Hostname: page-component-74656f9884-c52kr Total loading time: 0 Render date: 2026-11-08T13:09:22.435Z Has data issue: false hasContentIssue false
  • English
  • Français

Fixed points of elliptic reversible transformations with integrals

Published online by Cambridge University Press:  19 September 2008

Xianghong Gong
Xianghong Gong
Affiliation:
Department of Mathematics, University of Chicago, Chicago, IL 60637, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that for a certain family of integrable reversible transformations, the curves of periodic points of a general transformation cross the level curves of its integrals. This leads to the divergence of the normal form for a general reversible transformation with integrals. We also study the integrable holomorphic reversible transformations coming from real analytic surfaces in ℂ2 with non-degenerate complex tangents. We show the existence of real analytic surfaces with hyperbolic complex tangents, which are contained in a real hyperplane, but cannot be transformed into the Moser—Webster normal form through any holomorphic transformation.

Information

Type
Survey Article
Information
Ergodic Theory and Dynamical Systems , Volume 16 , Issue 4 , August 1996 , pp. 683 - 702
Copyright
Copyright © Cambridge University Press 1996

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

REFERENCES

[1]Arnol'd, V. I.. Reversible systems. Nonlinear and Turbulent Processes. Academic, New York, 1984, pp. 1161–1174.Google Scholar
[2]Bedford, E.. Levi flat hypersurfaces in ℂ2 with prescribed boundary. Ann. Scuola Norm. Sup. di Pisa 9 (1982), 529–570.Google Scholar
[3]Birkhoff, G. D.. The Restricted Problem of Three Bodies. Rendi di Palermo 39 (1915), 265–334 (in particular §20).Google Scholar
[4]Birkhoff, G. D.. Surface transformations and their dynamical applications. Acta Math. 43 (1920), 1–119 (in particular p. 40).Google Scholar
[5]Birkhoff, G. D.. Dynamical Systems. (AMS Coll. Publ. vol. 9), 1927, reprinted 1966. (in particular p. 177)CrossRefGoogle Scholar
[6]Bishop, E.. Differentiable manifolds in complex Euclidean space. Duke Math. J. 32 (1965), 1–22.Google Scholar
[7]Devaney, R. L.. Reversible diffeomorphisms and flows. Trans. Amer. Math. Soc. 218 (1976), 89–113.CrossRefGoogle Scholar
[8]Gong, X.. On the convergence of normalizations of real analytic surfaces near hyperbolic complex tangents. Comment. Math. Helv. 69 (1994), 549–574.Google Scholar
[9]Moser, J. K.. On invariant curves of area-preserving mappings of an annulus. Nachr. Akad. Wiss. Göttingen, math.-phys. Kl. 1962, 1–10.Google Scholar
[10]Moser, J. K.. Stable and random motions in dynamical systems. Annals of Math. Studies. 77, Princeton University Press, Princeton, 1973.Google Scholar
[11]Moser, J. K. and Webster, S. M.. Normal forms for real surfaces in ℂ2 near complex tangents and hyperbolic surface transformations. Acta Math. 150 (1983), 255–296.CrossRefGoogle Scholar
[12]Rüssmann, H.. Über die Existenz einer Normalform inhaltstreuer elliptischer Transformationen. Math. Ann. 137 (1959), 64–77.Google Scholar
[13]Sevryuk, M. B.. Reversible Systems. Springer Lecture Notes in Mathematics 1211 (1986).Google Scholar
[14]Siegel, C. L.. Über die Existenz einer Normalform analytischer Hamiltonscher Differentialgleichungen in der Nähe einer Gleichgewichtslösung. Math. Ann. 128 (1954), 144–170.CrossRefGoogle Scholar
[15]Siegel, C. L. and Moser, J. K.. Lectures on Celestial Mechanics. Springer, 1971.Google Scholar