Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Positive sequence topological entropy characterizes chaotic maps

Authors: N. Franzová and J. Smítal
Journal: Proc. Amer. Math. Soc. 112 (1991), 1083-1086
MSC: Primary 58F13; Secondary 28D20, 54H20, 58F11
MathSciNet review: 1062387
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that a continuous map $ f$ of the interval is chaotic (in the sense of Li and Yorke) iff its sequence topological entropy $ {h_A}(f)$ relative to a suitable increasing sequence $ A$ of times is positive. This result is interesting since the ordinary topological entropy $ h(f)$ of chaotic maps can be zero.

References [Enhancements On Off] (What's this?)

  • [1] V. V. Fedorenko, A. N. Šarkovskii, and J. Smítal, Characterizations of weakly chaotic maps of the interval, Proc. Amer. Math. Soc. 110 (1990), 141-148. MR 1017846 (91a:58148)
  • [2] T. N. T. Goodman, Topological sequence entropy, Proc. London Math. Soc. (3) 29 (1974), 331-350. MR 0356009 (50:8482)
  • [3] M. Lemańczyk, The sequence entropy for Morse shifts and some counterexamples, Studia Math. 82 (1985), 221-241. MR 825480 (87e:28034)
  • [4] A. N. Šarkovskii, Nonwandering points and the center of a continuous map of the line into itself, Dopovidi Akad. Nauk Ukrain. RSR Ser. A. (1964), 865-868. (Ukrainian) MR 0165178 (29:2467)
  • [5] -, On a theorem of G. Birkhoff, Dopovidi Akad. Nauk Ukrain. RSR. Ser. A 5 (1967), 429-432. (Ukrainian) MR 0212781 (35:3646)
  • [6] J. Smítal, Chaotic functions with zero topological entropy, Trans. Amer. Math. Soc. 297 (1986), 269-282. MR 849479 (87m:58107)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 58F13, 28D20, 54H20, 58F11

Retrieve articles in all journals with MSC: 58F13, 28D20, 54H20, 58F11

Additional Information

Article copyright: © Copyright 1991 American Mathematical Society

American Mathematical Society