Existence de cycles pour des multi-applications du cercle

Abstract

AbstractWe consider multi-applications Γ of the circleS1, the graphs of which are ‘degree (1, 1)’, continuous piece-wise monotonic curves ofS1×S1In general Γpis not a connected curve but it is a union of a degree (1, 1) continuous curve ΓpofS1×S1and of some other curves homotopic to a point. Using these Γpwe 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, onS1the 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 aC2curve with no degenerate critical points, or if Γ is aC∞ curve with no ‘flat’ points, there are always ‘cycles’, unless Γ is an homeomorphism.