Abstract
We consider an annular set of the form \(K=B\times \mathbb{T}^{\infty}\), where \(B\) is a closed ball of the Banach space \(E\), \(\mathbb{T}^{\infty}\) 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 \(\Pi\colon K\to K\), we establish sufficient conditions for the existence and stability of an invariant toroidal manifold of the form
where \(h(\varphi)\) is a continuous function of the argument \(\varphi\in\mathbb{T}^{\infty}\). We also study the question of the \(C^m\)-smoothness of this manifold for any natural \(m\).
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Translated from Matematicheskie Zametki, 2021, Vol. 109, pp. 508-528 https://doi.org/10.4213/mzm12912.
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Glyzin, S.D., Kolesov, A.Y. & Rozov, N.K. On the Existence and Stability of an Infinite-Dimensional Invariant Torus. Math Notes 109, 534–550 (2021). https://doi.org/10.1134/S0001434621030226
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DOI: https://doi.org/10.1134/S0001434621030226