Skip to main content
SpringerLink
Log in

Iterations and logarithms of formal automorphisms

  • Research Papers
  • Published:
aequationes mathematicae Aims and scope Submit manuscript

Abstract

Using the decomposition of an automorphism of the ring of formal power series in several variables, in a semisimple and a unipotent automorphism, I prove in this paper that an automorphism allows a continuous iteration if and only if it is the exponential of a derivation. This result implies a number of results recently obtained by Reich, Schwaiger, and Bucher.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Atiyah, M. F. andMacdonald, I. G.,Introduction to commutative algebra. Addison-Wesley, Reading, Mass.-London-Don Mills, Ont., 1969, 128.

    Google Scholar 

  2. Borel, A.,Linear algebraic groups. W. A. Benjamin Inc., New York-Amsterdam, 1969, 398.

    Google Scholar 

  3. Bourbaki, N.,Algèbre commutative. Ch. 3, Graduations, filtrations et topologies. Hermann, Paris, 1961.

    Google Scholar 

  4. Bourbaki, N.,Algèbre. Ch. 8, Modules et anneaux semi-simples. Hermann, Paris, 1958.

    Google Scholar 

  5. Bucher, W.,Kontinuierlichen Iterationen formal-biholomorpher Abbildungen. Ber. Math.-Statist. Sekt. Forsch. Graz No. 97 (1978), 45.

  6. Chen, K.-T.,Local diffeomorphisms — ℂ ∞ realization of formal properties. Amer. J. Math.87 (1965), 140–157.

    Google Scholar 

  7. Ecalle, J.,Théorie des invariants holomorphes. Thesis. Publications Mathématiques d'Orsay, Paris, 1974.

    Google Scholar 

  8. Gerard, R. andLevelt, A. H. M.,Sur les connexions à singularités régulières dans le cas aux plusieurs variables. Funkcial. Ekvac.19 (1976), 149–173.

    Google Scholar 

  9. Lewis, D. C., Jr.,Formal power series transformations. Duke Math. J.5 (1939), 794–805.

    Google Scholar 

  10. Praagman, C.,Formal decomposition of n commuting partial linear difference operators. Duke Math. J.51 (1984), 331–353.

    Google Scholar 

  11. Reich, L.,Iteration problems in power series rings. InThéorie de l'Itération et ses Applications. Coll. Int. du C.N.R.S. 332, Paris, 1982.

  12. Reich, L. andSchwaiger, J.,Uber ein Satz von Shl. Sternberg in der Theorie der analytischen Iterationen. Monatsh. Math.83 (1977), 207–221.

    Google Scholar 

  13. Reich, L. andSchwaiger, J.,Linearisierung formal-biholomorpher Abbildungen und Iterationsprobleme. Aequationes Math.20 (1980), 224–243.

    Google Scholar 

  14. Sternberg, S.,Infinite Lie groups and the formal aspects of dynamical systems. J. Math. Mech.10 (1961), 451–474.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematisch Instituut, Rijksuniversiteit Gronïgen, Postbus 800, 9700 AV, Gronïgen, The Netherlands

    C. Praagman

Authors
  1. C. Praagman
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Praagman, C. Iterations and logarithms of formal automorphisms. Aeq. Math. 30, 151–160 (1986). https://doi.org/10.1007/BF02189922

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02189922

AMS (1980) subject classification

Search

Navigation