On the topological dynamics and phase-locking renormalization of Lorenz-like maps

Alsedà, Lluis; Falcó, Antonio

Annales de l'Institut Fourier, Tome 53 (2003) no. 3, pp. 859-883.

L’objectif de cet article est double. D’abord, on donne une caractérisation de l’ensemble des invariants de pétrissage pour la classe des applications de type Lorenz considérées comme des applications du cercle de degré un avec une discontinuité. Dans une deuxième étape, on considère la sous-classe des applications de type Lorenz engendrées pour la classe des applications de Lorenz dans l’intervalle. Pour cette classe des applications on donne une caractérisation de l’ensemble des applications renormalisables avec intervalle de rotation dégénéré à un numéro rationnel, c’est-à-dire des applications de confinement de phase renormalisables. On obtient cette caractérisation en montrant l’équivalence entre la méthode de renormalisation géométrique et la méthode combinatoire (qu’on exprime en termes d’un produit du type $*$ défini sur l’ensemble des invariants de pétrissage. Finalement, on démontre l’existence au niveau combinatoire, de points périodiques de toutes les périodes de l’application de renormalisation.

The aim of this paper is twofold. First we give a characterization of the set of kneading invariants for the class of Lorenz–like maps considered as a map of the circle of degree one with one discontinuity. In a second step we will consider the subclass of the Lorenz– like maps generated by the class of Lorenz maps in the interval. For this class of maps we give a characterization of the set of renormalizable maps with rotation interval degenerate to a rational number, that is, of phase–locking renormalizable maps. This characterization is given by showing the equivalence between the geometric renormalization procedure and the combinatorial one (which is expressed in terms of an $*$–like product defined in the set of kneading invariants). Finally, we will prove the existence, at a combinatorial level, of periodic points of all periods for the renormalization map.

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