Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-28T06:47:32.576Z Has data issue: false hasContentIssue false
  • English
  • Français

Unfoldings of discrete dynamical systems

Published online by Cambridge University Press:  19 September 2008

Joel W. Robbin
Joel W. Robbin
Affiliation:
Mathematics Department, University of Wisconsin, Madison, WI 53706, USA
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A universal unfolding of a discrete dynamical system f0 is a manifold F of dynamical systems such that each system g sufficiently near f0 is topologically conjugate to an element f of F with the conjugacy φ and the element f depending continuously on f0. An infinitesimally universal unfolding of f0 is (roughly speaking) a manifold F transversal to the topological conjugacy class of f0. Using Nash-Moser iteration we show infinitesimally universal unfoldings are universal and (in part II) give a class of examples relating to moduli of stability introduced by Palis and De Melo.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 4 , Issue 3 , September 1984 , pp. 421 - 486
Copyright
Copyright © Cambridge University Press 1984

References

REFERENCES

[1]Abraham, R. & Robbin, J.. Transversal Mappings and Flows. Benjamin: New York, 1967.Google Scholar
[2]Conley, C. C.. Isolated Invariant Sets and the Morse Index; CBMS monographs 38 Amer. Math. Soc: Providence, R.I., 1978.Google Scholar
[3]DeMelo, W.. Moduli of stability of diffeomorphisms. Topology 19 (1978), 9–21.Google Scholar
[4]Golubitsky, M. & Guillemin, V.. Stable Mappings and Their Singularities. Springer Graduate Texts in Math. # 14. Springer-Verlag: Berlin-Heidelberg-New York, 1974.Google Scholar
[5]Hamilton, R. S.. The inverse function theorem of Nash and Moser. Bull. Amer. Math. Soc. 7 (1982), 65–222.Google Scholar
[6]Irwin, M. C.. On the smoothness of the composition map. Quart. J. Math. Oxford (2) 23 (1972), 113–33.CrossRefGoogle Scholar
[7]Knuth, D.. The Art of Computer Programming Vol. I. Addison-Wesley: Reading, Mass., 1973.Google Scholar
[8]Kopell, N.. Commuting diffeomorphisms. In Proc. of Symposia in Pure Math (Vol. XIV: Global Analysis) 165–184. Amer. Math. Soc: Providence, R.I. (1970).Google Scholar
[9]Lassalle, M. G.. Une demonstration du theoreme de division. Topology 12 (1973) 41–62.CrossRefGoogle Scholar
[10]Mañe, R.. Absolute stability and bifurcation theory. Invent. Math. 53 (1979), 59–75.CrossRefGoogle Scholar
[11]Moser, J.. A new technique for the construction of solutions of non-linear differentiation equations. Proc. Nat. Acad. Sci. U.S.A., 47 (1961), 1824–1831.Google Scholar
[12]Moser, J.. A rapidly convergent iteration method and non-linear partial differential equations. Ann. Scuola Norm. Sup. Pisa, Ser. III, Vol. XXI (1966), (pp. 265–315) II (pp. 499–535).Google Scholar
[13]Nash, J.. The embedding problem for Riemannian manifolds. Ann. of Math. 63 (1956), 20–63.CrossRefGoogle Scholar
[14]Palis, J.. A differentiate invariant of topological conjugacies and moduli of stability. Asterisque 51 (1978).Google Scholar
[15]Reed, M. & Simon, B.. Methods of Modern Mathematical Physics II, Fourier Analysis, Self-Adjointness. Academic Press: New York, 1975.Google Scholar
[16]Robbin, J.. A structural stability theorem. Ann. of Math. 94 (1971), 447–93.Google Scholar
[17]Robinson, C.. Structural stability of C 1 diffeomorphisms. J. Diff. Eq. 22 (1976), 28–73.CrossRefGoogle Scholar
[18]Schwartz, J.. On Nash's Implicit Function Theorem, Comm. Pure Appl. Math. 13 (1960), 509–530. See also his lecture notes ‘Nonlinear Functional Analysis’ NYU, 1964, [18a].CrossRefGoogle Scholar
[19]Sergeraert, F.. Un theoreme de fonctions implicites sur certains espace de Frechet et quelques applications. Ann. Sci. École Norm. Sup. (4), 5 (1972), 599–660.CrossRefGoogle Scholar
[20]Smale, S.. Differentiable dynamical systems. Bull. Amer. Math. Soc. 73 (1967), 747–817; (reprinted in Smale [22]).Google Scholar
[21]Smale, S.. Notes on differentiable dynamical systems. In Proc. of Symposia in Pure Math (Global Analysis) XIV, pp. 277–287. Amer. Math. Soc: Providence, RI, 1970.Google Scholar
[22]Smale, S.. The Mathematics of Time. Springer-Verlag: New York, 1980.CrossRefGoogle Scholar
[23]Sternberg, S.. Infinite Lie groups and formal aspects of dynamical systems. J. of Math. & Mech. 10 (1961), 451–474.Google Scholar
[24]Wall, C. T. C.. Introduction to the preparation theorem. In Proc. of Liverpool Singularities Symposium I, Springer lecture notes 192, pp. 90–95. Springer-Verlag: Berlin-Heidelberg-New York, 1971.Google Scholar
[25]Zeeman, E. C.. The classification of elementary catastrophes of codimension ≤ 5. (Lecture notes of a course given at Warwick in 1973). Published in: Structural Stability, the Theory of Catastrophes, and Applications in Science, Springer lecture notes in math. 525 (1976), pp. 263–327 reprinted in Catastrophe Theory, Selected papers 1972–1977, Addison-Wesley, 1977, pp. 497–561.CrossRefGoogle Scholar
[26]Zehnder, E.. Generalized implicit function theorems with applications to some small divisor problems, I. Comm. Pure Appl. Math 28 (1975), 91–140.Google Scholar