional torus (the direct product of a countable number of circles with the topology of coordinatewise uniform convergence). For a certain class of smooth maps , we establish sufficient conditio...

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Bibliographic Details
Published in:Mathematical Notes 2021-03, Vol.109 (3-4), p.534-550
Main Authors: Glyzin, S. D., Kolesov, A. Yu, Rozov, N. Kh
Format: Article
Language:English
Subjects:
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Banach spaces
Closed balls
Continuity (mathematics)
Dimensional stability
Invariants
Manifolds (mathematics)
Mathematics
Mathematics and Statistics
Smoothness
Topology
Toruses
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Summary:We consider an annular set of the form , where is a closed ball of the Banach space , is the infinite-dimensional torus (the direct product of a countable number of circles with the topology of coordinatewise uniform convergence). For a certain class of smooth maps , we establish sufficient conditions for the existence and stability of an invariant toroidal manifold of the form where is a continuous function of the argument . We also study the question of the -smoothness of this manifold for any natural .
ISSN:0001-4346
1067-9073
1573-8876
DOI:10.1134/S0001434621030226