Étude de la classification topologique des fonctions unimodales
Annales de l'Institut Fourier, Volume 35 (1985) no. 3, p. 59-77

Using the theory of itineraries and kneading sequences, we study the topological conjugacy of unimodal functions. We introduce the notion of macroscopical conjugacy, characterized by the equality of the kneading sequences. Then we present a theorem of classification of unimodal functions. In order to illustrate these results, we show that the set of solutions of Feigenbaum equation contains an infinite number of classes.

À l’aide de la théorie des itinéraires et des suites de tricotage, nous étudions la conjugaison topologique des fonctions unimodales. Nous introduisons la notion de conjugaison macroscopique, caractérisée par l’égalité des suites de tricotage. Puis nous présentons un théorème de classification des fonctions unimodales. Pour illustrer ces résultats, nous montrons que l’ensemble des solutions de l’équation de Feigenbaum contient une infinité de classes topologiques.

@article{AIF_1985__35_3_59_0,
     author = {Cosnard, Michel},
     title = {\'Etude de la classification topologique des fonctions unimodales},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Louis-Jean},
     address = {Gap},
     volume = {35},
     number = {3},
     year = {1985},
     pages = {59-77},
     doi = {10.5802/aif.1019},
     zbl = {0569.58004},
     mrnumber = {87i:58091},
     language = {fr},
     url = {http://www.numdam.org/item/AIF_1985__35_3_59_0}
}
Cosnard, Michel. Étude de la classification topologique des fonctions unimodales. Annales de l'Institut Fourier, Volume 35 (1985) no. 3, pp. 59-77. doi : 10.5802/aif.1019. http://www.numdam.org/item/AIF_1985__35_3_59_0/

[1] J.P. Allouche et M. Cosnard, Itération de fonctions unimodales et suites engendrées par automates, C.R.A.S., 286 (1982), 159-162. | Zbl 0547.58027

[2] M. Campanino et H. Epstein, On the existence of Feigenbaum's fixed point, Preprint IHES P/80/35 (1980). | Zbl 0474.58013

[3] P. Collet et J.P. Eckmann, Iterated maps on the interval as dynamical systems, Birkhäuser, Basel, (1980) | MR 82j:58078 | Zbl 0458.58002

[4] M. Cosnard, Etude des solutions de l'équation fonctionnelle de Feigenbaum, Actes du colloque de Dijon, Astérisque, (1983). | Zbl 0519.58031

[5] B. Derrida, A. Gervois et Y. Pomeau, Iteration of endomorphism on the real axis and representation of numbers, Ann. Inst. Henri Poincaré, 29, 3 (1978), 305-356. | Numdam | MR 80g:10056 | Zbl 0416.28012

[6] M.J. Feigenbaum, The universal metric properties of nonlinear transformation, J. Stat. Phys., 21, 6 (1979), 669-706. | MR 82e:58072 | Zbl 0515.58028

[7] M.J. Feigenbaum, The transition to aperiodic behavior in turbulent systems, Commun, Math. Phys., 77 (1980), 65-86. | MR 83e:58053 | Zbl 0465.76050

[8] C. Godbillon, Systèmes dynamiques sur les surfaces, IRMA Strasbourg, (1979). | Zbl 0515.58001

[9] J. Guckenheimer, Sensitive dependence to initial conditions for one dimensional maps, Commun. Math. Phys., 70 (1969), 133-160. | MR 82c:58037 | Zbl 0429.58012

[10] I. Gumowski et C. Mira, Recurrences and discrete dynamic systems, Lect. Notes Math., 809 (1980). | MR 81m:58052 | Zbl 0449.58003

[11] O.E. Lanford Iii, A computer-assisted proof of the Feigenbaum conjectures, Preprint IHES P/81/17 (1981).

[12] N. Metropolis, M.L. Stein et P.R. Stein, On the finite sets for transformations on the unit interval, J. Comb. Theory., 15 (1973), 25-44. | MR 47 #5183 | Zbl 0259.26003

[13] J. Milnor et W. Thurston, On iterated maps of the interval, Preprint Princeton, (1977). | Zbl 0664.58015

[14] M. Misiurewicz, Invariant measures for continuous transformations of [0, 1] with zero topological entropy, Lecture Notes Math., 729 (1980) 144-152. | MR 81a:28017 | Zbl 0415.28015

[15] G. Targonski, Topics in iteration theory, Vandenhoeck & Ruprecht, Göttingen, (1981). | MR 82j:39001 | Zbl 0454.39003

[16] S. Ulam, Sets, number and universes., M.I.T., Press Cambridge, (1974). | Zbl 0558.00017