Abstract
In this paper, we discuss a continuous self-map of an interval and the existence of an uncountable strongly chaotic set. It is proved that if a continuous self-map of an interval has positive topological entropy, then it has an uncountable strongly chaotic set in which each point is recurrent, but is not almost periodic.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
B. Schweizer and J. Smítal,Measures of chaos and a spectral decomposition of dynamical systems on the interval, Trans. Amer. Math. Soc.344 (1994), 737–754.
P. Erdő and A. H. Stone,Some remarks on almost periodic transformations, Bull. Amer. Math. Soc.51 (1945), 126–130.
T. Y. Li, M. Misiurewicz, G. Pianigiam and J. Yorke,No division implies chaos, Trans. Amer. Math. Soc.273 (1982), 191–199.
R. S. Yang,Pseudo-shift-invariant sets and chaos, Chinese Ann. Math. Ser. A13 (1992), 22–25. (in chinese)
W. H. Gottschalk,Orbit-closure decompositions and almost periodic properties, ibid.50 (1944), 915–919.
T. Y. Li, M. Misiurewicz, G. Pianigiam and J. Yorke,Odd chaos, Phys. Lett. A87 (1982), 271–273.
G. F. Liao,Chian recurrent orbits of mapping of the interval, Northeast Math. J.2 (1986), 240–244.
P. Walters,An Introduction to Ergodic Theory, Springer-Verlag, New York Heidelberg Berlin, 1982.
Z. L. Zhou and W. H. He,Administrative levels of orbit structure and topological semiconjugate, Science of China (A),25 (5), (1995), 457–464.
L. D. Wang, Z. Y. Chu and G. F. Liao,Recurrence, strong chaos and unique ergodicity, Journal of Jilin Normal University (Natural Science Edition),24 (1) (2003), 11–15. (in Chinese)
L. S. Block and W. A. Coppel,Dynamics in One Dimension, Springer-Verlag, New York Heidelberg Berlin, (1992), 35–36.
J. C. Xiong,Set of almost periodic points of a continuous self-map of an interval, Acta. Math. Sci., New Series,2 (1986), 73–77.
B. S. Du,Every chaotic interval map has a scrambled set in the recurrent set, Bull. Austral. Math. Soc.30 (1989), 259–264.
T. Y. Li and J. A. Yorke,Period 3 implies chaos, Amer. Math. Monthly82 (1975), 985–992.
Y. Oono,Period ≠ 2 n implies chaos, Prog. Theor. Phys. F59 (1978), 1028–1030.
Eleonor Ciurea,An algorithm for minimal dynamic flow, J. Appl. Math. & Computing(old KJCAM)7(2) (2000), 259–270.
Author information
Authors and Affiliations
Corresponding author
Additional information
Lidong Wang is a professor and doctor in the Research Institute of Nonlinear Information & Technology, Dalian Nationalities University. His research interests are major in chaos, topological dynamics and ergodic theory.
Gongfu Liao is a professor of the Research Institute of Mathematics in Jilin University. His research interests are major in chaos, topological dynamics and ergodic theory.
Zhenyan Chu is a graduate student of Jilin Normal University and a teacher in the Research Institute of Nonlinear Information & Technology, Dalian Nationalities University. Her research interests are major in chaos, topological dynamics and ergodic theory.
Xiaodong Duan is a professor and doctor in the Research Institute of Nonlinear Information & Technology, Dalian Nationalities University. His research interests are major in chaos and fractal.
Rights and permissions
About this article
Cite this article
Wang, L., Liao, G., Chu, Z. et al. The set of recurrent points of a continuous self-map on an interval and strong chaos. JAMC 14, 277–288 (2004). https://doi.org/10.1007/BF02936114
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02936114