Hostname: page-component-7479d7b7d-wxhwt Total loading time: 0 Render date: 2024-07-10T13:32:25.836Z Has data issue: false hasContentIssue false
  • English
  • Français

Existence de cycles pour des multi-applications du cercle

Published online by Cambridge University Press:  19 September 2008

Jean-Paul Dufour
Jean-Paul Dufour
Affiliation:
Gétodim, Institut de Mathématiques, U.S.T.L., Pl. E. Bataillon, 34000 Montpellier, France
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.

We consider multi-applications Γ of the circle S1, the graphs of which are ‘degree (1, 1)’, continuous piece-wise monotonic curves of S1 × S1 In general Γp is not a connected curve but it is a union of a degree (1, 1) continuous curve Γp of S1 × S1 and of some other curves homotopic to a point. Using these Γp we are able to study dynamics of Γ. We focus on the case where Γ has no periodic points and we see, for instance, that all ‘regular’ orbits have, on S1 the same order as orbits of an irrational rotation. Using this we prove that such F without ‘cycles’ are obtained from a Denjoy's counter-example, perturbing it in the holes of the invariant set. Finally we generalize the classical result of Block and Franke showing that if Γ is a C2 curve with no degenerate critical points, or if Γ is a C∞ curve with no ‘flat’ points, there are always ‘cycles’, unless Γ is an homeomorphism.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 9 , Issue 1 , March 1989 , pp. 47 - 65
Copyright
Copyright © Cambridge University Press 1989

References

REFERENCES

[1]Arnol'd, V. I.. Small denominators I. A.M.S. Transl. 2 (46) 213–284.Google Scholar
[2]Auslander, J. & Katznelson, Y.. Continuous maps of the circle without periodic points. Israel J. Math. 32 (4) (1979), 375–381.Google Scholar
[3]Bamon, R., Malta, I. P., Pacifico, M. J. & Takens, F.. Rotation intervals of endomorphisms of the circle. Ergod. Th. & Dynam. Sys. 4 (1984), 493–498.CrossRefGoogle Scholar
[4]Block, L. & Franke, J.. Existence of periodic points for maps of S 1. Inventiones Math. 22 (1973), 69–73.CrossRefGoogle Scholar
[5]Dufour, J. P.. Dynamique de couples de fonctions. Preprint du Séminaire de Géométrie Différentielle de Montpellier (1985–1986) France.Google Scholar
[6]Dufour, J. P.. Stabilité simultanée de deux fonctions différentiables. Ann. Inst. Fourier, Grenoble 29 (1) (1979), 263–282.CrossRefGoogle Scholar
[7]Dufour, J. P.. Intervalles de rotation pour les multi-applications du cercle. Comptes rendus du Colloque Géométrie Symplectique et Physique Mathématique (1986) (Lyon), à paraître.Google Scholar
[8]Hofbauer, F.. Periodic points for piecewise monotonic transformations. Ergod. Th. & Dynam. Sys.(1985), 237–256.CrossRefGoogle Scholar
[9]Misiurewicz, M.. Rotation intervals for a class of maps of the real line into itself. Ergod. Th. & Dynam. Sys. 6 (1986), 117–132.CrossRefGoogle Scholar
[10]Przytycki, F.. Chaos after bifurcation of a Morse-Smale diffeomorphism through a one-cycle saddle-node and iterations of multivalued mappings of an interval and a circle. Preprint 347 (85) Inst. of Math., Polish Acad. of Science.Google Scholar
[11]Yoccoz, J. C.. II n'y a pas de contre-exemple de Denjoy analytique. C.R. Acad. Sc. Paris, t.298, Serie I, 7 (1984), 141–144.Google Scholar