Regular and Chaotic Dynamics

, Volume 23, Issue 1, pp 1–11 | Cite as

Uniform Boundedness of Iterates of Analytic Mappings Implies Linearization: a Simple Proof and Extensions

  • Rafael de la LlaveEmail author


A well-known result in complex dynamics shows that if the iterates of an analytic map are uniformly bounded in a complex domain, then the map is analytically conjugate to a linear map. We present a simple proof of this result in any dimension. We also present several generalizations and relations to other results in the literature.


analytic maps linearization 

MSC2010 numbers

30D05 37F50 39-02 


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  1. 1.
    Arnold, V. I., Geometrical Methods in the Theory of Ordinary Differential Equations, 2nd ed., Grundlehren Math. Wiss., vol. 250, New York: Springer, 1988.CrossRefGoogle Scholar
  2. 2.
    Blanchard, P., Complex Analytic Dynamics on the Riemann Sphere, Bull. Amer. Math. Soc. (N. S.), 1984, vol. 11, no. 1, pp. 85–141.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bochner, S. and Martin, W. T., Several Complex Variables, Princeton Math. Ser., vol. 10, Princeton, N.J.: Princeton Univ. Press, 1948.zbMATHGoogle Scholar
  4. 4.
    Brjuno, A.D., Analytic Form of Differential Equations: 1, Trans. Moscow Math. Soc., 1971, vol. 25, pp. 131–288; see also: Tr. Mosk. Mat. Obs., 1971, vol. 25, pp. 119–262.MathSciNetGoogle Scholar
  5. 4a.
    Brjuno, A.D., Analytic Form of Differential Equations: 2, Trans. Moscow Math. Soc., 1972, vol. 26, pp. 199–239; see also: Tr. Mosk. Mat. Obs., 1972, vol. 26, pp. 199–239.Google Scholar
  6. 5.
    Carathéodory, C., Über die Abbildungen, die durch Systeme von analytischen Funktionen von mehreren Veränderlichen erzeugt werden, Math. Z., 1932, vol. 34, no. 1, pp. 758–792.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 6.
    Cartan, H., Sur les fonctions de plusieurs variables complexes. L’iteration des transformations intérieures d’un domaine borné, Math. Z., 1932, vol. 35, no. 1, pp. 760–773.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 7.
    Cremer, H., Zum Zentrumproblem, Math. Ann., 1928, vol. 98, no. 1, pp. 151–163.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 8.
    Cremer, H., Über die Häufigkeit der Nichtzentren, Math. Ann., 1938, vol. 115, no. 1, pp. 573–580.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 9.
    de la Llave, R., Simple Proofs and Extensions of a Result of L.D.Pustyl’nikov on Non-Autonomous Siegel Center Theorem: Preprint (2017).Google Scholar
  11. 10.
    He, X. and de la Llave, R., Construction of Quasi-Periodic Solutions of State-Dependent Delay Differential Equations by the Parameterization Method: 2. Analytic Case, J. Differential Equations, 2016, vol. 261, no. 3, pp. 2068–2108.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 11.
    Herman, M.-R., Recent Results and Some Open Questions on Siegel’s Linearization Theorem of Germs of Complex Analytic Diffeomorphisms of Cn near a Fixed Point, in Proc. of the 8th Internat. Congr. on Mathematical Physics (Marseille, 1986), Singapore: World Sci., 1987, pp. 138–184.Google Scholar
  13. 12.
    Herman, M.-R., Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Inst. Hautes Études Sci. Publ. Math., 1979, no. 49, pp. 5–233.CrossRefzbMATHGoogle Scholar
  14. 13.
    Moser, J., Stable and Random Motions in Dynamical Systems: With Special Emphasis on Celestial Mechanics, Ann. of Math. Stud., No. 77, Princeton, N.J.: Princeton Univ. Press, 1973.zbMATHGoogle Scholar
  15. 14.
    Pustyl’nikov, L.D., Stable and Oscillating Motions in Nonautonomous Dynamical Systems: A Generalization of C.L. Siegel’s Theorem to the Nonautonomous Case, Math. USSR-Sb., 1974, vol. 23, no. 3, pp. 382–404; see also: Mat. Sb. (N. S.), 1974, vol. 94(136), no. 3(7), pp. 407–429.CrossRefzbMATHGoogle Scholar
  16. 15.
    Saks, S. and Zygmund, A., Analytic Functions, 2nd ed., enl., Monogr. Matem., vol. 28, Warsaw: PWN, 1965.zbMATHGoogle Scholar
  17. 16.
    Siegel, C. L., Iteration of Analytic Functions, Ann. of Math. (2), 1942, vol. 43, pp. 607–612.MathSciNetCrossRefzbMATHGoogle Scholar
  18. 17.
    Siegel, C. L. and Moser, J. K., Lectures on Celestial Mechanics, Grundlehren Math. Wiss., vol. 187, New York: Springer, 1971.CrossRefzbMATHGoogle Scholar
  19. 18.
    Sinaĭ, Ya.G., Topics in Ergodic Theory, Princeton Math. Ser., vol. 44, Princeton, N.J.: Princeton Univ. Press, 1994.zbMATHGoogle Scholar
  20. 19.
    Sternberg, Sh., Infinite Lie Groups and the Formal Aspects of Dynamical Systems, J. Math. Mech., 1961, vol. 10, pp. 451–474.MathSciNetzbMATHGoogle Scholar
  21. 20.
    Yoccoz, J.-Ch., Théorème de Siegel, nombres de Bruno et polynômes quadratiques, in Petits diviseurs en dimension 1, Astérisque, vol. 231, Paris: Soc. Math. France, 1995, pp. 3–88.Google Scholar

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© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.School of MathematicsGeorgia Institute of TechnologyAtlantaUSA

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