Topological sequence entropy for maps of the circle
Hric, Roman
Commentationes Mathematicae Universitatis Carolinae, Tome 41 (2000), p. 53-59 / Harvested from Czech Digital Mathematics Library
Access to full text
Full (PDF)

A continuous map $f$ of the interval is chaotic iff there is an increasing sequence of nonnegative integers $T$ such that the topological sequence entropy of $f$ relative to $T$, $h_T(f)$, is positive ([FS]). On the other hand, for any increasing sequence of nonnegative integers $T$ there is a chaotic map $f$ of the interval such that $h_T(f)=0$ ([H]). We prove that the same results hold for maps of the circle. We also prove some preliminary results concerning topological sequence entropy for maps of general compact metric spaces.

Publié le : 2000-01-01
Classification:  26A18,  37B40,  37D45,  37E10,  54H20,  58F13 
@article{119140,
     author = {Roman Hric},
     title = {Topological sequence entropy for maps of the circle},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     volume = {41},
     year = {2000},
     pages = {53-59},
     zbl = {1039.37007},
     mrnumber = {1756926},
     language = {en},
     url = {http://dml.mathdoc.fr/item/119140}
}

Hric, Roman. Topological sequence entropy for maps of the circle. Commentationes Mathematicae Universitatis Carolinae, Tome 41 (2000) pp. 53-59. http://gdmltest.u-ga.fr/item/119140/

Alsedà L.; Llibre J.; Misiurewicz M. Combinatorial Dynamics and Entropy in Dimension One, World Scientific Publ. Singapore (1993). (1993) | MR 1255515

Auslander J.; Katznelson Y. Continuous maps of the circle without periodic points, Israel. J. Math. 32 (1979), 375-381. (1979) | MR 0571091 | Zbl 0442.54011

Balibrea F.; Cánovas J.S.; Jiménez López V. Commutativity and noncommutativity of topological sequence entropy, preprint.

Block L.S.; Coppel W.A. Dynamics in One Dimension, Lecture Notes in Math., vol. 1513 Springer Berlin (1992). (1992) | MR 1176513 | Zbl 0746.58007

Franzová N.; Smítal J. Positive sequence entropy characterizes chaotic maps, Proc. Amer. Math. Soc. 112 (1991), 1083-1086. (1991) | MR 1062387

Goodman T.N.T. Topological sequence entropy, Proc. London Math. Soc. 29 (1974), 331-350. (1974) | MR 0356009 | Zbl 0293.54043

Hocking J.G.; Young G.S. Topology, Dover New York (1988). (1988) | MR 0939613 | Zbl 0718.55001

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

Janková K.; Smítal J. A characterization of chaos, Bull. Austral. Math. Soc. 34 (1986), 283-292. (1986) | MR 0854575

Kolyada S.; Snoha Ł. Topological entropy of nonautonomous dynamical systems, Random and Comp. Dynamics 4 (1996), 205-233. (1996) | MR 1402417 | Zbl 0909.54012

Kuchta M. Characterization of chaos for continuous maps of the circle, Comment. Math. Univ. Carolinae 31 (1990), 383-390. (1990) | MR 1077909 | Zbl 0728.26011

Kuchta M.; Smítal J. Two point scrambled set implies chaos, European Conference on Iteration Theory ECIT'87 World Sci. Publishing Co. Singapore. | MR 1085314

Lemańczyk M. The sequence entropy for Morse shifts and some counterexamples, Studia Math. 52 (1985), 221-241. (1985) | MR 0825480

Li T Y.; Yorke J.A. Period three implies chaos, Amer. Math. Monthly 82 (1975), 985-992. (1975) | MR 0385028 | Zbl 0351.92021

Smítal J. Chaotic functions with zero topological entropy, Trans. Amer. Math. Soc. 297 (1986), 269-281. (1986) | MR 0849479