Computing explicitly topological sequence entropy: the unimodal case
[Calcul explicite de l'entropie topologique séquentielle; le cas unimodal]
Annales de l'Institut Fourier, Tome 52 (2002) no. 4, pp. 1093-1133.

Nous considérons $W\left(I\right)$ la famille de fonctions $f$ continues de l’intervalle $I=\left[a,b\right]$ sur lui–même, telles que (1) $f\left(a\right)=f\left(b\right)\in \left\{a,b\right\}$; (2) elles sont constituées de deux morceaux monotones; et (3) elles ont des points périodiques de périodes toutes les puissances de $2$ exactement. L’objectif principal de ce travail est de calculer explicitement l’entropie topologique séquentielle ${h}_{D}\left(f\right)$ de tout élément $f$ de $W\left(I\right)$ par rapport à la suite $D={\left({2}^{m-1}\right)}_{m=1}^{\infty }$.

Let $W\left(I\right)$ denote the family of continuous maps $f$ from an interval $I=\left[a,b\right]$ into itself such that (1) $f\left(a\right)=f\left(b\right)\in \left\{a,b\right\}$; (2) they consist of two monotone pieces; and (3) they have periodic points of periods exactly all powers of $2$. The main aim of this paper is to compute explicitly the topological sequence entropy ${h}_{D}\left(f\right)$ of any map $f\in W\left(I\right)$ respect to the sequence $D={\left({2}^{m-1}\right)}_{m=1}^{\infty }$.

DOI : https://doi.org/10.5802/aif.1913
Classification : 37B40,  26A18,  54H20
Mots clés : fonction de type ${2}^{\infty }$, entropie séquentielle topologique, fonction unimodale
@article{AIF_2002__52_4_1093_0,
author = {Jim\'enez L\'opez, Victor and C\'anovas Pe\~na, Jose Salvador},
title = {Computing explicitly topological sequence entropy: the unimodal case},
journal = {Annales de l'Institut Fourier},
pages = {1093--1133},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {52},
number = {4},
year = {2002},
doi = {10.5802/aif.1913},
zbl = {1083.37012},
mrnumber = {1926675},
language = {en},
url = {http://archive.numdam.org/articles/10.5802/aif.1913/}
}
Jiménez López, Victor; Cánovas Peña, Jose Salvador. Computing explicitly topological sequence entropy: the unimodal case. Annales de l'Institut Fourier, Tome 52 (2002) no. 4, pp. 1093-1133. doi : 10.5802/aif.1913. http://archive.numdam.org/articles/10.5802/aif.1913/

 R.L. Adler; A.G. Konheim; M.H. McAndrew Topological entropy, Trans. Amer. Math. Soc., Volume 114 (1965), pp. 309-319 | Article | MR 175106 | Zbl 0127.13102

 F. Balibrea; J.S. Cánovas Peña Commutativity and non-commutativity of topological sequence entropy, Ann. Inst. Fourier, Volume 49 (1999) no. 5, pp. 1693-1709 | Article | Numdam | MR 1723832 | Zbl 0990.37010

 F. Balibrea; V. Jiménez López The measure of scrambled sets: a survey, Acta Univ. Mathaei Belii Nat. Sci., Ser. Math., Volume 7 (1999), pp. 3-11 | MR 1766951 | Zbl 0967.37021

 R. Bowen; J. Franks The periodic points of maps of the circle and the interval, Topology, Volume 15 (1976), pp. 337-342 | Article | MR 431282 | Zbl 0346.58010

 H. Bruin An algorithm to compute the topological entropy of a unimodal map, Internat. J. Bifur. Chaos Appl. Sci. Engrg., Volume 9 (1999), p. 1881-1882 | Article | MR 1728747 | Zbl 1089.37515

 J.S. Cánovas On topological sequence entropy of piecewise monotonic mappings, Bull. Austral. Math. Soc., Volume 62 (2000), pp. 21-28 | Article | MR 1775883 | Zbl 0965.37033

 N. Franzová; J. Smital Positive sequence topological entropy characterizes chaotic maps, Proc. Amer. Math. Soc., Volume 112 (1991), pp. 1083-1086 | MR 1062387 | Zbl 0735.26005

