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.
@article{AIF_2003__53_3_859_0,
author = {Alsed\`a, Lluis and Falc\'o, Antonio},
title = {On the topological dynamics and phase-locking renormalization of Lorenz-like maps},
journal = {Annales de l'Institut Fourier},
volume = {53},
year = {2003},
pages = {859-883},
doi = {10.5802/aif.1963},
mrnumber = {2008444},
zbl = {1027.37021},
language = {en},
url = {http://dml.mathdoc.fr/item/AIF_2003__53_3_859_0}
}
Alsedà, Lluis; Falcó, Antonio. On the topological dynamics and phase-locking renormalization of Lorenz-like maps. Annales de l'Institut Fourier, Tome 53 (2003) pp. 859-883. doi : 10.5802/aif.1963. http://gdmltest.u-ga.fr/item/AIF_2003__53_3_859_0/
[1] A characterization of the kneading pair for bimodal degree one circle maps, Ann. Inst. Fourier, Tome 47 (1997) no. 1, pp. 273-304 | Article | Numdam | MR 1437186 | Zbl 0861.58014
[2] Combinatorial dynamics and entropy in dimension one, World Scientific, Singapore, Advanced Series in Nonlinear Dynamics, Tome 5 (1993) | MR 1255515 | Zbl 0843.58034
[3] Periods and entropy for Lorentz-like maps, Ann. Inst. Fourier, Tome 39 (1989) no. 4, pp. 929-952 | Article | Numdam | MR 1036338 | Zbl 0678.34047
[4] Kneading theory and rotation interval for a class of circle maps of degree one, Nonlinearity, Tome 3 (1990), pp. 413-452 | Article | MR 1054582 | Zbl 0735.54026
[5] Kneading theory for a family of circle maps with one discontinuity, Acta Math. Univ. Comeniae, Tome LXV (1996), pp. 11-22 | MR 1422291 | Zbl 0863.34046
[6] Iterated maps on the interval as dynamical systems, Birkhäuser, Progress in Physics (1980) | MR 613981 | Zbl 0458.58002
[7] Bifurcations and symbolic dynamics for bimodal degree one circle maps: The Arnol'd tongues and the Devil's staircase (1995) (Ph.D. Thesis, Universitat Autònoma de Barcelona)
[8] Topological conjugation of Lorenz maps by -transformations, Math. Proc. Camb. Phil. Soc., Tome 107 (1990), pp. 401-413 | Article | MR 1027793 | Zbl 0705.58035
[9] Prime and renormalisable kneading invariants and the dynamics of expanding Lorenz maps, Physica D, Tome 62 (1993), pp. 22-50 | Article | MR 1207415 | Zbl 0783.58046
[10] A strange, strange attractor. The Hopf bifurcations and its applications, Springer-Verlag, Appl. Math. Sci., Tome 19 (1976)
[11] Structural stability of Lorenz attractors, Publ. Math. IHES, Tome 50 (1979), pp. 307-320 | Numdam | Zbl 0436.58018
[12] The Classification of Topologically Expansive Lorenz Maps, Comm. Pure Appl. Math., Tome XLIII (1990), pp. 431-443 | Article | MR 1047331 | Zbl 0714.58041
[13] Deterministic non-periodic flow, J. Atmos. Sci., Tome 20 (1963), pp. 130-141 | Article
[14] The periodic points of renormalization, Ann. of Math., Tome 147 (1998), pp. 543-584 | Article | MR 1637651 | Zbl 0936.37017
[15] Universal models for Lorenz maps (1997) (Preprint, IMPA)
[16] On iterated maps on the interval I, II, Dynamical Systems, Springer (Lecture Notes in Math.) Tome 1342 (1988) | Zbl 0664.58015
[17] Rotation intervals for a class of maps of the real line into itself, Ergod. Th. \& Dynam. Sys., Tome 6 (1986), pp. 117-132 | MR 837979 | Zbl 0615.54030
[18] Singular strange attractors on the boundary of Morse-Smale systems, Ann. Sci. École Norm. Sup., Tome 30 (1997), pp. 693-717 | Numdam | MR 1476293 | Zbl 0911.58022
[19] Sur les courbes définies par les équations différentielles, Œuvres complètes, Gauthiers-Villars, Paris, Tome vol. 1 (1952), pp. 137-158
[20] Rotation numbers for monotone functions on the circle, J. London Math. Soc., Tome 34 (1986), pp. 360-368 | Article | MR 856518 | Zbl 0623.58008
[21] The Lorenz equations: Bifurcations, chaos and strange attractors, Springer-Verlag, Appl. Math. Sci., Tome 41 (1982) | MR 681294 | Zbl 0504.58001
[22] The structure of Lorenz attractors, Publ. Math. IHES, Tome 50 (1979), pp. 321-347 | Numdam | MR 556583 | Zbl 0484.58021