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Maximal Tori in Infinite-Dimensional Hamiltonian Systems: a Renormalisation Group Approach

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Abstract

We study the existence of infinite-dimensional invariant tori in a mechanical system of infinitely many rotators weakly interacting with each other. We consider explicitly interactions depending only on the angles, with the aim of discussing in a simple case the analyticity properties to be required on the perturbation of the integrable system in order to ensure the persistence of a large measure set of invariant tori with finite energy. The proof we provide of the persistence of the invariant tori implements the renormalisation group scheme based on the tree formalism, i. e., the graphical representation of the solutions of the equations of motion in terms of trees, which has been widely used in finite-dimensional problems. The method is very effectual and flexible: it naturally extends, once the functional setting has been fixed, to the infinite-dimensional case with only minor technical-natured adaptations.

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Notes

  1. Here and in the following \(\|\cdot\|_{p}\) denotes as usual the \(\ell^{p}\) norm, while \(\|\cdot\|\) is the sup-norm.

  2. We mention also that, in the infinite-dimensional case, Berti and Biasco provided a super-exponential bound of the maximum size of the perturbation allowed for the persistence of finite-dimensional tori in terms of the dimensions of the tori [3]. The bound was slightly improved subsequently by Li and Liu [35].

  3. The subscript \(f\) stands for “finite support”, and it has nothing to do with the perturbation \(f\) we introduce in (2.4).

  4. Here and in the following \(\langle j\rangle=\max\{1,|j|\}\).

  5. In fact, the topology induced by the metric (2.2) makes \(\mathbb{T}^{\mathbb{Z}}\) a Banach manifold modeled on \(\ell^{\infty}(\mathbb{R})\).

  6. This is the model which is studied by Chierchia and Perfetti [16].

  7. In Lemma 4 we prove only \(s_{1}<s\) due to the fact that, merely to simplify the exposition, we arbitrarily fix the loss of regularity to be \(s\).

  8. This is an analytic change of variables provided one imposes some lower bounds on the sequence \(\{I_{j}\}_{j\in\mathbb{Z}}\) and assumes that \(u=\{u_{j}\}_{j\in\mathbb{Z}}\) lives in an \(\ell^{\infty}\)-based Banach space.

  9. The name is borrowed from quantum field theory [26, 28].

  10. Also called self-energy cluster or self-energy graph in the literature, again by analogy with quantum field theory.

  11. Given two Banach spaces \(X\) and \(Y\), \({\mathcal{L}}(X,Y)\) denotes the space of continuous linear operators \(L:X\to Y\).

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ACKNOWLEDGMENTS

We thank Luca Biasco, Ruben Bottini, Roberto Feola, Emanuele Haus and Jessica Elisa Massetti for useful discussions and comments.

Funding

L. C. has been supported by the research projects PRIN 2020XBFL “Hamiltonian and dispersive PDEs” and PRIN 2022HSSYPN “Turbulent Effects vs Stability in Equations from Oceanography” (TESEO) of the Italian Ministry of Education and Research (MIUR). G. G. has been supported by the research project PRIN 20223J85K3 “Mathematical Interacting Quantum Fields” of the Italian Ministry of Education and Research (MIUR). M. P. has been supported by the research projects PRIN 2020XBFL “Hamiltonian and Dispersive PDEs” and PRIN 2022FPZEES “Stability in Hamiltonian Dynamics and beyond” of the Italian Ministry of Education and Research (MIUR).

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  1. Dipartimento di Matematica e Fisica, Università Roma Tre, 00146, Roma, Italy

    Livia Corsi, Guido Gentile & Michela Procesi

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  1. Livia Corsi
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Correspondence to Livia Corsi, Guido Gentile or Michela Procesi.

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Corsi, L., Gentile, G. & Procesi, M. Maximal Tori in Infinite-Dimensional Hamiltonian Systems: a Renormalisation Group Approach. Regul. Chaot. Dyn. (2024). https://doi.org/10.1134/S1560354724540025

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