Abstract
We establish Écalle’s mould calculus in an abstract Lie-theoretic setting and use it to solve a normalization problem, which covers several formal normal form problems in the theory of dynamical systems. The mould formalism allows us to reduce the Lie-theoretic problem to a mould equation, the solutions of which are remarkably explicit and can be fully described by means of a gauge transformation group. The dynamical applications include the construction of Poincaré–Dulac formal normal forms for a vector field around an equilibrium point, a formal infinite-order multiphase averaging procedure for vector fields with fast angular variables (Hamiltonian or not), or the construction of Birkhoff normal forms both in classical and quantum situations. As a by-product we obtain, in the case of harmonic oscillators, the convergence of the quantum Birkhoff form to the classical one, without any Diophantine hypothesis on the frequencies of the unperturbed Hamiltonians.
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Arnol’d, V. I., Kozlov, V.V., and Neǐshtadt, A. I., Mathematical Aspects of Classical and Celestial Mechanics, 3rd ed., Encyclopaedia Math. Sci., vol. 3, Berlin: Springer, 2006.
Broer, H., Formal Normal Form Theorems for Vector Fields and Some Consequences for Bifurcations in the Volume Preserving Case, in Dynamical Systems and Turbulence, Warwick 1980 (Coventry, 1979/1980), Lecture Notes in Math., vol. 898, New York: Springer, 1981, pp. 54–74.
Broer, H., Normal Forms in Perturbation Theory, in Mathematics of Complexity and Dynamical Systems: Vols. 1–3, R.A. Meyers (Ed.), New York: Springer, 2012, pp. 1152–1171.
Esposti, M. D., Graffi, S., and Herczynski, J., Quantization of the Classical Lie Algorithm in the Bargmann Representation, Ann. Physics, 1991, vol. 209, no. 2, pp. 364–392.
Écalle, J., Les fonctions résurgentes: Vols. 1–3, Paris: Publ. Math. Orsay, 1981, 1985.
Écalle, J., Six Lectures on Transseries, Analysable Functions and the Constructive Proof of Dulac’s Conjecture, in Bifurcations and Periodic Orbits of Vector Fields, D. Schlomiuk (Ed.), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 408, Dordrecht: Kluwer, 1993, pp. 75–184.
Écalle, J. and Vallet, B., Prenormalization, Correction, and Linearization of Resonant Vector Fields or Diffeomorphisms: Prepub. Orsay, no. 32, 1995, 90 pp.
Folland, G. B., Harmonic Analysis in Phase Space, Ann. of Math. Stud., vol. 122, Princeton,N.J.: Princeton Univ. Press, 1989.
Graffi, S. and Paul, Th., The Schrödinger Equation and Canonical Perturbation Theory, Comm. Math. Phys., 1987, vol. 108, no. 1, pp. 25–40.
Kato, T., Perturbation Theory of Linear Operators, 2nd ed., Classics Math., vol. 132, Berlin: Springer, 1995.
Lochak, P. and Meunier, C., Multiphase Averaging for Classical Systems: With Applications to Adiabatic Theorems, Appl. Math. Sci., vol. 72, New York: Springer, 1988.
Menous, F., Formal Differential Equations and Renormalization, in Renormalization and Galois Theories, A. Connes, F. Fauvet, J.-P. Ramis (Eds.), IRMA Lect. Math. Theor. Phys., vol. 15, Zürich: Eur. Math. Soc., 2009, pp. 229–246.
Menous, F., From Dynamical Systems to Renormalization, J. Math. Phys., 2013, vol. 54, no. 9, 092702, 24 pp.
Marco, J.-P. and Sauzin, D., Stability and Instability for Gevrey Quasi-Convex Near-Integrable Hamiltonian Systems, Publ. Math. Inst. Hautes Études Sci., 2002, no. 96, pp. 199–275.
Murua, A. and Sanz-Serna, J.M., Word Series for Dynamical Systems and Their Numerical Integrators, Found. Comput. Math., 2017, vol. 17, no. 3, pp. 675–712.
Murua, A. and Sanz-Serna, J.M., Computing Normal Forms and Formal Invariants of Dynamical Systems by Means of Word Series, Nonlinear Anal., 2016, vol. 138, pp. 326–345.
Paul, T. and Sauzin, D., Normalization in Banach Scale Lie Algebras via Mould Calculus and Applications, arXiv:1607.00780v2 (2017).
Reed, M. and Simon, B., Methods of Modern Mathematical Physics: Vol. 1. Functional Analysis, New York: Acad. Press, 1972.
Reed, M. and Simon, B., Methods of Modern Mathematical Physics: Vol. 3. Scattering Theory, New York: Acad. Press, 1979.
Reutenauer, Ch., Free Lie Algebras, London Math. Soc. Monogr. (N. S.), vol. 7, New York: Oxford Univ. Press, 1993.
Sauzin, D., Mould Expansions for the Saddle-Node and Resurgence Monomials, in Renormalization and Galois Theories, A. Connes, F. Fauvet, J.-P. Ramis (Eds.), IRMA Lect. Math. Theor. Phys., vol. 15, Zürich: Eur. Math. Soc., 2009, pp. 83–163.
Thibon, J.-Y., Noncommutative Symmetric Functions and Combinatorial Hopf Algebras, in Asymptotics in Dynamics, Geometry and PDEs; Generalized Borel Summation: Vol. 1, O. Costin, F. Fauvet, F. Menous, D. Sauzin (Eds.), CRM Ser., vol. 12, Pisa: Ed. Norm., 2011, pp. 219–258.
von Waldenfels, W., Zur Charakterisierung Liescher Elemente in freien Algebren, Arch. Math. (Basel), 1966, vol. 17, pp. 44–48.
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Paul, T., Sauzin, D. Normalization in Lie algebras via mould calculus and applications. Regul. Chaot. Dyn. 22, 616–649 (2017). https://doi.org/10.1134/S1560354717060041
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DOI: https://doi.org/10.1134/S1560354717060041