A Rigidity Property of Perturbations of n Identical Harmonic Oscillators
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Villarini Massimo [1]
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[1] Universitá di Modena e Reggio Emilia
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- Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 19, Nº 3, 2020
- Idioma: inglés
- DOI: 10.1007/s12346-020-00426-w
- Enlaces
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Resumen
Let Xε:S2n-1→TS2n-1 be a smooth perturbation of X0, the vector field associated to the dynamical system defined by n identical uncoupled harmonic oscillators constrained to their 1-energy level. We are dealing with the case when any orbit of every Xε is closed: while in general is false that the vector fields of the perturbation are orbitally equivalent to the unperturbed X0 (Villarini in Ergod Theory Dyn Syst 39:1–32, 2019), we prove that this rigidity behaviour is indeed true if each Xε restricted to a codimension 2 sphere in S2n-1 is orbitally conjugated to a subsystem of X0 made by n-1 harmonic oscillators. In other words: to have a non-rigid, or truly non-linear, perturbation of X0 at least two harmonic oscillators must be destroyed by the perturbation. We use this rigidity result to prove a linearization theorem for real analytic multicentres. Finally we give an example of a real analytic perturbation of X0 showing discontinuous changing of integer invariants of the vector fields of the perturbation.
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Referencias bibliográficas
- 1. Brunella, M., Villarini, M.: On the Poincaré–Lyapunov centre theorem. Bol. Soc. Mat. Mex. (3) 5, 155–161 (1999)
- 2. Bruhat, F., Cartan, H.: Sur la structure des sous-ensembles anlytiques reels. C. R. Acad. Sci. Paris Ser. I Math. 244, 988–996 (1957)
- 3. Besse, A.: Manifolds All of Whose Geodesics are Closed, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 93. Springer, Berlin (1978)
- 4. Chern, S.: Circle bundles. In: Palis, J., do Carmo, M. (eds.) Geometry and Topology. LNM, vol. 597, pp. 114–131. Springer, Berlin (1977)
- 5. Dumortier, F.: Local study of planar vector fields. In: Broer, H.W., et al. (eds.) Structures in Dynamics. Finite Detrministic Studies....
- 6. Godbillon, C.: Feuilletages: Etude Geometriques, Progress in Mathematics, vol. 98. Birkhauser, Boston (1991)
- 7. Griffiths, P., Harris, J.: Principles of Algebraic Geometry. Wiley, New York (1978)
- 8. Montgomery, D., Zippin, L.: Topological Transformation Groups. Interscience, New York (1955)
- 9. Poincaré, H.: Sur les courbes definies par les equations differentielles. J. Math. Pures Appl. (4) 1 (1885) or Oeuvres, Tome I, Gauthier-Villars,...
- 10. Sullivan, D.: A counterexample to the periodic orbit conjecture. Publ. Math. IHES 46, 5–14 (1976)
- 11. Takens, F.: Singularities of vector fields. Publ. Math. IHES 43, 48–100 (1974)
- 12. Villarini, M.: On a rigidity property of perturbations of circle bundles on 3-manifolds. arXiv:1708.00718 [math.DS]
- 13. Villarini, M.: Smooth foliations by circles of S7 with unbounded periods and nonlinearizable multicentres. Ergod. Theory Dyn. Syst. 39,...
