Topological sequence entropy for maps of the interval

Hric, Roman

doi:10.1090/S0002-9939-99-04799-1

Topological sequence entropy for maps of the interval
HTML articles powered by AMS MathViewer

by Roman Hric PDF

Proc. Amer. Math. Soc. 127 (1999), 2045-2052 Request permission

Abstract:

A result by Franzová and Smítal shows that a continuous map of the interval into itself is chaotic if and only if its topological sequence entropy relative to a suitable increasing sequence of nonnegative integers is positive. In the present paper we prove that for any increasing sequence of nonnegative integers there exists a chaotic continuous map with zero topological sequence entropy relative to this sequence.

References

A. M. Bruckner and J. Smítal, A characterization of $\omega$-limit sets of maps of the interval with zero topological entropy, Ergodic Theory Dynam. Systems 13 (1993), no. 1, 7–19. MR 1213076, DOI 10.1017/S0143385700007173
V. V. Fedorenko, A. N. Šarkovskii, and J. Smítal, Characterizations of weakly chaotic maps of the interval, Proc. Amer. Math. Soc. 110 (1990), no. 1, 141–148. MR 1017846, DOI 10.1090/S0002-9939-1990-1017846-5
N. Franzová and J. Smítal, Positive sequence topological entropy characterizes chaotic maps, Proc. Amer. Math. Soc. 112 (1991), no. 4, 1083–1086. MR 1062387, DOI 10.1090/S0002-9939-1991-1062387-3
T. N. T. Goodman, Topological sequence entropy, Proc. London Math. Soc. (3) 29 (1974), 331–350. MR 356009, DOI 10.1112/plms/s3-29.2.331
Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3
J. Harrison, Wandering intervals, Dynamical systems and turbulence, Warwick 1980 (Coventry, 1979/1980), Lecture Notes in Math., vol. 898, Springer, Berlin-New York, 1981, pp. 154–163. MR 654888
Kolyada, S. and Snoha, L’., Topological entropy of nonautonomous dynamical systems, Random and Comp. Dynamics 4 (1996), 205 – 233.
T. Y. Li and James A. Yorke, Period three implies chaos, Amer. Math. Monthly 82 (1975), no. 10, 985–992. MR 385028, DOI 10.2307/2318254
MichałMisiurewicz, Invariant measures for continuous transformations of $[0,1]$ with zero topological entropy, Ergodic theory (Proc. Conf., Math. Forschungsinst., Oberwolfach, 1978) Lecture Notes in Math., vol. 729, Springer, Berlin, 1979, pp. 144–152. MR 550417
J. Smítal, Chaotic functions with zero topological entropy, Trans. Amer. Math. Soc. 297 (1986), no. 1, 269–282. MR 849479, DOI 10.1090/S0002-9947-1986-0849479-9