Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-27T21:52:37.805Z Has data issue: false hasContentIssue false
  • English
  • Français

Diffeomorphisms on surfaces with a finite number of moduli

Published online by Cambridge University Press:  19 September 2008

W. de Melo  and
S.J. van Strien
W. de Melo
Affiliation:
IMPA, Estrada Dona Castorina, 110, Rio de Janeiro, Brazil;
S.J. van Strien
Affiliation:
Department of Mathematics, Techniche Hogeschool Delft, 2628 BL Delft, Julianalaan, 134, Holland
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.

This paper generalises the well known structural stability theorem which says that a diffeomorphism is structurally stable if it is Axiom A and if all invariant manifolds are transversal to each other.

If these transversality conditions are not satisfied then the diffeomorphism not only fails to be stable, but also this gives rise to the appearance of moduli. That is, one needs several real parameters to parameterise all conjugacy classes of nearby diffeomorphisms. (The minimum number of parameters needed is called the number of moduli).

Here we deal with diffeomorphisms on two dimensional manifolds, whose asymptotic dynamics are well understood (the class of Axiom A diffeomorphisms). The main result characterises those Axiom A diffeomorphisms which have a finite number of moduli. This result can be regarded as a generalisation of the structural stability theorem. From the proofs it follows that the dynamics of these diffeomorphisms can also be well understood.

In the proof of our main theorem we need certain invariant foliations to be quite smooth. In an appendix we prove a differentiable version of the Lambda Lemma.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 7 , Issue 3 , September 1987 , pp. 415 - 462
Copyright
Copyright © Cambridge University Press 1987

References

REFERENCES

[Ha]Hartman, P.. On local homeomorphisms of Euclidean spaces. Bol. Soc. Math. Mexicana 5(1960).Google Scholar
[HP]Hirsch, M. & Pugh, C.. Stable manifolds and hyperbolic sets. In Global Analysis. Proc. Symp. in Pure Math. XIV. Amer. Math. Soc. (1970), 133–164.Google Scholar
[Me1]de Melo, W.. Structural stability of diffeomorphisms on two manifolds. Inv. Math. 21 (1973), 233–246.CrossRefGoogle Scholar
[Me2]de Melo, W. Moduli of stability of two-dimensional diffeomorphisms. Topology 19 (1980), 9–21.CrossRefGoogle Scholar
[MP]de Melo, W. & Palis, J.. Moduli of stability for diffeomorphisms. Lecture Notes in Math. 819. Springer-Verlag (1980), 315–339.Google Scholar
[MPS]de Melo, W., Palis, J. & van Strien, S. J.. Characterizing diffeomorphisms with modulus of stability one. In Dynamical Systems and Turbulence, Warwick 1980. Springer Lecture Notes in Math. 898 (1981), 266–285.CrossRefGoogle Scholar
[NPT]Newhouse, S., Palis, J. & Takens, F.. Bifurcations and stability of families of diffeomorphisms. Publ. Math. Inst. Hautes Études Scientifiques 57 (1983).Google Scholar
[P1]Palis, J.. On Morse-Smale dynamical systems. Topology 8 (1969), 385–404.CrossRefGoogle Scholar
[P2]Palis, J.. Moduli of stability and bifurcation theory. Proc. Int. Congress of Mathematicians Helsinki (1978).Google Scholar
[PS]Palis, J. & Smale, S.. Structural stability theorems. In. Global Analysis. Proc. Symp. in Pure Math. XIV. Amer. Math. Soc. (1970), 223–232.Google Scholar
[R]Robinson, C.. Cr structural stability implies Kupka-Smale. In Dynamical Systems (ed. Peixoto, M.) Academic Press (1973), 443–449.CrossRefGoogle Scholar
[R1]Robbin, J.. A structural stability theorem. Ann. of Math. 94 (1971), 447–493.CrossRefGoogle Scholar
[R2]Robbin, J.. Unfoldings of discrete dynamical systems. Ergod. Th. & Dynam. Sys. 4 (1984), 421–486.CrossRefGoogle Scholar
[Sm1]Smale, S.. Differentiable dynamical systems. Bull. Amer. Math. Soc. 73 (1967), 747–817.CrossRefGoogle Scholar
[Sm2]Smale, S.. The Ω-stability theorem. In Global Analysis. Proc. Symp. in Pure Math. XIV. Amer. Math. Soc. (1970), 289–297.Google Scholar
[S1]van Strien, S.J.. Saddle connections of arcs of diffeomorphisms, moduli of stability. In Dynamical Systems and Turbulence, Warwick 1980. Springer Lecture Notes in Math. 898 (1981), 352–366.CrossRefGoogle Scholar
[S2]van Strien, S. J.. One parameter families of vectorfields. Bifurcations near saddle-connections. Ph.D. Thesis, Utrecht (1982). (An extended version will appear as a monograph.)Google Scholar