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Hidden toric symmetry and structural stability of singularities in integrable systems

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Abstract

The goal of the paper is to develop a systematic approach to the study of (possibly degenerate) singularities of integrable systems and their structural stability. Following Nguyen Tien Zung, as the main tool, we use “hidden” system-preserving torus actions near singular orbits. We give sufficient conditions for the existence of such actions and show that they are persistent under integrable perturbations. We find toric symmetries for several infinite series of singularities and prove, as an application, structural stability of Kalashnikov’s parabolic orbits with resonances in the real-analytic case. We also classify all Hamiltonian k-torus actions near a singular orbit on a symplectic manifold \(M^{2n}\) (or on its complexification) and prove that the normal forms of these actions are persistent under small perturbations. As a by-product, we prove an equivariant version of the Vey theorem (Amer J Math 100(3):591–614, 1978) about local symplectic normal form of nondegenerate singularities.

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Notes

  1. In practice, one can verify condition (ii) of Theorems  2.1 and 2.2 using Lemma 2.3.

  2. Indeed, it follows from Theorem 3.10 that the action of generators \(\psi ^a :=p(\gamma ^a)\) of \(\Gamma \) on V has the form

    for some . Here is a generating set of the lattice \(p^{-1}(\Gamma )\subset {\mathbb {R}}^r\), \(p:{\mathbb {R}}^r\rightarrow (S^1)^r\) denotes the projection. Since \(p^{-1}(\Gamma )\) is a lattice in \({\mathbb {R}}^r\), there exists a unique linear map \({\mathbb {R}}^r\rightarrow {\mathbb {R}}^{n-r}\) sending , \(1\leqslant a\leqslant r\). Clearly, this linear map has the form \(\gamma \mapsto (\langle m_1,\gamma \rangle ,\dots ,\langle m_{n-r},\gamma \rangle )\), \(\gamma \in {\mathbb {R}}^r\), for some \(m_1,\dots ,m_{n-r}\in {\mathbb {R}}^r\). From the short exact sequence \(0\rightarrow 2\pi {{\mathbb {Z}}}^r\rightarrow p^{-1}(\Gamma )\rightarrow \Gamma \rightarrow 0\), we conclude that , provided that \(\gamma \in 2\pi {{\mathbb {Z}}}^r\). Therefore . This proves (11).

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Acknowledgements

The author is grateful to Alexey Bolsinov for helpful comments on Cartan subalgebras of the Lie algebra and valuable suggestions on a preliminary version of the paper, to Andrey Oshemkov for the useful discussion on proving extendability of homomorphisms to a circle from a finite subgroup of a torus (cf. (11)), to Stefan Nemirovski for helpful comments on topologies on the spaces of analytic functions, and to the referees for useful comments which helped to improve the paper.

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  1. Faculty of Mechanics and Mathematics, Moscow State University, Moscow, Russia, 119991

    Elena A. Kudryavtseva

  2. Moscow Center for Fundamental and Applied Mathematics, Moscow, Russia

    Elena A. Kudryavtseva

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Kudryavtseva, E.A. Hidden toric symmetry and structural stability of singularities in integrable systems. European Journal of Mathematics 8, 1487–1549 (2022). https://doi.org/10.1007/s40879-021-00501-9

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