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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
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Abstract

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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

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