Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-28T02:31:43.410Z Has data issue: false hasContentIssue false
  • English
  • Français

On Cherry flows

Published online by Cambridge University Press:  19 September 2008

Marco Martens ,
Sebastian Van Strien ,
Welington De Melo  and
Pedro Mendes
Marco Martens
Affiliation:
Mathematics Department, Technical University of Delft, Julianalaan 132, 2628 BL Delft, The Netherlands
Sebastian Van Strien
Affiliation:
Mathematics Department, Technical University of Delft, Julianalaan 132, 2628 BL Delft, The Netherlands
Welington De Melo
Affiliation:
IMPA, Estrada Dona Castorina 110, 22460 Rio de Janeiro RJ, Brazil
Pedro Mendes
Affiliation:
Departamento de Matemática, ICEx-UFMG, Cidade Universitária, Pampulha, 30000 Belo Horizonte MG, Brazil
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.

The purpose of this research is to describe all smooth vector fields on the torus T2 with a finite number of singularities, no periodic orbits and no saddleconnections. In this paper we are able to complete the description within the class of vector fields which are area contracting near all singularities. In particular we give a large class of analytic vector fields on the torus T2 which have non-trivial recurrence and also sinks.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 10 , Issue 3 , September 1990 , pp. 531 - 554
Copyright
Copyright © Cambridge University Press 1990

References

REFERENCES

[Bel]Belitskii, G. R.. Equivalence and normal forms of germs of smooth mappings. Russian Math. Surveys 33 (1978), 107–177.CrossRefGoogle Scholar
[BD]Bonckaert, P. and Dumoriter, F.. On a linearization theorem of Sternberg for germs of diffeomorphisms. Math. Z. 185 (1984), 115–135.CrossRefGoogle Scholar
[Ch]Cherry, T. M.. Analytic quasi–periodic curves of discontinuous type on the torus. Proc. London Math. Soc. 44 (1938), 175–215.CrossRefGoogle Scholar
[Den]Denjoy, A.. Sur les courbes dèfinies par les èquations diffèrentielles à la surface du tore. J. de Math. Pures et Appl. 11 (1932), 333–375.Google Scholar
[Ha]Hall, G.. A C ∞ Denjoy counter example. Ergod. Th. & Dynam. Sys. 1 (1981), 261–272.CrossRefGoogle Scholar
[He]Herman, M. R.. Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations. Publ. Math. I.H.E.S. 49 (1979), 5–233.CrossRefGoogle Scholar
[MS1]de Melo, W. and van Strien, S.. A structure theorem in one dimensional dynamics. Ann. Math. 189 (1989), 519–546.CrossRefGoogle Scholar
[MS2]de Melo, W. and van Strien, S.. One–dimensional dynamics: The Schwarzian derivative and beyond. Bull. Amer. Math. Soc. 18 (1988), 159–162.CrossRefGoogle Scholar
[PaMe]Palis, J. and de Melo, W.. Geometric Theory of Dynamical Systems, an Introduction. Springer Verlag: New York, 1982.CrossRefGoogle Scholar
[Sch]Schwartz, A. J.. A generalization of a Poincaré–Bendixson Theorem to closed two dimensional manifolds. Amer. J. Math. 85 (1963), 453–458.CrossRefGoogle Scholar
[Ster]Sternberg, S.. On the structure of local homeomorphisms in Euclidean n–space, II. Amer. J. Math. 80 (1958), 623–631.CrossRefGoogle Scholar
[Sto]Stowe, D.. Linearisation in two dimensions. J. Diff. Eq. 63 (1986) 183–226.CrossRefGoogle Scholar
[Ta]Takens, F.. Singularities of vector fields. Publ. Math. I.H.E.S. 43 (1974), 47–100.CrossRefGoogle Scholar
[Y]Yoccoz, J. C.. II n'y a pas de contre–exemples de Denjoy analytique. C.R. Acad. Sci. Paris 298, ser. 1, no. 7 (1984), 141–144.Google Scholar