A characterization of the kneading pair for bimodal degree one circle maps
Annales de l'Institut Fourier, Volume 47 (1997) no. 1, p. 273-304

For continuous maps on the interval with finitely many monotonicity intervals, the kneading theory developed by Milnor and Thurston gives a symbolic description of the dynamics of a given map. This description is given in terms of the kneading invariants which essentially consists in the symbolic orbits of the turning points of the map under consideration. Moreover, this theory also describes a classification of all such maps through theses invariants. For continuous bimodal degree one circle maps, similar invariants were introduced by Alsedà and Mañosas, where the first part of the program just described was carried through, and where relations between the circle maps invariants and the rotation interval were elucidated. The main theorem of this paper characterizes the set of kneading invariants for all bimodal degree one circle maps.

La théorie des itinéraires symboliques développée par Milnor et Thurston donne, pour les applications de l’intervalle dans lui-même avec un nombre fini de morceaux monotones, une caractérisation de la dynamique de ces applications. Dans cet article nous apportons une caractérisation de toutes les “paires de pétrissage” pour l’ensemble des relèvements des applications continues du cercle dans lui-même de degré un bimodales.

@article{AIF_1997__47_1_273_0,
     author = {Alsed\`a, Lluis and Falc\'o, Antonio},
     title = {A characterization of the kneading pair for bimodal degree one circle maps},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {47},
     number = {1},
     year = {1997},
     pages = {273-304},
     doi = {10.5802/aif.1567},
     zbl = {0861.58014},
     mrnumber = {98h:58055},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1997__47_1_273_0}
}
Alsedà, Lluis; Falcó, Antonio. A characterization of the kneading pair for bimodal degree one circle maps. Annales de l'Institut Fourier, Volume 47 (1997) no. 1, pp. 273-304. doi : 10.5802/aif.1567. http://www.numdam.org/item/AIF_1997__47_1_273_0/

[1] Ll. Alsedà, A. Falcó, An entropy formula for a class of circle maps, C. R. Acad. Sci. Paris, 314, Série I (1992), 677-682. | MR 93e:58103 | Zbl 0755.54011

[2] Ll. Alsedà, A. Falcó, Devil's staircase route to chaos in a forced relaxation oscillator, Ann. Inst. Fourier, 44-1 (1994), 109-128. | Numdam | MR 95b:58098 | Zbl 0793.34028

[3] Ll. Alsedà, J. Llibre, M. Misiurewicz, Combinatorial dynamics and entropy in dimension one, Advanced Series Nonlinear Dynamics, vol. 5, World Scientific, Singapore, 1993. | MR 95j:58042 | Zbl 0843.58034

[4] Ll. Alsedà, F. Mañosas, Kneading theory and rotation interval for a class of circle maps of degree one, Nonlinearity, 3 (1990), 413-452. | MR 91e:58090 | Zbl 0735.54026

[5] Ll. Alsedà, F. Mañosas, W. Szlenk, A Characterization of the uniquely ergodic endomorphisms of the circle, Proc. Amer. Math. Soc., 117 (1993), 711-714. | MR 93d:58086 | Zbl 0773.34036

[6] P.L. Boyland, Bifurcations of circle maps : Arnol'd Tongues, bistability and rotation intervals, Commun. Math. Phys., 106 (1986), 353-381. | MR 88c:58045 | Zbl 0612.58032

[7] P. Collet, J. P. Eckmann, Iterated maps on the interval as dynamical systems, Progress in Physics, Birkhäuser, 1980. | MR 82j:58078 | Zbl 0458.58002

[8] A. Chenciner, J.M. Gambaudo, Ch. Tresser, Une remarque sur la structure des endomorphismes de degré 1 du cercle, C. R. Acad. Sci., Paris series I, 299 (1984), 145-148. | MR 86b:58102 | Zbl 0584.58004

[9] M. Denker, C. Grillenberger, K. Sigmund, Ergodic Theory in compact spaces, Lecture Notes in Math., 527, Springer, Berlin, 1976. | MR 56 #15879 | Zbl 0328.28008

[10] A. Falcó, Bifurcations and symbolic dynamics for bimodal degree one circle maps : The Arnol'd tongues and the Devil's staircase, Ph. D. Thesis, Universitat Autònoma de Barcelona, 1995.

[11] L. Glass, M.C. Mackey, From clocks to chaos, Princeton University Press, 1988. | MR 90c:92012 | Zbl 0705.92004

[12] R. Ito, Rotation sets are closed, Math. Proc. Camb. Phil. Soc., 89 (1981), 107-111. | MR 82i:58061 | Zbl 0484.58027

[13] M. Levi, Qualitative analysis of the periodically forced relaxation oscillations, Mem. Amer. Math. Soc., 244 (1981). | MR 82g:58052 | Zbl 0448.34032

[14] I. Malta, On Denjoy's theorem for endomorphisms, Ergod. Th. & Dynam. Sys., 6 (1986), 259-264. | MR 88d:58092 | Zbl 0657.58018

[15] W. De Melo, S. Van Strien, One dimensional dynamics, Springer-Verlag, 1993. | MR 95a:58035 | Zbl 0791.58003

[16] J. Milnor, P. Thurston, On iterated maps on the interval, I, II, Dynamical Systems, Lecture Notes in Math. 1342, Springer, (1988), 465-563. | Zbl 0664.58015

[17] M. Misiurewicz, Rotation intervals for a class of maps of the real line into itself, Ergod. Theor. Dynam. Sys., 6 (1986), 117-132. | MR 87k:58131 | Zbl 0615.54030

[18] S.E. Newhouse, J. Palis, F. Takens, Bifurcations and stability of families of diffeomorphisms, Inst. Hautes Études Sci., Publ. Math., 57 (1983), 5-71. | Numdam | MR 84g:58080 | Zbl 0518.58031

[19] H. Poincaré, Sur les curves définies par les équations differentielles, Œuvres complètes, vol 1, 137-158, Gauthiers-Villars, Paris, 1952.

[20] J.C. Yoccoz, Il n'y a pas de contre-exemple de Denjoy analytique, C. R. Acad. Sci. Paris, 298, Série I (1984), 141-144. | MR 85j:58134 | Zbl 0573.58023