Shadowing property of continuous maps
Authors:
Tomáš Gedeon and Milan Kuchta
Journal:
Proc. Amer. Math. Soc. 115 (1992), 271-281
MSC:
Primary 58F12; Secondary 58F20
DOI:
https://doi.org/10.1090/S0002-9939-1992-1086325-3
MathSciNet review:
1086325
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Abstract | References | Similar Articles | Additional Information
Abstract: We study continuous maps of an interval into itself. We find the necessary and sufficient condition for the maps of the type 
to have the shadowing property. Further we show that any chaotic map, which has only cycles of order a power of 2, does not have the shadowing property.
- [1] Ethan M. Coven, Ittai Kan, and James A. Yorke, Pseudo-orbit shadowing in the family of tent maps, Trans. Amer. Math. Soc. 308 (1988), no. 1, 227–241. MR 946440, https://doi.org/10.1090/S0002-9947-1988-0946440-2
- [2] K. Janková and J. Smítal, A characterization of chaos, Bull. Austral. Math. Soc. 34 (1986), no. 2, 283–292. MR 854575, https://doi.org/10.1017/S0004972700010157
- [3] Timothy Pennings and Jeffrey Van Eeuwen, Pseudo-orbit shadowing on the unit interval, Real Anal. Exchange 16 (1990/91), no. 1, 238–244. MR 1087487
- [4] O. M. Šarkovs′kiĭ, On cycles and the structure of a continuous mapping, Ukrain. Mat. Ž. 17 (1965), no. 3, 104–111 (Russian). MR 0186757
- [5] J. Smítal, Chaotic functions with zero topological entropy, Trans. Amer. Math. Soc. 297 (1986), no. 1, 269–282. MR 849479, https://doi.org/10.1090/S0002-9947-1986-0849479-9
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Additional Information
DOI:
https://doi.org/10.1090/S0002-9939-1992-1086325-3
Keywords:
Iteration,
shadowing property,
topological dynamics
Article copyright:
© Copyright 1992
American Mathematical Society