 T.N.T. Goodman Topological sequence entropy, Proc. London Math. Soc., Volume 29 (1974), pp. 331-350 | Article | MR 356009 | Zbl 0293.54043

 W.H. Gottschalk; G.A. Hedlund Topological dynamics, American Mathematical Society, Providence, 1955 | MR 74810 | Zbl 0067.15204

 R. Hric Topological sequence entropy for maps of the interval, Proc. Amer. Math. Soc., Volume 127 (1999), pp. 2045-2052 | Article | MR 1487372 | Zbl 0923.26004

 V. Jiménez López Large chaos in smooth functions of zero topological entropy, Bull. Austral. Math. Soc., Volume 46 (1992), pp. 271-285 | Article | MR 1183783 | Zbl 0758.26004

 V. Jiménez López An explicit description of all scrambled sets of weakly unimodal functions of type ${2}^{\infty }$, Real. Anal. Exchange, Volume 21 (1995/1996), pp. 664-688 | MR 1407279 | Zbl 0879.58044

 V. Jiménez López; L. Snoha There are no piecewise linear maps of type ${2}^{\infty }$, Trans. Amer. Math. Soc., Volume 349 (1997), pp. 1377-1387 | Article | MR 1389785 | Zbl 0947.37025

 M. Kutcha; J. SmÍtal Two-point scrambled set implies chaos, ECIT 87 (Caldes de Malavella, Spain, 1987) (Proceedings of the European Conference on Iteration Theory) (1989), pp. 427-430

 Z.L. Leibenzon Investigation of some properties of a continuous pointwise mapping of an interval onto itself, having an application in the theory of nonlinear oscilations, Prikl. Mat. i Mekh (Russian), Volume 17 (1953), pp. 351-360 | MR 55429

 T.Y. Li; J.A. Yorke Period three implies chaos, Amer. Math. Monthly, Volume 82 (1975), pp. 985-992 | Article | MR 385028 | Zbl 0351.92021

 M. Martens; W. de Melo; S. van Strien Julia-Fatou-Sullivan theory for real one-dimensional dynamics, Acta Math., Volume 168 (1992), pp. 271-318 | MR 1161268 | Zbl 0761.58007

 M. Martens; C. Tresser Forcing of periodic orbits for interval maps and renormalization of piecewise affine maps, Proc. Amer. Math. Soc., Volume 124 (1996), pp. 2863-2870 | Article | MR 1343712 | Zbl 0864.58035

 J. Milnor; W. Thurston On iterated maps of the interval, Dynamical systems (College Park, MD, 1986/1987) (Lecture Notes in Math.), Volume 1342 (1988), pp. 465-563 | Zbl 0664.58015

 M. Misiurewicz Horseshoes for mappings of an interval, Bull. Acad. Pol. Sci., Sér. Sci. Math., Volume 27 (1979), pp. 167-169 | MR 542778 | Zbl 0459.54031

 M. Misiurewicz; J. Smital Smooth chaotic functions with zero topological entropy, Ergodic Theory Dynam. Systems, Volume 8 (1988), pp. 421-424 | Article | MR 961740 | Zbl 0689.58028

 M. Misiurewicz; W. Szlenk Entropy of piecewise monotone mappings, Studia Math., Volume 67 (1980), pp. 45-63 | MR 579440 | Zbl 0445.54007

 J. Rothschild On the computation of topological entropy (1971) (Ph. D. Thesis, CUNY)

 A.N. Sharkovsky Coexistence of cycles of a continuous map of the line into itself, Ukrain. Mat. Ž, Volume 16 (1964), pp. 61-71 | MR 159905

 A.N. Sharkovsky Coexistence of cycles of a continuous map of the line into itself, Internat. J. Bifur. Chaos Appl. Sci. Engrg. (English transl.), Volume 5 (1995), pp. 1263-1273 | Article | MR 1361914 | Zbl 0890.58012

 A.N. Sharkovsky; S.F. Kolyada; A.G. Sivak Dynamics of one-dimensional maps, Kluwer, Dordrecht, 1997

 P. Walters An introduction to ergodic theory, Springer-Verlag, New York-Berlin, 1982 | MR 648108 | Zbl 0475.28009